**Content-Based Image Feature Description and Retrieving**

Nai-Chung Yang, Chung-Ming Kuo and Wei-Han Chang

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/45841

## **1. Introduction**

22 Search Algorithms

44 Search Algorithms for Engineering Optimization

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*presented at a talk at the Stanford Artificial Project, unpublished but often cited* .

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pp. 197–200.

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*Science and Technology (IJEST)* 3(12): 8211–8218.

*M. Middendorf, T. Stüzle (Eds.)*, Brussels, Belgium, pp. 113–116.

With the growth in the number of color images, developing an efficient image retrieval sys‐ tem has received much attention in recent years. The first step to retrieve relevant informa‐ tion from image and video databases is the selection of appropriate feature representations (e.g. color, texture, shape) so that the feature attributes are both consistent in feature space and perceptually close to the user [1]. There are many CBIR systems, which adopt different low level features and similarity measure, have been proposed in the literature [2-5]. In gen‐ eral, perceptually similar images are not necessarily similar in terms of low-level features [6]. Hence, these content-based systems capture pre-attentive similarity rather than semantic similarity [7]. In order to achieve more efficient CBIR system, active researches are currently focused on the two complemented approaches: region-based approach [4, 8-10] and rele‐ vance feedback [6, 11-13].

Typically, the region-based approaches segment each image into several regions with homo‐ genous visual prosperities, and enable users to rate the relevant regions for constructing a new query. In general, an incorrect segmentation may result in inaccurate representation. However, automatically extracting image objects is still a challengeing issue, especially for a database containing a collection of heterogeneous images. For example, Jing et al. [8] inte‐ grate several effective relevance feedback algorithms into a region-based image retrieval system, which incorporates the properties of all the segmented regions to perform many-tomany relationships of regional similarity measure. However, some semantic information will be disregarded without considering similar regions in the same image. In another study [10], Vu et al. proposed a region-of-interest (ROI) technique which is a sampling-based ap‐

© 2013 Yang et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Yang et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

proach called SamMatch for matching framework. This method can prevent incorrectly de‐ tecting the visual features.

**1.** Computational cost increases as the selected features increased. However, an algorithm with large number of features does not guarantee an improvement of retrieval perform‐ ance. In theory, the retrieval performance can be enhanced by choosing more compact

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**2.** The CBIR systems retrieve similar images according to the user-defined feature vectors [10]. To improve the accuracy, the region-based approaches [14, 15] segment each image into several regions, and then extract the image features, such as the dominant color, texture or shape. However, the correct detection of semantic objects involves many con‐ ditions [16] such as lighting conditions, occlusion and inaccurate segmentation. Since no automatic segmentation algorithm achieves satisfactory performance currently, seg‐ mented regions are commonly provided by the user to support the image retrieval. However, semantically correct segmentation is a strict challenge to the user, even some

**3.** The CBIR technique helps the system to learn how to retrieve the results that users are looking for. Therefore, there is an urgent need to develop a convient technique for re‐

Color is one of the most widely used visual features for retrieving images from common se‐ mantic categories [12]. MPEG-7 specifies several color descriptors [17], such as dominant col‐ ors, scalable color histogram, color structure, color layout and GoF/GoP color. The human visual system captures dominant colors in images and eliminates the fine details in small areas [18]. In MPEG-7, DCD provides a compact color representation, and describes the color distri‐

In order to extract the dominant colors from an image, a color quantization algorithm has to be predetermined. A commonly used approach is the modified generalized Lloyd algorithm (GLA) [19], which is a color quantization algorithm with clusters merging. This method can simplify the large number of colors to a small number of representative colors. However, the GLA has several intrinsic problems associated with the existing algorithm as follows [20].

**5.** A correct initialization of the centroid of cluster is a crucial issue because some clusters

**4.** It may give different clustering results when the number of clusters is changed.

may be empty if their initial centers lie far from the distribution of data.

{{ , }, 1, }, *i i F cp i N* = = L (1)

is a 3-D dominant color vector,

bution in an image[16]. The dominant color descriptor in MPEG-7 is defined as

where *N* is the total number of dominant colors in image, *c <sup>i</sup>*

is the percentage for each dominant color, and *Σpi* =1 .

feature vectors.

systems provide segmentation tools.

**3. A modified dominant color descriptor**

gion-of-interest analysis.

*p i*

On the other hand, the mechanism of relevance feedback is an online-learning technique that can capture the inherent subjectivity of user's perception during a retrieval session. In Power Tool [11], the user is allowed to give the relevance scores to the best matched images, and the system adjusts the weights by putting more emphasis on the specific features. Cox et al. [11] propose an alternative way to achieve CBIR that predicts the possible image tar‐ gets by Bayes' rule rather than provides with segmented regions of the query image. How‐ ever, the feedback information in [12] could be ignored if the most likely images and irrelevant images have similar features.

In this Chapter, a novel region-based relevance feedback system is proposed that incorporates several feature vectors. First, unsupervised texture segmentation for natural images is used to partition an image to several homogeneous regions. Then we propose an efficient dominant color descriptor (DCD) to represent the partitioned regions in image. Next, a regional similari‐ ty matrix model is introduced to rank the images. In order to attack the possible fails of seg‐ mentation and to simplify the user operations, we propose a foreground assumption to separate an image into two parts: foreground and background. The background could be re‐ garded as the irrelevant region that confuses with the query semantics for retrieval. It should be noted that the main objectives of this approach could exclude irrelevant regions (back‐ ground) from contributing to image-to-image similarity model. Furthermore, the global fea‐ tures extracted from entire image are used to compensate the inaccuracy due to imperfect segmentations. The details will be presented in the following Sections. Experimental results show that our framework improves the accuracy of relevance-feedback retrieval.

The Chapter is organized as follows. Section 2 describes the key observations which explain the basis of our algorithm. In Section 3, we first present a quantization scheme for extracting the representative colors from images, and then introduce a modified similarity measure for DCD. In Section 4, image segmentation and region representation based on our modified dominant color descriptor and local binary pattern are described. Then the image represen‐ tation and the foreground assumption are explained in Section 5. Our integrated regionbased relevance feedback strategies, which consider pseudo query image and relevant images as the relevance information, are introduced in Section 6. Experimental results and discussions of the framework are made in Section 7. Finally, a short conclusion is presented in Section 8.

## **2. Problem statement**

The major goal in region-based relevance feedback for image retrieval is to search perceptu‐ ally similar images with good accuracy in short response time. For nature image retrieval, conversional region-based relevance feedback systems use multiple features (e.g., color, shape, texture, size) and update weighting scheme. In this context, our algorithm is motivat‐ ed by the following viewpoints.


## **3. A modified dominant color descriptor**

proach called SamMatch for matching framework. This method can prevent incorrectly de‐

On the other hand, the mechanism of relevance feedback is an online-learning technique that can capture the inherent subjectivity of user's perception during a retrieval session. In Power Tool [11], the user is allowed to give the relevance scores to the best matched images, and the system adjusts the weights by putting more emphasis on the specific features. Cox et al. [11] propose an alternative way to achieve CBIR that predicts the possible image tar‐ gets by Bayes' rule rather than provides with segmented regions of the query image. How‐ ever, the feedback information in [12] could be ignored if the most likely images and

In this Chapter, a novel region-based relevance feedback system is proposed that incorporates several feature vectors. First, unsupervised texture segmentation for natural images is used to partition an image to several homogeneous regions. Then we propose an efficient dominant color descriptor (DCD) to represent the partitioned regions in image. Next, a regional similari‐ ty matrix model is introduced to rank the images. In order to attack the possible fails of seg‐ mentation and to simplify the user operations, we propose a foreground assumption to separate an image into two parts: foreground and background. The background could be re‐ garded as the irrelevant region that confuses with the query semantics for retrieval. It should be noted that the main objectives of this approach could exclude irrelevant regions (back‐ ground) from contributing to image-to-image similarity model. Furthermore, the global fea‐ tures extracted from entire image are used to compensate the inaccuracy due to imperfect segmentations. The details will be presented in the following Sections. Experimental results

show that our framework improves the accuracy of relevance-feedback retrieval.

The Chapter is organized as follows. Section 2 describes the key observations which explain the basis of our algorithm. In Section 3, we first present a quantization scheme for extracting the representative colors from images, and then introduce a modified similarity measure for DCD. In Section 4, image segmentation and region representation based on our modified dominant color descriptor and local binary pattern are described. Then the image represen‐ tation and the foreground assumption are explained in Section 5. Our integrated regionbased relevance feedback strategies, which consider pseudo query image and relevant images as the relevance information, are introduced in Section 6. Experimental results and discussions of the framework are made in Section 7. Finally, a short conclusion is presented

The major goal in region-based relevance feedback for image retrieval is to search perceptu‐ ally similar images with good accuracy in short response time. For nature image retrieval, conversional region-based relevance feedback systems use multiple features (e.g., color, shape, texture, size) and update weighting scheme. In this context, our algorithm is motivat‐

tecting the visual features.

46 Search Algorithms for Engineering Optimization

in Section 8.

**2. Problem statement**

ed by the following viewpoints.

irrelevant images have similar features.

Color is one of the most widely used visual features for retrieving images from common se‐ mantic categories [12]. MPEG-7 specifies several color descriptors [17], such as dominant col‐ ors, scalable color histogram, color structure, color layout and GoF/GoP color. The human visual system captures dominant colors in images and eliminates the fine details in small areas [18]. In MPEG-7, DCD provides a compact color representation, and describes the color distri‐ bution in an image[16]. The dominant color descriptor in MPEG-7 is defined as

$$F = \{ \{ c\_i, p\_i \} \,, \ i = 1 \downarrow \mathbb{L} \ N \} \,\tag{1}$$

where *N* is the total number of dominant colors in image, *c <sup>i</sup>* is a 3-D dominant color vector, *p i* is the percentage for each dominant color, and *Σpi* =1 .

In order to extract the dominant colors from an image, a color quantization algorithm has to be predetermined. A commonly used approach is the modified generalized Lloyd algorithm (GLA) [19], which is a color quantization algorithm with clusters merging. This method can simplify the large number of colors to a small number of representative colors. However, the GLA has several intrinsic problems associated with the existing algorithm as follows [20].


**6.** The criterion of the GLA depends on the cluster "distance"; therefore, different initial parameters of an image may cause different clustering results.

In general, the conventional clustering algorithms are very time consuming [2, 21-24]. On the other hand, the quadratic-like measure [2, 17, 25] for dominant color descriptor in MPEG7 does not matching human perception very well, and it could cause incorrect ranks for images with similar color distribution [3, 20, 26]. In this Chapter, we adopt the linear block algorithm (LBA) [20] to extract the representative colors, and measure the perceptual similar dominant colors by the modified similarity measure.

Considering two dominant color features *F*<sup>1</sup> ={{*ci* , *pi* }, *i* =1, ⋯, *N*1} and *F*<sup>2</sup> ={{*bj* , *qj* }, *j* =1, ...*N*2} , the quadratic-like dissimilarity measure between two images *F*<sup>1</sup> and *F*2 is calculated by:

$$D^2(F1, F2) = \sum\_{i=1}^{N\_1} p\_i^2 + \sum\_{j=1}^{N\_2} q\_j^2 - \sum\_{i=1}^{N\_1} \sum\_{j=1}^{N\_2} 2a\_{i,j} p\_i q\_j \tag{2}$$

, [1- ( ) - ( ) ] min( ( ), ( )), *ij q t q t S pi pj pipj* = ´ (5)

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where *pq*(*i*) and *pt*( *j*) are the percentages of the *i*th dominant color in query image and the *j*th dominant color in target image, respectively. The term in bracket, 1− | *pq*(*i*)− *pt*( *j*)| is used to measure the difference between two colors in percentage, and the term min(*pq*(*i*), *pt*( *j*)) is the intersection of *pq*(*i*) and *pt*( *j*) that represents the similarity between two colors in percentage. In Fig. 1, we use two real images selected from Corel as our exam‐

**Figure 1.** Example images with the dominant colors and their percentage values. First row: 3-D dominant color vector

In Fig. 1, we calculate this example by using the modified measure and quadratic-like meas‐ ure for comparison. In order to properly reflect similarity coefficient between two color clus‐ ters, the parameter is set to 2 and *Td* =25 in Eq(3). Since the pair-wised distance between *Q* and *F*1 in Fig. 1 is exceed Td, the quadratic-like dissimilarity measure can be determined by

However, using the quadratic-like dissimilarity measure between the Q and *F*2 is:

(*Q*, *F*2)=0.6732 + 0.4489 - 2×(1 - 22 / 50)×0.20576×0.548096=0.9958.

man perception. Whereas, using the dissimilarity measure in [19], we have

for each dominant color. Middle row: the original images. Bottom row: the corresponding

(*Q*, *<sup>F</sup>*2)>*<sup>D</sup>* <sup>2</sup>

(*Q*, *F*1) is not consistent with hu‐

*ci*

*D* <sup>2</sup>

*D* <sup>2</sup>

and the percentage *p <sup>i</sup>*

(*Q*, *F*1)=0.6732 + 0.249=0.9222.

It can be seen that the comparison result of *D* <sup>2</sup>

quantized images.

ple, where the color and percentage values are given for comparison.

where *ai*, *<sup>j</sup>* is the similarity coefficient between color clusters *ci* and *bj* , and it is given by

$$a\_{i,j} = \begin{cases} 1 - d\_{i,j} / d\_{\text{max}} & d\_{i,j} \le T\_d \\ 0 & d\_{i,j} > T\_d \end{cases} \tag{3}$$

The threshold *Td* is the maximum distance used to judge whether two color clusters are sim‐ ilar, and *di*, *<sup>j</sup>* is Euclidean distance between two color clusters *ci* and *bj* ; *d*max =*αTd* , notation *α* is a parameter that is set to 2.0 in this work.

The quadratic-like distance measure in Eq. (2) may incorrectly reflect the distance between two images. The improper results are mainly caused by two reasons. 1) If the number of dominant colors N2 in target image increases, it might cause incorrect results. 2) If one dom‐ inant color can be found both in target images and query image, a high percentage *qj* of the color in target image might cause improper results. In our earlier work [19], we proposed a modified distance measure that considers not only the similarity of dominant colors but also the difference of color percentages between images. The experimental results show that the measure in [20] provides better match to human perception in judging image similarity than the MPEG-7 DCD. The modified similarity measure between two images *F*1 and *F*2 is calcu‐ lated by:

$$\text{tr}\,D^2(F1\_\prime, F2\_\prime) = 1 - \sum\_{i=1}^{N\_1} \sum\_{j=1}^{N\_2} a\_{i,j} \mathbb{S}\_{i,j\prime} \tag{4}$$

$$S\_{i,j} = \left[1 \cdot \left| p\_q(i) \cdot p\_t(j) \right|\right] \times \min(p\_q(i), p\_t(j))\_\prime \tag{5}$$

where *pq*(*i*) and *pt*( *j*) are the percentages of the *i*th dominant color in query image and the *j*th dominant color in target image, respectively. The term in bracket, 1− | *pq*(*i*)− *pt*( *j*)| is used to measure the difference between two colors in percentage, and the term min(*pq*(*i*), *pt*( *j*)) is the intersection of *pq*(*i*) and *pt*( *j*) that represents the similarity between two colors in percentage. In Fig. 1, we use two real images selected from Corel as our exam‐ ple, where the color and percentage values are given for comparison.


**Figure 1.** Example images with the dominant colors and their percentage values. First row: 3-D dominant color vector *ci* and the percentage *p <sup>i</sup>* for each dominant color. Middle row: the original images. Bottom row: the corresponding quantized images.

In Fig. 1, we calculate this example by using the modified measure and quadratic-like meas‐ ure for comparison. In order to properly reflect similarity coefficient between two color clus‐ ters, the parameter is set to 2 and *Td* =25 in Eq(3). Since the pair-wised distance between *Q* and *F*1 in Fig. 1 is exceed Td, the quadratic-like dissimilarity measure can be determined by

*D* <sup>2</sup> (*Q*, *F*1)=0.6732 + 0.249=0.9222.

**6.** The criterion of the GLA depends on the cluster "distance"; therefore, different initial

In general, the conventional clustering algorithms are very time consuming [2, 21-24]. On the other hand, the quadratic-like measure [2, 17, 25] for dominant color descriptor in MPEG7 does not matching human perception very well, and it could cause incorrect ranks for images with similar color distribution [3, 20, 26]. In this Chapter, we adopt the linear block algorithm (LBA) [20] to extract the representative colors, and measure the perceptual

> 1 2 12 2 22 1 2 ,

( , ) 2 *N N NN*

is the similarity coefficient between color clusters *ci*

is Euclidean distance between two color clusters *ci*

,

*a*

*α* is a parameter that is set to 2.0 in this work.

*i j*

1 1 11

, max ,

<sup>ì</sup> - £ <sup>ï</sup> <sup>=</sup> <sup>í</sup> <sup>&</sup>gt; ïî

1 / . <sup>0</sup> *i j ij d*

The threshold *Td* is the maximum distance used to judge whether two color clusters are sim‐

The quadratic-like distance measure in Eq. (2) may incorrectly reflect the distance between two images. The improper results are mainly caused by two reasons. 1) If the number of dominant colors N2 in target image increases, it might cause incorrect results. 2) If one dom‐

color in target image might cause improper results. In our earlier work [19], we proposed a modified distance measure that considers not only the similarity of dominant colors but also the difference of color percentages between images. The experimental results show that the measure in [20] provides better match to human perception in judging image similarity than the MPEG-7 DCD. The modified similarity measure between two images *F*1 and *F*2 is calcu‐

1 2

1 1 (, )1 , *N N*

*i j*

= =

*ij ij*

<sup>2</sup> 1 2 , ,

*D FF aS*

inant color can be found both in target images and query image, a high percentage *qj*

*dd d T*

*i j ij D F F p q a pq* = = ==

}, *j* =1, ...*N*2} , the quadratic-like dissimilarity measure between two images *F*<sup>1</sup>

*i j ij i j*

,

*ij d*

*d T*

, *pi*

=+- å å åå (2)

and *bj*

and *bj*

= -åå (4)

}, *i* =1, ⋯, *N*1} and

, and it is given by

; *d*max =*αTd* , notation

(3)

of the

parameters of an image may cause different clustering results.

similar dominant colors by the modified similarity measure.

*F*<sup>2</sup> ={{*bj*

where *ai*, *<sup>j</sup>*

ilar, and *di*, *<sup>j</sup>*

lated by:

, *qj*

and *F*2 is calculated by:

48 Search Algorithms for Engineering Optimization

Considering two dominant color features *F*<sup>1</sup> ={{*ci*

However, using the quadratic-like dissimilarity measure between the Q and *F*2 is:

*D* <sup>2</sup> (*Q*, *F*2)=0.6732 + 0.4489 - 2×(1 - 22 / 50)×0.20576×0.548096=0.9958.

It can be seen that the comparison result of *D* <sup>2</sup> (*Q*, *<sup>F</sup>*2)>*<sup>D</sup>* <sup>2</sup> (*Q*, *F*1) is not consistent with hu‐ man perception. Whereas, using the dissimilarity measure in [19], we have

$$D^2(Q\_{\prime}, F\_1) = 1 - 0 = 1$$

and

*D* <sup>2</sup> (*Q*, *F*2)=1−{(1−22 / 50)×(1− |0.20576−0.548096|)×0.20576}=0.9242

**4. Image segmentation and region representation**

rithm are used to support segmented regions.

It has been mentioned that segmentation is necessary for those region-based image retrieval systems. Nevertheless, automatic segmentation is still unpractical for the applications of re‐ gion-based image retrieval (RBIR) systems [8, 30-32]. Although many systems provide seg‐ mentation tools, they usually need complicated user interaction to achieve image retrieval. Therefore, the processing is very inefficient and time consuming to the user. In the follow‐ ing, the new approach will propose to overcome this problem. In our algorithm, the user does not need to provide precisely segmented regions, instead, the boundary checking algo‐

Content-Based Image Feature Description and Retrieving

http://dx.doi.org/10.5772/45841

51

**Figure 3.** a), (b) and (c) are the results by using the method of T. Ojala et. al. (a'), (b') and (c') are the results by using

For region-based image retrieval, we adopt the unsupervised texture segmentation meth‐ od [30, 33]. In [30], Ojala et al. use the nonparametric log-likelihood-ratio test and the G statistic to compare the similarity of feature distributions. The method is efficient for finding homogeneously textured image regions. Based on this method, a boundary checking algorithm [34] has been proposed to improve the segmentation accuracy and computational cost. For more details about our segmentation algorithm, we refer the reader to [33]. In this Chapter, the weighted distribution of global information CIH (col‐ or index histogram) and local information LBP (local binary pattern) are applied to meas‐

**4.1. Image segmentation**

our earlier segmentation method.

ure the similarity of two adjacent regions.

**Figure 2.** Example images with the dominant colors and their percentage values. First row: 3-D dominant color vector *ci* and the percentage *p <sup>i</sup>* for each dominant color. Middle row: the original images. Bottom row: the corresponding quantized images.

In DCD, the quadratic-like measure results incorrect matches due to the existence of high percentage of the same color in target image. For example, consider the quantized images in Fig. 2. We can see that the percentage of dominant colors of *F*<sup>1</sup> (rose) and *F*2 (gorilla) are 82.21% and 92.72%, respectively. In human perception, Q is more similar to *F*2 . However, the quadratic-like similarity measure is *D* <sup>2</sup> (*Q*, *<sup>F</sup>*2)>*<sup>D</sup>* <sup>2</sup> (*Q*, *F*1) . Obviously, the result causes a wrong rank. The robust similarity measure [19] is more accurate to capture human percep‐ tion than that of MPEG-7 DCD. In our experiments, the modified DCD achieves 16.7% and 3% average retrieval rate (ARR) [27] improvements than Ma [28] and Mojsilovic [29], respec‐ tively. In this Chapter, the modified dominant color descriptor is chosen to support the pro‐ posed CBIR system.

## **4. Image segmentation and region representation**

#### **4.1. Image segmentation**

*D* <sup>2</sup>

and

*D* <sup>2</sup>

*ci*

and the percentage *p <sup>i</sup>*

posed CBIR system.

the quadratic-like similarity measure is *D* <sup>2</sup>

quantized images.

(*Q*, *F*1)=1−0=1

50 Search Algorithms for Engineering Optimization

(*Q*, *F*2)=1−{(1−22 / 50)×(1− |0.20576−0.548096|)×0.20576}=0.9242

**Figure 2.** Example images with the dominant colors and their percentage values. First row: 3-D dominant color vector

In DCD, the quadratic-like measure results incorrect matches due to the existence of high percentage of the same color in target image. For example, consider the quantized images in Fig. 2. We can see that the percentage of dominant colors of *F*<sup>1</sup> (rose) and *F*2 (gorilla) are 82.21% and 92.72%, respectively. In human perception, Q is more similar to *F*2 . However,

a wrong rank. The robust similarity measure [19] is more accurate to capture human percep‐ tion than that of MPEG-7 DCD. In our experiments, the modified DCD achieves 16.7% and 3% average retrieval rate (ARR) [27] improvements than Ma [28] and Mojsilovic [29], respec‐ tively. In this Chapter, the modified dominant color descriptor is chosen to support the pro‐

(*Q*, *<sup>F</sup>*2)>*<sup>D</sup>* <sup>2</sup>

for each dominant color. Middle row: the original images. Bottom row: the corresponding

(*Q*, *F*1) . Obviously, the result causes

It has been mentioned that segmentation is necessary for those region-based image retrieval systems. Nevertheless, automatic segmentation is still unpractical for the applications of re‐ gion-based image retrieval (RBIR) systems [8, 30-32]. Although many systems provide seg‐ mentation tools, they usually need complicated user interaction to achieve image retrieval. Therefore, the processing is very inefficient and time consuming to the user. In the follow‐ ing, the new approach will propose to overcome this problem. In our algorithm, the user does not need to provide precisely segmented regions, instead, the boundary checking algo‐ rithm are used to support segmented regions.

**Figure 3.** a), (b) and (c) are the results by using the method of T. Ojala et. al. (a'), (b') and (c') are the results by using our earlier segmentation method.

For region-based image retrieval, we adopt the unsupervised texture segmentation meth‐ od [30, 33]. In [30], Ojala et al. use the nonparametric log-likelihood-ratio test and the G statistic to compare the similarity of feature distributions. The method is efficient for finding homogeneously textured image regions. Based on this method, a boundary checking algorithm [34] has been proposed to improve the segmentation accuracy and computational cost. For more details about our segmentation algorithm, we refer the reader to [33]. In this Chapter, the weighted distribution of global information CIH (col‐ or index histogram) and local information LBP (local binary pattern) are applied to meas‐ ure the similarity of two adjacent regions.

An example is shown in Fig. 3. It can be seen that boundary checking algorithm segments the test image correctly, and it costs only about 1/20 processing time of the method in [30]. For color image segmentation, another example is shown in Fig. 4. In Fig. 4(c) Fig. 4(c'), we can see that the boundary checking algorithm achieves robustness segmentation for test im‐ age "Akiyo" and another nature image.

\_ , *<sup>K</sup> LBP h <sup>K</sup> <sup>R</sup> <sup>n</sup>*

region. Therefore, the texture feature of region *R* is defined as

**4.3. Image representation and definition of the foreground assumption**

gions. For an image *I* that contains *N* non-overlaping regions, i.e., *I* ={*I*

ture) and area are similar.

∪*i*=1 *<sup>N</sup> I*

*<sup>R</sup> <sup>i</sup>* and *I*

the main subject.

*<sup>R</sup> <sup>i</sup>* ∩ *I*

*<sup>R</sup> <sup>j</sup>* =0 , where *I*

the importance of central region of an image, we define

spectively; *h* and *w* is height and width of the image.

*foreground*

*background*

where *nK* represents the frequency of LBP value at *k*th bin, and *P* is the number of pixels in a

In addition, we define a feature *Rpoa* to represent the *percentage of area* for region *R* in the im‐ age. Two regions are considered to be visual similar if both of their content (color and tex‐

For image retrieval, each image in database is described by a set of its non-overlapping re‐

approaches perform well in [9, 11], their retrieval performances are strongly depends on success of image segmentation because segmentation techniques are still far from reliable for heterogeneous images database. In order to address the possible fails of segmentation, we propose a foreground assumption to "guess" the foreground and background regions in images. For instance, we can readily find a gorilla sitting on the grass as shown in Fig. 5. If Fig. 5 is the query image, the user could be interested in the main subject (gorilla) rather than grass-like features (color, texture, etc). In most case, user would pay more attention to

The main goal of foreground assumption is to simply distinguish main objects and irrele‐ vant regions in images. Assume that we can divide an image into two parts: foreground and background. In general, the foreground stands the central region of an image. To emphasize

1 71 7 {( , ) : , } 8 88 8

= < >< >

where *Rforeground* and *Rbackground* are the occupied regions of foreground and background, re‐

*R xy x h x hy w y w*

= ££ ££

*R xy h x h w y w*

1 71 7 {( , ) : or , or }, 8 88 8

*<sup>P</sup>* <sup>=</sup> (7)

Content-Based Image Feature Description and Retrieving

http://dx.doi.org/10.5772/45841

53

*<sup>R</sup>* 1, *I*

*<sup>R</sup>* 2, ...., *I*

*<sup>R</sup> <sup>N</sup>* } ,

(9)

*RR k texture* <sup>=</sup> {{ *LBP h* \_ *<sup>k</sup>*}, 1 256 . £ £ } (8)

*<sup>R</sup> <sup>i</sup>* represents the *i*th region in *I*. Although the region-based

**Figure 4.** The segmentation processes for test image "Akiyo" and a nature image. (a), (a') Original image. (b), (b') Split‐ ting and Merging. (c), (c') Boundary checking and modification.

#### **4.2. Region representation**

To achieve region-based image retrieval, we use two compact and intuitive visual features to describe a segmented region: dominate color descriptor (DCD) and texture. For the first one, we use our modified dominant color descriptor in [19, 26]. The feature representation of a segmented region *R* is defined as

$$R\_{DCD} = \left( \left\{ R\_{c\_i}, R\_{p\_i} \right\}\_{\prime}, \ 1 \le i \le 8 \right\}\_{\prime} \tag{6}$$

where *Rci* and *Rpi* are the *i*th dominant color and its percentage in *R*, respectively.

For the second one, the texture feature of a region is characterized by the weighted distribu‐ tion of local binary pattern (LBP) [6, 25, 32]. The advantages of LBP include its invariant property to illumination change and its low computational cost [32]. The value of *k*th bin in LBP histogram is given by:

$$R\_{LBP\\_h\_K} = \frac{n\_K}{P} \tag{7}$$

where *nK* represents the frequency of LBP value at *k*th bin, and *P* is the number of pixels in a region. Therefore, the texture feature of region *R* is defined as

$$R\_{texture} = \left( \left\langle R\_{LBP\\_h\_k} \right\rangle\_{\prime} \mid 1 \le k \le 256 \right). \tag{8}$$

In addition, we define a feature *Rpoa* to represent the *percentage of area* for region *R* in the im‐ age. Two regions are considered to be visual similar if both of their content (color and tex‐ ture) and area are similar.

#### **4.3. Image representation and definition of the foreground assumption**

An example is shown in Fig. 3. It can be seen that boundary checking algorithm segments the test image correctly, and it costs only about 1/20 processing time of the method in [30]. For color image segmentation, another example is shown in Fig. 4. In Fig. 4(c) Fig. 4(c'), we can see that the boundary checking algorithm achieves robustness segmentation for test im‐

**Figure 4.** The segmentation processes for test image "Akiyo" and a nature image. (a), (a') Original image. (b), (b') Split‐

To achieve region-based image retrieval, we use two compact and intuitive visual features to describe a segmented region: dominate color descriptor (DCD) and texture. For the first one, we use our modified dominant color descriptor in [19, 26]. The feature representation of

are the *i*th dominant color and its percentage in *R*, respectively.

For the second one, the texture feature of a region is characterized by the weighted distribu‐ tion of local binary pattern (LBP) [6, 25, 32]. The advantages of LBP include its invariant property to illumination change and its low computational cost [32]. The value of *k*th bin in

*R R ,R i DCD c p* <sup>=</sup> {{ *i i*}, 1 8 , £ £ } (6)

age "Akiyo" and another nature image.

52 Search Algorithms for Engineering Optimization

ting and Merging. (c), (c') Boundary checking and modification.

**4.2. Region representation**

and *Rpi*

LBP histogram is given by:

where *Rci*

a segmented region *R* is defined as

For image retrieval, each image in database is described by a set of its non-overlapping re‐ gions. For an image *I* that contains *N* non-overlaping regions, i.e., *I* ={*I <sup>R</sup>* 1, *I <sup>R</sup>* 2, ...., *I <sup>R</sup> <sup>N</sup>* } , ∪*i*=1 *<sup>N</sup> I <sup>R</sup> <sup>i</sup>* and *I <sup>R</sup> <sup>i</sup>* ∩ *I <sup>R</sup> <sup>j</sup>* =0 , where *I <sup>R</sup> <sup>i</sup>* represents the *i*th region in *I*. Although the region-based approaches perform well in [9, 11], their retrieval performances are strongly depends on success of image segmentation because segmentation techniques are still far from reliable for heterogeneous images database. In order to address the possible fails of segmentation, we propose a foreground assumption to "guess" the foreground and background regions in images. For instance, we can readily find a gorilla sitting on the grass as shown in Fig. 5. If Fig. 5 is the query image, the user could be interested in the main subject (gorilla) rather than grass-like features (color, texture, etc). In most case, user would pay more attention to the main subject.

The main goal of foreground assumption is to simply distinguish main objects and irrele‐ vant regions in images. Assume that we can divide an image into two parts: foreground and background. In general, the foreground stands the central region of an image. To emphasize the importance of central region of an image, we define

$$\begin{aligned} R\_{foreground} &= \{ (\mathbf{x}, y) : \frac{1}{8}h \le \mathbf{x} \le \frac{7}{8}h, \quad \frac{1}{8}w \le y \le \frac{7}{8}w \} \\ R\_{background} &= \{ (\mathbf{x}, y) : \mathbf{x} < \frac{1}{8}h \text{ or } \mathbf{x} > \frac{7}{8}h, y < \frac{1}{8}w \text{ or } y > \frac{7}{8}w \} \end{aligned} \tag{9}$$

where *Rforeground* and *Rbackground* are the occupied regions of foreground and background, re‐ spectively; *h* and *w* is height and width of the image.

On the other hand, we extract the global features for an image to compensate the inaccuracy of segmentation algorithms. The features *F <sup>I</sup>* includes three feature sets: 1) dominant color

*DCD* {{{{ *i i*}, 1 8 , , , 1 } } } *<sup>I</sup> j j <sup>j</sup>*

*texture* {{{ \_ *<sup>k</sup>*}, 1 256 , 1 } } *<sup>I</sup> <sup>j</sup>*

{ , , } *II I I*

*<sup>I</sup>* describes the texture distribution for each region; *Fglobal*

*<sup>I</sup>* represent the global, foreground and background color features, respectively. In brief, the images are first segmented using the fast color quantization scheme. Then, the dominant colors, texture distribution and the three color features are extracted in the image.

In region-based image retrieval, an image is considered as relevant if it contains some re‐ gions with satisfactory similarity to the query image. The retrieval system can recon‐ struct a new query that includes only the relevant regions according to user's feedback. In this way, the system can capture the user's query concept automatically. For exam‐ ple, Jing et al. [8] suggest that information in every region could be helpful in retrieval, and group all regions of positive examples by K-means algorithm iteratively to ensure the distance between all the clusters not exceeding a predefined threshold. Then, all re‐ gions within a cluster are merged into a new region. However, the computational cost for merging new regions is proportional to the number of positive examples. Moreover, users might be more interested in some specified regions or main objects rather than the

To speed up the system, we introduce a similarity matrix model to infer the region-of-inter‐ est sets. Inspired by the query-point movement method [8, 31], the proposed system per‐ forms similarity comparisons by analyzing the salient region in pseudo query image and

where *N* is the number of partitioned regions in image *I*; *FRDCD*

**5. Integrated region-based relevance feedback framework**

relevant images based on user's feedback information.

*<sup>I</sup>* for each region, and 3) dominant color *F <sup>I</sup>*

*<sup>R</sup> c p poa j F R ,R i R BV j N* = £ £ £ £ (11)

*<sup>R</sup> LBP h F R k jN* = £ £ ££ (12)

*global foreground background FF F F* = (13)

.

http://dx.doi.org/10.5772/45841

55

Content-Based Image Feature Description and Retrieving

*<sup>I</sup>* represents the dominant col‐

*<sup>I</sup>* , *<sup>F</sup> foreground*

*<sup>I</sup>* and

*FRDCD*

or vectors; *FRtexture*

positive examples.

*Fbackground*

*<sup>I</sup>* for each region, 2) texture *FRtexture*

**Figure 5.** The definition of foreground and background based on foreground assumption.

In region-based retrieval procedure, segmented regions are required. It can be provided by the users or be generated by the system automatically. However, the criterion for similarity measure is based on the overall distances between feature vectors. If an image in database has background regions that is similar to the foreground object of the query image, this im‐ age will be considered as similar image based on the similarity measure. In this case, the ac‐ curacy of region-based retrieval system decreases. Therefore, we modify our region representation by adding a Boolean model *BV* ∈{0, 1} to determine whether the segmented region *R* belongs to the background of the query image or not.

$$BV = \begin{cases} 1 & \mathcal{R} \in \mathcal{R}\_{background} \\ 0 & \mathcal{R} \notin \mathcal{R}\_{background} \end{cases} \tag{10}$$

Note that the variable is designed to reduce the segmentation error.

On the other hand, we extract the global features for an image to compensate the inaccuracy of segmentation algorithms. The features *F <sup>I</sup>* includes three feature sets: 1) dominant color *FRDCD <sup>I</sup>* for each region, 2) texture *FRtexture <sup>I</sup>* for each region, and 3) dominant color *F <sup>I</sup>* .

$$F\_{R\_{\rm DCD}}^{I} = \left\{ \left| \left\langle \left( \boldsymbol{\mathcal{R}}\_{c\_i}^j, \boldsymbol{\mathcal{R}}\_{p\_i}^j \right)\_{\prime}, 1 \le i \le 8 \right\rangle\_{\prime} \; \boldsymbol{\mathcal{R}}\_{p\rm uu}^j \; \boldsymbol{\mathcal{B}} \boldsymbol{V}\_j \right\}\_{\prime} \; 1 \le j \le N \right\} \tag{11}$$

$$F\_{R\_{\text{tartw}}}^I = \left\langle \left| \left( R\_{LBP\\_h\_k}^j \right)\_{\prime} \right. \right. \left. 1 \le k \le 256 \right\rangle\_{\prime} \; 1 \le j \le N \right\} \tag{12}$$

$$F^I = \begin{Bmatrix} F\_{global\ \prime}^I & F\_{foreground\ \prime}^I & F\_{background}^I \end{Bmatrix} \tag{13}$$

where *N* is the number of partitioned regions in image *I*; *FRDCD <sup>I</sup>* represents the dominant col‐ or vectors; *FRtexture <sup>I</sup>* describes the texture distribution for each region; *Fglobal <sup>I</sup>* , *<sup>F</sup> foreground <sup>I</sup>* and *Fbackground <sup>I</sup>* represent the global, foreground and background color features, respectively. In brief, the images are first segmented using the fast color quantization scheme. Then, the dominant colors, texture distribution and the three color features are extracted in the image.

## **5. Integrated region-based relevance feedback framework**

**Figure 5.** The definition of foreground and background based on foreground assumption.

54 Search Algorithms for Engineering Optimization

region *R* belongs to the background of the query image or not.

*BV*

0 1 

Note that the variable is designed to reduce the segmentation error.

<sup>ì</sup> <sup>Î</sup> <sup>ï</sup> <sup>=</sup> <sup>í</sup> <sup>ï</sup> <sup>Ï</sup> <sup>î</sup>

In region-based retrieval procedure, segmented regions are required. It can be provided by the users or be generated by the system automatically. However, the criterion for similarity measure is based on the overall distances between feature vectors. If an image in database has background regions that is similar to the foreground object of the query image, this im‐ age will be considered as similar image based on the similarity measure. In this case, the ac‐ curacy of region-based retrieval system decreases. Therefore, we modify our region representation by adding a Boolean model *BV* ∈{0, 1} to determine whether the segmented

> *background background*

(10)

*R R*

*R R*

In region-based image retrieval, an image is considered as relevant if it contains some re‐ gions with satisfactory similarity to the query image. The retrieval system can recon‐ struct a new query that includes only the relevant regions according to user's feedback. In this way, the system can capture the user's query concept automatically. For exam‐ ple, Jing et al. [8] suggest that information in every region could be helpful in retrieval, and group all regions of positive examples by K-means algorithm iteratively to ensure the distance between all the clusters not exceeding a predefined threshold. Then, all re‐ gions within a cluster are merged into a new region. However, the computational cost for merging new regions is proportional to the number of positive examples. Moreover, users might be more interested in some specified regions or main objects rather than the positive examples.

To speed up the system, we introduce a similarity matrix model to infer the region-of-inter‐ est sets. Inspired by the query-point movement method [8, 31], the proposed system per‐ forms similarity comparisons by analyzing the salient region in pseudo query image and relevant images based on user's feedback information.

#### **5.1. The formation of region-of-interest set**

#### *5.1.1. Region-based similarity measure*

In order to perform region-of-interest (ROI) queries, the relevant regions are obtained by the measurement of region-based color similarity *R* \_*S*(*R*, *R* ′ ) and region based texture similarity *<sup>R</sup>* \_*ST* (*R*, *<sup>R</sup>* ′ ) in Eq. (14) and (15), respectively. This similarity measure allows users to select their relevant regions accurately. Note that the conventional color histogram could not be applied on DCD directly because the images do not have exact numbers of dominant colors [12]. The regionbased color similarity between two segmented regions *R* and *R* ' can be calculated by

$$\begin{aligned} R\_{-}S\left(R,R'\right) &= R\_{-}S\_{c}\left(R,R'\right) \times R\_{-}S\_{\text{pos}}\left(R,R'\right) \\ R\_{-}S\_{c}\left(R,R'\right) &= \sum\_{i=1}^{m}\sum\_{j=1}^{n} \min\left(R\_{p\_{i}},R'\_{p\_{j}}\right) \quad \text{if} \quad d\left(R\_{c\_{i}},R'\_{c\_{j}}\right) < T\_{d}, \end{aligned} \tag{14}$$

The region similarity measure is performed for all regions. The relevant image set is denoted

*I* <sup>2</sup> *I* <sup>3</sup> *I NI*

1 2 *RI*

2 2 *R I*

3 2 *R I*

4 2 *R I*

5 2 *R I*

, *I* <sup>2</sup>

<sup>2</sup> } and *I* <sup>3</sup> ={*I*

*R* <sup>2</sup> <sup>3</sup> , *I R* <sup>1</sup> <sup>1</sup> , *I R* <sup>1</sup>

*R* <sup>1</sup> <sup>3</sup> , *I R* <sup>2</sup>

fer the user's query concept is shown in Fig. 7, where the symbol "1" means that two re‐ gions are regarded as similar. On the contrary, the symbol "0" represents that two regions

To support ROI queries, we perform the one-to-many relationships to find a collection of

gion-of-interest sets can be obtained by merging all similar region sets. For example, the

with the above eight similar region sets. In this example, three region-of-interest sets can

Since user may be interested in some repeated similar regions, the single region set {*I*

could be assumed to be irrelevant in our approach. Therefore, we have

*R* <sup>1</sup> <sup>1</sup> , *I R* <sup>1</sup> <sup>2</sup> , *I R* <sup>2</sup> <sup>3</sup> } , {*I*

*R* <sup>2</sup> <sup>1</sup> , *I R* <sup>2</sup> <sup>2</sup> } , {*I*

; *i* =1, ..., *N* } , where *N* represents the number of positive images from user's feed‐

1 3 *RI*

2 3 *R I*

3 3 *R I*

4 3 *R I*

contains several segmented regions. See Fig. 6.

1 *N RI*

Content-Based Image Feature Description and Retrieving

http://dx.doi.org/10.5772/45841

57

2 *N R I*

3 *N R I*

4 *N R I*

<sup>3</sup> } . Our similarity matrix model to in‐

<sup>2</sup> } , see Fig. 8. After this step, several re‐

*R* <sup>1</sup> <sup>2</sup> , *I R* <sup>1</sup> <sup>1</sup> , *I R* <sup>2</sup> <sup>3</sup> } ,

> *R* <sup>1</sup> <sup>3</sup> } .

> > *R* <sup>1</sup> 3 }

, *I* 3} contains three relevant images, where

*R* <sup>3</sup> <sup>1</sup> , *I R* <sup>2</sup> <sup>2</sup> , *I R* <sup>3</sup> <sup>2</sup> } , {*I*

> *R* <sup>2</sup> <sup>1</sup> , *I R* <sup>3</sup> <sup>1</sup> , *I R* <sup>2</sup> <sup>2</sup> , *I R* <sup>3</sup> <sup>2</sup> } and {*I*

<sup>3</sup> } contains three similar regions. Each region will be merged together

as *Rs* ={*<sup>I</sup> <sup>i</sup>*

back, and each positive image *I <sup>i</sup>*

1

1 1 *RI*

2 1 *RI*

3 1 *R I*

4 1 *R I*

> *R* <sup>1</sup> <sup>2</sup> , *I R* <sup>2</sup> <sup>2</sup> , *I R* <sup>3</sup>

be obtained by the merging operation, i.e., {*I*

*R* <sup>1</sup> <sup>1</sup> , *I R* <sup>1</sup> <sup>2</sup> , *I R* <sup>2</sup> <sup>3</sup> } , {*I*

**Figure 6.** The similarity matching for region pairs.

As an example, let *Rs* ={*<sup>I</sup>* <sup>1</sup>

are non-similar in content.

*R* <sup>1</sup> <sup>1</sup> , *I R* <sup>1</sup> <sup>2</sup> , *I R* <sup>2</sup>

similar region sets, e.g., {*I*

*I* <sup>1</sup> ={*I R* <sup>1</sup> <sup>1</sup> , *I R* <sup>2</sup> <sup>1</sup> , *I R* <sup>3</sup> <sup>1</sup> } , *I* <sup>2</sup> ={*I*

{*I R* <sup>2</sup> <sup>2</sup> , *I R* <sup>2</sup> <sup>1</sup> , *I R* <sup>3</sup> <sup>1</sup> } , {*I R* <sup>3</sup> <sup>2</sup> , *I R* <sup>3</sup> <sup>1</sup> } , {*I R* <sup>1</sup> <sup>3</sup> } and {*I*

first set {*I*

where *m* and *n* are the number of dominate colors in *R* and *R* ' , respectively; *R* \_*Sc* (*R*, *R* ′ ) is the maximum similarity between two regions in similar color percentage. If the pair-wise Eucli‐ dean distance of two dominate color vector *ci* and *cj* is less than a predefined threshold *Td* , it is set to 25 in our work. The notation *R*\_*Spoa* (*R*, *R* ′ ) is used to measure the similarity of the area percent‐ age for region pair (*R*, *R* ′ ) . To measure the texture similarity between two regions, we define

$$R\_{-}S\_{\Gamma}\left(R, R'\right) = \frac{\sum\_{k=1}^{256} \min\left(R\_{LBP\\_hk'}, R'\_{LBP\\_hk}\right)}{\min\left(R\_{Pxl'}, R'\_{Pxl}\right)},\tag{15}$$

where *RPxl* and *R* ′ *Pxl* represent the number of pixels in regions *R* and *R*', respectively; min(*RLBP*\_*<sup>h</sup> <sup>k</sup>* , *R*' *LBP*\_*<sup>h</sup> <sup>k</sup>* ) is the intersection of LBA histogram for the *k*th bin.

Theoretically, visual similar is achieved when both color and texture are similar. For exam‐ ple, two regions should be considered as non-similar if they are similar in terms of color but not texture. This can be achieved by imposing

$$R\_\\_S > 0.8 \text{ and } R\_\\_S > 0.9. \tag{16}$$

#### *5.1.2. Similarity matrix model*

In the following, we introduce a region-based similarity matrix model. The regions of positive examples, which helps the system to find the intention of user's query, are able to exclude the irrelevant regions flexibly. The proposed similarity matrix model is described as follows.

The region similarity measure is performed for all regions. The relevant image set is denoted as *Rs* ={*<sup>I</sup> <sup>i</sup>* ; *i* =1, ..., *N* } , where *N* represents the number of positive images from user's feed‐ back, and each positive image *I <sup>i</sup>* contains several segmented regions. See Fig. 6.

**Figure 6.** The similarity matching for region pairs.

**5.1. The formation of region-of-interest set**

dean distance of two dominate color vector *ci*

to 25 in our work. The notation *R*\_*Spoa*

and *R* ′

, *R*'

*5.1.2. Similarity matrix model*

age for region pair (*R*, *R* ′

where *RPxl*

min(*RLBP*\_*<sup>h</sup> <sup>k</sup>*

measurement of region-based color similarity *R* \_*S*(*R*, *R* ′

In order to perform region-of-interest (ROI) queries, the relevant regions are obtained by the

relevant regions accurately. Note that the conventional color histogram could not be applied on DCD directly because the images do not have exact numbers of dominant colors [12]. The region-

based color similarity between two segmented regions *R* and *R* ' can be calculated by

*c poa*

where *m* and *n* are the number of dominate colors in *R* and *R* ' , respectively; *R* \_*Sc*

(*R*, *R* ′

256

1

<sup>=</sup> ¢ <sup>=</sup> ¢

*LBP*\_*<sup>h</sup> <sup>k</sup>* ) is the intersection of LBA histogram for the *k*th bin.

Theoretically, visual similar is achieved when both color and texture are similar. For exam‐ ple, two regions should be considered as non-similar if they are similar in terms of color but

In the following, we introduce a region-based similarity matrix model. The regions of positive examples, which helps the system to find the intention of user's query, are able to exclude the irrelevant regions flexibly. The proposed similarity matrix model is described as follows.

*k*

*R\_S R,R R ,R*

¢¢ ¢ = ´

( ) ( ) ( )

*R\_S R,R R ,R if d R ,R T*

*c p p cc d*

maximum similarity between two regions in similar color percentage. If the pair-wise Eucli‐

and *cj*

min , '

*R R*

( ) ( ) ( )

*R\_S R,R R\_S R,R R\_S R,R*

1 1

*i j*

( )

*T*

not texture. This can be achieved by imposing

= =

*m n*

) in Eq. (14) and (15), respectively. This similarity measure allows users to select their

min , , *i j i j*

¢ ¢¢ = < åå (14)

) . To measure the texture similarity between two regions, we define

*Pxl* represent the number of pixels in regions *R* and *R*', respectively;

\_ 0.8 and \_ 0.9. *RS RST* > > (16)

( )

( )

, min *LBP\_hk LBP\_hk*

*Pxl Pxl*

) and region based texture similarity

is less than a predefined threshold *Td* , it is set

) is used to measure the similarity of the area percent‐

å (15)

(*R*, *R* ′

) is the

*5.1.1. Region-based similarity measure*

56 Search Algorithms for Engineering Optimization

*<sup>R</sup>* \_*ST* (*R*, *<sup>R</sup>* ′

As an example, let *Rs* ={*<sup>I</sup>* <sup>1</sup> , *I* <sup>2</sup> , *I* 3} contains three relevant images, where *I* <sup>1</sup> ={*I R* <sup>1</sup> <sup>1</sup> , *I R* <sup>2</sup> <sup>1</sup> , *I R* <sup>3</sup> <sup>1</sup> } , *I* <sup>2</sup> ={*I R* <sup>1</sup> <sup>2</sup> , *I R* <sup>2</sup> <sup>2</sup> , *I R* <sup>3</sup> <sup>2</sup> } and *I* <sup>3</sup> ={*I R* <sup>1</sup> <sup>3</sup> , *I R* <sup>2</sup> <sup>3</sup> } . Our similarity matrix model to in‐ fer the user's query concept is shown in Fig. 7, where the symbol "1" means that two re‐ gions are regarded as similar. On the contrary, the symbol "0" represents that two regions are non-similar in content.

To support ROI queries, we perform the one-to-many relationships to find a collection of similar region sets, e.g., {*I R* <sup>1</sup> <sup>1</sup> , *I R* <sup>1</sup> <sup>2</sup> , *I R* <sup>2</sup> <sup>3</sup> } , {*I R* <sup>2</sup> <sup>1</sup> , *I R* <sup>2</sup> <sup>2</sup> } , {*I R* <sup>3</sup> <sup>1</sup> , *I R* <sup>2</sup> <sup>2</sup> , *I R* <sup>3</sup> <sup>2</sup> } , {*I R* <sup>1</sup> <sup>2</sup> , *I R* <sup>1</sup> <sup>1</sup> , *I R* <sup>2</sup> <sup>3</sup> } , {*I R* <sup>2</sup> <sup>2</sup> , *I R* <sup>2</sup> <sup>1</sup> , *I R* <sup>3</sup> <sup>1</sup> } , {*I R* <sup>3</sup> <sup>2</sup> , *I R* <sup>3</sup> <sup>1</sup> } , {*I R* <sup>1</sup> <sup>3</sup> } and {*I R* <sup>2</sup> <sup>3</sup> , *I R* <sup>1</sup> <sup>1</sup> , *I R* <sup>1</sup> <sup>2</sup> } , see Fig. 8. After this step, several re‐ gion-of-interest sets can be obtained by merging all similar region sets. For example, the first set {*I R* <sup>1</sup> <sup>1</sup> , *I R* <sup>1</sup> <sup>2</sup> , *I R* <sup>2</sup> <sup>3</sup> } contains three similar regions. Each region will be merged together with the above eight similar region sets. In this example, three region-of-interest sets can be obtained by the merging operation, i.e., {*I R* <sup>1</sup> <sup>1</sup> , *I R* <sup>1</sup> <sup>2</sup> , *I R* <sup>2</sup> <sup>3</sup> } , {*I R* <sup>2</sup> <sup>1</sup> , *I R* <sup>3</sup> <sup>1</sup> , *I R* <sup>2</sup> <sup>2</sup> , *I R* <sup>3</sup> <sup>2</sup> } and {*I R* <sup>1</sup> <sup>3</sup> } . Since user may be interested in some repeated similar regions, the single region set {*I R* <sup>1</sup> 3 } could be assumed to be irrelevant in our approach. Therefore, we have *ROI* <sup>1</sup> ={*I R* <sup>1</sup> <sup>1</sup> , *I R* <sup>1</sup> <sup>2</sup> , *I R* <sup>2</sup> <sup>3</sup> } and *ROI* <sup>2</sup> ={*I R* <sup>2</sup> <sup>1</sup> , *I R* <sup>3</sup> <sup>1</sup> , *I R* <sup>2</sup> <sup>2</sup> , *I R* <sup>3</sup> <sup>2</sup> } as shown in Fig. 8. The two sets are considered as region-of-interests that reflect user's query perception.

*5.1.3. Salient region model*

where *<sup>C</sup>*¯

where *Nci*

*i*

RGB color space as shown in Fig. 9.

**Figure 9.** The division of *RGB* color space.

To improve retrieval performance, all the region-of-interest sets from the relevant image set *Rs* will be integrated for the next step during relevance feedback. As described in previous subsection, each region-of-interest set could be regarded as a collection of regions, and ex‐ tracted information can be used to identify the user's query concept. However, correctly capturing the semantic concept from the similar regions is still a difficult task. In this stage, we define salient region as all similar regions within each ROI set. The features of the new

In order to emphasize the percentage of area feature, we modified the dominant color de‐

All similar regions in ROI can be determined from the eight uniformly divided partitions in

5

1

3 2

4

111

is the number of dominant colors in cluster *i* ; *Rci*

*ccc iii*

*<sup>i</sup> NNN jjj jjj*

ååå

111

è ø

ååå

*i i i i i i ccc iii*

*R RR R RG R RB C i RRR* === ===

æ ö ç ÷ ´´´

*NNN jj jj jj pc pc pc*

() () ()

*iii*

the dominant color components of R, G and B located within partition *i* for the region *j*, re‐

*ppp jjj*

= £ £

, , , 1 8

*j* (*R*) , *Rci j*

(*G*) and *Rci*

*j*

(*B*) represent

(18)

7

6

*F CP i R SR* = {{{ *<sup>i</sup>* , , 1 8 , , *<sup>i</sup>*} £ £ } *poa*} (17)

Content-Based Image Feature Description and Retrieving

http://dx.doi.org/10.5772/45841

59

scriptor in Eq. (1). The feature representation of the salient region *SR* is described as

region are equal to the weighted average features of individual regions.

is the *i* th average dominant color of similar region.


**Figure 7.** Our proposed matrix structure comparison. ×: no comparison for those regions in the same image, 1: similar regions and 0: non-similar regions.

**Figure 8.** The region-of-interest sets based on the proposed matrix structure comparison.

If users are interested in many regions, the simple merging process can be used to capture the query concept. In Fig. 8, for example, {*I R* <sup>2</sup> <sup>1</sup> , *I R* <sup>3</sup> <sup>1</sup> } and {*I R* <sup>2</sup> <sup>2</sup> , *I R* <sup>3</sup> <sup>2</sup> } are the regions belong to the same relevant image *I* <sup>1</sup> and *I* <sup>2</sup> , respectively. It can be seen that the similar matrix ap‐ proach is consistent with human perception and is efficient for region-based comparison.

#### *5.1.3. Salient region model*

*ROI* <sup>1</sup> ={*I*

*R* <sup>1</sup> <sup>1</sup> , *I R* <sup>1</sup> <sup>2</sup> , *I R* <sup>2</sup>

58 Search Algorithms for Engineering Optimization

regions and 0: non-similar regions.

<sup>3</sup> } and *ROI* <sup>2</sup> ={*I*

*R* <sup>2</sup> <sup>1</sup> , *I R* <sup>3</sup> <sup>1</sup> , *I R* <sup>2</sup> <sup>2</sup> , *I R* <sup>3</sup>

considered as region-of-interests that reflect user's query perception.

1 1 *R I* <sup>2</sup> 1 *R I* <sup>1</sup> 2 *R I* <sup>3</sup> 2 *R I* <sup>2</sup> 3 *R I*

´ ´ ´ ´ ´ ´

0 01 1 10

0 00 0 01

3 1 *R I* <sup>2</sup> 2 *R I* <sup>1</sup> 3 *R I*

*I* ´ ´ ´ 0

´ ´ ´ ´ ´ ´ ´ ´ ´

0 00 0 01

<sup>2</sup>*I* <sup>3</sup>*I*

0 10 0 0

**Figure 7.** Our proposed matrix structure comparison. ×: no comparison for those regions in the same image, 1: similar

1 2 *R I*

2 2 *R I*

3 2 *R I*

If users are interested in many regions, the simple merging process can be used to capture

*R* <sup>2</sup> <sup>1</sup> , *I R* <sup>3</sup> <sup>1</sup> } and {*I*

proach is consistent with human perception and is efficient for region-based comparison.

01 10 1 00 0 0 1 10 0 0

> ´ ´ ´ ´

1 2 *R I*

2 2 *R I*

> *R* <sup>2</sup> <sup>2</sup> , *I R* <sup>3</sup>

, respectively. It can be seen that the similar matrix ap‐

<sup>1</sup> *ROI*

<sup>2</sup> } are the regions belong to

<sup>2</sup> *ROI*

2 1 *R I*

1 1 *R*

3 1 *R I*

1 2 *R I*

2 2 *R I*

3 2 *R I*

1 3 *R I*

2 3 *R I*

1 *I*

1 1 *R I*

2 1 *R I*

3 1 *R I*

the query concept. In Fig. 8, for example, {*I*

the same relevant image *I* <sup>1</sup>

**Figure 8.** The region-of-interest sets based on the proposed matrix structure comparison.

and *I* <sup>2</sup>

<sup>2</sup> } as shown in Fig. 8. The two sets are

To improve retrieval performance, all the region-of-interest sets from the relevant image set *Rs* will be integrated for the next step during relevance feedback. As described in previous subsection, each region-of-interest set could be regarded as a collection of regions, and ex‐ tracted information can be used to identify the user's query concept. However, correctly capturing the semantic concept from the similar regions is still a difficult task. In this stage, we define salient region as all similar regions within each ROI set. The features of the new region are equal to the weighted average features of individual regions.

In order to emphasize the percentage of area feature, we modified the dominant color de‐ scriptor in Eq. (1). The feature representation of the salient region *SR* is described as

$$F\_{SR} = \left\langle \left| \left\langle \overline{\mathbb{C}}\_{i} \,\, \overline{P}\_{i} \right\rangle\_{\prime} \,\, \mathbf{1} \le i \le 8 \right\rangle\_{\prime} \,\, \overline{R}\_{\text{pau}} \right\rangle\_{\prime} \tag{17}$$

where *<sup>C</sup>*¯ *i* is the *i* th average dominant color of similar region.

All similar regions in ROI can be determined from the eight uniformly divided partitions in RGB color space as shown in Fig. 9.

**Figure 9.** The division of *RGB* color space.

$$\overline{\mathbf{C}}\_{i} = \left( \sum\_{j=1}^{N\_{c\_{i}}} \frac{\mathbf{R}\_{p\_{i}}^{j} \times \mathbf{R}\_{c\_{i}}^{j} \text{(\ $\mathbf{R}\$ )}}{\sum\_{j=1}^{N\_{c\_{i}}} \mathbf{R}\_{p\_{i}}^{j}}, \sum\_{j=1}^{N\_{c\_{i}}} \frac{\mathbf{R}\_{p\_{i}}^{j} \times \mathbf{R}\_{c\_{i}}^{j} \text{(\ $\mathbf{G}\$ )}}{\sum\_{j=1}^{N\_{c\_{i}}} \mathbf{R}\_{p\_{i}}^{j}}, \sum\_{j=1}^{N\_{c\_{i}}} \frac{\mathbf{R}\_{p\_{i}}^{j} \times \mathbf{R}\_{c\_{i}}^{j} \text{(\ $\mathbf{B}\$ )}}{\sum\_{j=1}^{N\_{c\_{i}}} \mathbf{R}\_{p\_{i}}^{j}} \right), 1 \le i \le 8 \tag{18}$$

where *Nci* is the number of dominant colors in cluster *i* ; *Rci j* (*R*) , *Rci j* (*G*) and *Rci j* (*B*) represent the dominant color components of R, G and B located within partition *i* for the region *j*, re‐ spectively; *Rpi j* represents the percentage of its corresponding 3-D dominant color vector in *R <sup>j</sup>* ; *<sup>P</sup>*¯ *i* is the average percentage of dominant color in the *i*th coarse partition, i.e.,

$$\overline{P}\_{i} = \frac{\sum\_{j=1}^{N\_{c\_{i}}} R\_{p\_{i}}^{j}}{N\_{c\_{i}}};\\\overline{R}\_{pos} \text{ is the average percentage of area for all similar regions in ROI.}$$

#### **5.2. The pseudo query image and region weighting scheme**

To capture the inherent subjectivity of user perception, we define a pseudo image *I* <sup>+</sup> as the set of salient regions, *I* <sup>+</sup> ={*SR* <sup>1</sup> , *SR* <sup>2</sup> , ..., *SR <sup>n</sup>*} . The feature representation of *I* <sup>+</sup> can be writ‐ ten as

$$F\_{SR}^{\top} = \left\{ \left| \left\{ (\overline{\mathbf{C}}\_i^1, \overline{\mathbf{P}}\_i^1), \ 1 \le i \le 8 \right\} \right\} \begin{array}{c} \overline{\mathbf{R}}\_{pu}^1 \\ \end{array} \right\} \dots \left. \left\{ \left| (\overline{\mathbf{C}}\_i^n, \overline{\mathbf{P}}\_i^n), \ 1 \le i \le 8 \right\} \right\} \begin{array}{c} \overline{\mathbf{R}}\_{pu}^n \\ \end{array} \right\}.\tag{19}$$

During retrieval, the user chooses the best matched regions what he/she is looking for. How‐ ever, the retrieval system cannot precisely capture the user's query intention at the first or second steps of relevance feedback. With the increasing of the returned positive images, query vectors are then constructed to perform better results. Taking average [8] from all the feedback information could introduce redundant, i.e., information from irrelevant regions. Motivated by this observation, we suggest that each similar region in ROI should be proper‐ ly weighted according to the amount of similar regions. For example, the *ROI* <sup>2</sup> in Fig. 8 is more important than in *ROI* <sup>1</sup> . The weights associated with the significance of SR in *I* <sup>+</sup> can be dynamically updated as

$$w\_l = \frac{\left|ROI^l\right|}{\sum\_{l=1}^{n} ROI^l} \tag{20}$$

For the initial query, the similarity measure *S*(*FentireImage <sup>I</sup>* , *FentireImage <sup>I</sup>* '

age *I* and target image *I* ′

, *I* <sup>2</sup>

characterized by salient regions.

is calculated by

\_

where *n* is the number of salient region sets in *I* <sup>+</sup>

*<sup>I</sup>* and *F*backgrounde

The similarity between the relevant image set *Rs* ={*I*, *<sup>I</sup>* <sup>1</sup>

*s*

*s*

*s*

; *wl*

ground of the image will be excluded for matching in Eq. (21).

It should be noted that *I* <sup>+</sup>

ed regions in target image *I* ′

resentation of *I* <sup>+</sup>

image *FRDCD j I* '

*F*entireImage

*<sup>I</sup>* , *F*foreground

database is calculated by

where *F*entireImage

*Rs* , *F*foreground

*Rs* ={*I*, *<sup>I</sup>* <sup>1</sup>

) for the initial query im‐

http://dx.doi.org/10.5772/45841

61

Content-Based Image Feature Description and Retrieving

*I* +

. In Eq. (21), the image-

, ..., *I <sup>N</sup>* } and target image *I* ′ in

(22)

and target

in database are compared by using Eq. (4). Therefore, a coarse rele‐

and *Rs* defined above both contain the relevance information that

*DCD*

; *m* is the number of color/texture segment‐

¢ = ´ åå (21)

*<sup>I</sup>* in Eq. (13) are extracted to compensate the inaccuracy.

, *I* <sup>2</sup>

¢

¢

*Rs* are dominant colors, foreground and back‐

¢

vant-image set can be obtained. Then, all regions in the initial query image *I* and the positive images based on the user's feedback information are merged into relevant image set

and (15) to find the collection of the similar regions. The similar regions can be determined by Eq. (16), and then be merged into salient region *SR*. For the next iteration, the feature rep‐

( ) ( ) '

to-image similarity matching maximizes the value of region based color similarity by using Eq. (14). If the Boolean model *BV* =1 for a partitioned region in target image, then the back‐

On the other hand, *Rs* is a collection of relevant images based on the user's feedback infor‐ mation. Since poor matches arise from inaccurate image segmentations, three global features

> entireImage entireImage entireImage 1

( , ) max ( , )

*s*

*s*

*R I*

*R I*

*R I*

( , ) max ( , )

ground for the *i*th relevant image in *Rs* , respectively. In Eq. (22), the similarity measure

*s*

( , ) max ( , )

background background background 1

forground foreground forground 1

*N*

*S R I SF F*

å

*i N*

¢ =

¢ =

=

*S R I SF F*

*i N*

¢ =

*Rs* and *F*background

=

å

*i*

=

*S R I SF F*

å

, max , , *l j*

*n m I I*

is the weight of salient region *SR <sup>l</sup>*

reflects human semantics. The similarity measure for pseudo query image *FS <sup>R</sup> <sup>l</sup>*

1j 1

*region based l SR R l S I I w R\_S F F* <sup>+</sup> <sup>+</sup> = =

, ..., *I <sup>N</sup>* } . The proposed region-based similarity matrix model performs Eq. (14)

in Eq. (19) could be regarded as an optimal pseudo query image that is

where |*ROI <sup>l</sup>* | represents the number of similar regions in region-of-interest set *l* , and *n* is the number of region-of-interest sets.

#### **5.3. Region-based relevance feedback**

In reality, inaccurate segmentation leads to poor matching result. However, it is difficult to ask for precise segmented regions from users. Based on the foreground assumption, we de‐ fine three feature vectors, which are extracted from entire image (i.e., global dominant col‐ or), foreground and background, respectively. The advantage of this approach is that it provides an estimation that minimizes the influence of inaccurate segmentation. To inte‐ grate the two regional approaches, we summarize our relevance feedback as follows.

For the initial query, the similarity measure *S*(*FentireImage <sup>I</sup>* , *FentireImage <sup>I</sup>* ' ) for the initial query im‐ age *I* and target image *I* ′ in database are compared by using Eq. (4). Therefore, a coarse rele‐ vant-image set can be obtained. Then, all regions in the initial query image *I* and the positive images based on the user's feedback information are merged into relevant image set *Rs* ={*I*, *<sup>I</sup>* <sup>1</sup> , *I* <sup>2</sup> , ..., *I <sup>N</sup>* } . The proposed region-based similarity matrix model performs Eq. (14) and (15) to find the collection of the similar regions. The similar regions can be determined by Eq. (16), and then be merged into salient region *SR*. For the next iteration, the feature rep‐ resentation of *I* <sup>+</sup> in Eq. (19) could be regarded as an optimal pseudo query image that is characterized by salient regions.

spectively; *Rpi*

∑ *j*=1

*Nci*

*Nc i Rpi j*

*R <sup>j</sup>* ; *<sup>P</sup>*¯ *i*

*P*¯ *i* =

ten as

*j*

60 Search Algorithms for Engineering Optimization

; *R*¯

set of salient regions, *I* <sup>+</sup> ={*SR* <sup>1</sup>

more important than in *ROI* <sup>1</sup>

the number of region-of-interest sets.

**5.3. Region-based relevance feedback**

be dynamically updated as

represents the percentage of its corresponding 3-D dominant color vector in

, ..., *SR <sup>n</sup>*} . The feature representation of *I* <sup>+</sup>

. The weights associated with the significance of SR in *I* <sup>+</sup> can

<sup>å</sup> (20)

as the

can be writ‐

in Fig. 8 is

is the average percentage of dominant color in the *i*th coarse partition, i.e.,

*poa* is the average percentage of area for all similar regions in ROI.

To capture the inherent subjectivity of user perception, we define a pseudo image *I* <sup>+</sup>

( , ), 1 8 , ,.., ( , ), 1 8 , . *n n <sup>n</sup> <sup>I</sup>*

ly weighted according to the amount of similar regions. For example, the *ROI* <sup>2</sup>

1

*l*

=

*w*

=

*<sup>l</sup> <sup>n</sup> <sup>l</sup>*

*ROI*

*ROI*

where |*ROI <sup>l</sup>* | represents the number of similar regions in region-of-interest set *l* , and *n* is

In reality, inaccurate segmentation leads to poor matching result. However, it is difficult to ask for precise segmented regions from users. Based on the foreground assumption, we de‐ fine three feature vectors, which are extracted from entire image (i.e., global dominant col‐ or), foreground and background, respectively. The advantage of this approach is that it provides an estimation that minimizes the influence of inaccurate segmentation. To inte‐

grate the two regional approaches, we summarize our relevance feedback as follows.

, *l*

During retrieval, the user chooses the best matched regions what he/she is looking for. How‐ ever, the retrieval system cannot precisely capture the user's query intention at the first or second steps of relevance feedback. With the increasing of the returned positive images, query vectors are then constructed to perform better results. Taking average [8] from all the feedback information could introduce redundant, i.e., information from irrelevant regions. Motivated by this observation, we suggest that each similar region in ROI should be proper‐

*i i <sup>i</sup> poa <sup>i</sup> poa SR F CP i R CP i R* <sup>+</sup> ì ü ìì üì ü ü ì <sup>ü</sup> <sup>=</sup> ííí ýí ý £ £ ý í £ £ ýý î þ îî þî þ þ î <sup>þ</sup> (19)

**5.2. The pseudo query image and region weighting scheme**

1 1 1

, *SR* <sup>2</sup>

It should be noted that *I* <sup>+</sup> and *Rs* defined above both contain the relevance information that reflects human semantics. The similarity measure for pseudo query image *FS <sup>R</sup> <sup>l</sup> I* + and target image *FRDCD j I* ' is calculated by

$$\mathcal{S}\_{\text{region\\_based}}\left(I^+, I^\prime\right) = \sum\_{l=1}^n \sum\_{j=1}^m w\_l \times \max R\\_S\left(F\_{\text{SR}^{l\prime}}^{I^+}, F\_{R\_{\text{DCD}}^{l}}^{I^-}\right),\tag{21}$$

where *n* is the number of salient region sets in *I* <sup>+</sup> ; *m* is the number of color/texture segment‐ ed regions in target image *I* ′ ; *wl* is the weight of salient region *SR <sup>l</sup>* . In Eq. (21), the imageto-image similarity matching maximizes the value of region based color similarity by using Eq. (14). If the Boolean model *BV* =1 for a partitioned region in target image, then the back‐ ground of the image will be excluded for matching in Eq. (21).

On the other hand, *Rs* is a collection of relevant images based on the user's feedback infor‐ mation. Since poor matches arise from inaccurate image segmentations, three global features *F*entireImage *<sup>I</sup>* , *F*foreground *<sup>I</sup>* and *F*backgrounde *<sup>I</sup>* in Eq. (13) are extracted to compensate the inaccuracy. The similarity between the relevant image set *Rs* ={*I*, *<sup>I</sup>* <sup>1</sup> , *I* <sup>2</sup> , ..., *I <sup>N</sup>* } and target image *I* ′ in database is calculated by

$$\begin{aligned} \mathcal{S}\_{\text{entireImage}}(R\_{s}, I') &= \sum\_{i=1}^{N} \max \mathcal{S}(F\_{\text{entireImage}}^{R\_{s}}, F\_{\text{entireImage}}^{I'})\\ \mathcal{S}\_{\text{forground}}(R\_{s}, I') &= \sum\_{i=1}^{N} \max \mathcal{S}(F\_{\text{foreground}}^{R\_{s}}, F\_{\text{foreground}}^{I'})\\ \mathcal{S}\_{\text{background}}(R\_{s}, I') &= \sum\_{i=1}^{N} \max \mathcal{S}(F\_{\text{background}}^{R\_{s}}, F\_{\text{background}}^{I'}) \end{aligned} \tag{22}$$

where *F*entireImage *Rs* , *F*foreground *Rs* and *F*background *Rs* are dominant colors, foreground and back‐ ground for the *i*th relevant image in *Rs* , respectively. In Eq. (22), the similarity measure maximizes the similarity score using Eq. (5). To reflect the difference between *Rs* and target image *I* ′ , the average similarity measure is given by

$$\mathcal{S}\_{\text{avg}}(R\_{s'}I') = \frac{\langle \mathcal{S}\_{\text{entirelmage}}(R\_{s'}I') + \mathcal{S}\_{\text{foreground}}(R\_{s'}I') + \mathcal{S}\_{\text{background}}(R\_{s'}I') \rangle}{\mathfrak{3}}.\tag{23}$$

It is worth to mention that our region-based relevance feedback approach defined above is able to reflect human semantics. In other words, user might aware some relevant image from the initial query, and then provides some positive image.

Considering the ability to capture the user's perceptions more precisely, the system deter‐ mines the retrieved rank according to average of region-based image similarity measure in Eq. (21) and foreground-based similarity measure in Eq. (23).

$$S = \frac{S\_{\text{region\\_based}}\left(I^+, I^\prime\right) + S\_{\text{avg}}\left(R\_{s'}, I^\prime\right)}{2}. \tag{24}$$

Class 1 (gorilla)

Class 8 (cake)

Class 15 (duck)

Class 22 (seaelephant)

Class 29 (building) Class 2 (bird)

Class 9 (dinosaur)

Class 16 (leopard)

Class 23 (horse)

Class 30 (stadium)

**Table 1.** The labels and examples of the test database.

Class 3 (potted plant)

Class 10 (dolphin)

Class 17 (leaf)

Class 24 (helicopter)

Class 31 (people) Class 4 (card)

Class 11 (elephant)

Class 18 (car)

Class 25 (boat)

( ) MRR( ) AVR( ) 0.5 , <sup>2</sup>

Class 5 (cloud)

Class 12 (firework)

Class 19 (cactus)

Class 26 (snow)

*NG q q q* = -- (27)

Class 6 (sunset)

Content-Based Image Feature Description and Retrieving

Class 13 (flower)

Class 20 (airplane)

Class 27 (balloon) Class 7 (pumpkin) 63

http://dx.doi.org/10.5772/45841

Class 14 (food)

Class 21 (painting)

Class 28 (waterfall)

## **6. Experimental results**

We use an image database (31 categories about 3991 images) for general-purpose from Cor‐ el's photo to evaluate the performance of the proposed framework. The database has a vari‐ ety of images including animal, plant, vehicle, architecture, scene, etc. It has the advantages of large size and wide coverage [11]. Table 1 lists the labels for 31 classes. The effectiveness of our proposed region-based relevance feedback approach is evaluated.

In order to make a comparison on the retrieval performance, both average retrieval rate (ARR) and average normalized modified retrieval rank (ANMRR) [26] are applied. An ideal performance will consist of ARR values equal to 1 for all values of recall. A high ARR value represents a good performance for retrieval rate, and a low ANMRR value indicates a good performance for retrieval rank. The brief definitions are given as follows. For a query q, the ARR and ANMRR are defined as:

$$\text{ARRR}(\eta) = \frac{1}{NQ} \sum\_{q=1}^{NQ} \frac{NF(\beta, q)}{NG(q)},\tag{25}$$

$$\text{AVR}(q) = \sum\_{k=1}^{\text{NC}(q)} \frac{\text{Rank}(k)}{\text{NG}(q)},\tag{26}$$

**Table 1.** The labels and examples of the test database.

maximizes the similarity score using Eq. (5). To reflect the difference between *Rs* and target

entireImage foreground background

It is worth to mention that our region-based relevance feedback approach defined above is able to reflect human semantics. In other words, user might aware some relevant image

Considering the ability to capture the user's perceptions more precisely, the system deter‐ mines the retrieved rank according to average of region-based image similarity measure in

> region\_based ( ) avg , ( ,). <sup>2</sup> *<sup>s</sup> S I I S RI*

We use an image database (31 categories about 3991 images) for general-purpose from Cor‐ el's photo to evaluate the performance of the proposed framework. The database has a vari‐ ety of images including animal, plant, vehicle, architecture, scene, etc. It has the advantages of large size and wide coverage [11]. Table 1 lists the labels for 31 classes. The effectiveness

In order to make a comparison on the retrieval performance, both average retrieval rate (ARR) and average normalized modified retrieval rank (ANMRR) [26] are applied. An ideal performance will consist of ARR values equal to 1 for all values of recall. A high ARR value represents a good performance for retrieval rate, and a low ANMRR value indicates a good performance for retrieval rank. The brief definitions are given as follows. For a query q, the

> 1 <sup>1</sup> ( ,) ARR( ) , ( ) *NQ*

=

b

*Rank k*

<sup>=</sup> å (25)

<sup>=</sup> å (26)

*q NF q <sup>q</sup> NQ NG q*

( )

1 ( ) AVR( ) , ( ) *NG q*

*<sup>q</sup> NG q* <sup>=</sup>

*k*

of our proposed region-based relevance feedback approach is evaluated.

<sup>+</sup> ¢ ¢ +

= (24)

*S RI S RI S RI S RI* ¢¢ ¢ + + ¢ <sup>=</sup> (23)

*ss s*

( ( , ) ( , ) ( , )) ( ,) . <sup>3</sup>

, the average similarity measure is given by

from the initial query, and then provides some positive image.

Eq. (21) and foreground-based similarity measure in Eq. (23).

*S*

image *I* ′

avg

*s*

62 Search Algorithms for Engineering Optimization

**6. Experimental results**

ARR and ANMRR are defined as:

$$\text{MRR}(q) = \text{AVR}(q) - 0.5 - \frac{\text{NG}(q)}{2} \,\text{.}\tag{27}$$

$$\text{NMRR}(\eta) = \frac{\text{MRR}(\eta)}{K + 0.5 - 0.5 \times \text{NG}(\eta)} \,\text{}\tag{28}$$

For better understanding of the retrieval results, the DCD vectors of the query image, rank 6th image and rank 8th image are listed, respectively. See Fig. 11. It can be seen that the query image and the image "lemon" are very similar in the first dominant color (marked by box). If we use the global DCD as the only feature for image retrieval, the system only re‐ turns eleven correct matches. Therefore, further investigation on extracting comprehensive

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**Figure 11.** Example images with the dominant colors and their percentage values. First row: 3-D dominant color vec‐ tor c*<sup>i</sup>* and the percentage p*<sup>i</sup>* for each dominant color. Middle row: the original images. Bottom row: the corresponding

Assume that the user has selected five best matched images, marked by red box, as shown in Fig. 10(a). In conventional region-based relevance feedback approach, all regions in the initial query image *I* and the five positive images are merged into relevant image set

image features is needed.

quantized images.

$$\text{ANMRR}(q) = \frac{1}{NQ} \sum\_{q=1}^{NQ} \text{NMRR}(q), \tag{29}$$

where *NQ* is total number of queries; *NG*(*q*) is the number of the ground truth images for a query. The notation is a factor, and *NF* (*β*, *q*) is number of ground truth images found within the first *β* ⋅ *NG*(*q*) retrievals. *Rank*(*k*) is the rank of the retrieved signature image in the ground truth. In eq.(28), *K* =min(4⋅ *NG*(*q*);2⋅*GTM* ) , where *GTM* is max{*NG*(*q*)} for all queries. The NMRR and its average (ANMRR) are normalized to the range of [0 1].

To test the performance of our integrated approach for region-based relevance feedback, we first query an image with a gorilla sits on grass as shown Fig. 10(a).

As mentioned in Section 5.4, the dominant color between query image *I* and target image *I* ′ is used for similarity measure in the initial query. The retrieval results are shown in Fig. 10(b), the top 20 matching images are arranged from left to right and top to bottom in order of decreasing similarity score.

**Figure 10.** The initial query image and positive images. (a) Query image. (b) The 5 positive images in the first row are selected by user.

For better understanding of the retrieval results, the DCD vectors of the query image, rank 6th image and rank 8th image are listed, respectively. See Fig. 11. It can be seen that the query image and the image "lemon" are very similar in the first dominant color (marked by box). If we use the global DCD as the only feature for image retrieval, the system only re‐ turns eleven correct matches. Therefore, further investigation on extracting comprehensive image features is needed.

MRR( ) NMRR( ) , 0.5 0.5 ( )

<sup>1</sup> ANMRR( ) NMRR( ), *NQ*

queries. The NMRR and its average (ANMRR) are normalized to the range of [0 1].

first query an image with a gorilla sits on grass as shown Fig. 10(a).

of decreasing similarity score.

64 Search Algorithms for Engineering Optimization

(a)

selected by user.

1

where *NQ* is total number of queries; *NG*(*q*) is the number of the ground truth images for a query. The notation is a factor, and *NF*(*β*, *q*) is number of ground truth images found within the first *β* ⋅ *NG*(*q*) retrievals. *Rank*(*k*) is the rank of the retrieved signature image in the ground truth. In eq.(28), *K* =min(4⋅ *NG*(*q*);2⋅*GTM* ) , where *GTM* is max{*NG*(*q*)} for all

To test the performance of our integrated approach for region-based relevance feedback, we

As mentioned in Section 5.4, the dominant color between query image *I* and target image *I* ′ is used for similarity measure in the initial query. The retrieval results are shown in Fig. 10(b), the top 20 matching images are arranged from left to right and top to bottom in order

(b)

**Figure 10.** The initial query image and positive images. (a) Query image. (b) The 5 positive images in the first row are

*q q q NQ* <sup>=</sup>

*<sup>q</sup> <sup>q</sup> <sup>K</sup> NG q* <sup>=</sup> +-´ (28)

<sup>=</sup> å (29)

**Figure 11.** Example images with the dominant colors and their percentage values. First row: 3-D dominant color vec‐ tor c*<sup>i</sup>* and the percentage p*<sup>i</sup>* for each dominant color. Middle row: the original images. Bottom row: the corresponding quantized images.

Assume that the user has selected five best matched images, marked by red box, as shown in Fig. 10(a). In conventional region-based relevance feedback approach, all regions in the initial query image *I* and the five positive images are merged into relevant image set *Rs* ={*I*, *<sup>I</sup>* <sup>1</sup> , *I* <sup>2</sup> , ..., *I* 5} . The proposed similarity matrix model is able to find the region-of-in‐ terest region sets. For the next query, *I* <sup>+</sup> could be regarded as a new query image which is composed of some salient regions. The retrieval results based on the new query image *I* <sup>+</sup> are shown in Fig. 12. The following are discussions.

box) of the image "cucumber". In addition, the percentages of area (0.393911, 0.316813, 0.289276) of initial image "gorilla" are similar to the percentage of area (region#2, 0.264008) of the image "cucumber". The other similarity comparisons between "gorilla" and "cucumber" image are not presented here because the maximum similarity be‐ tween two regions in Eq. (14) is very small. In brief, without considering the exclusion of irrelevant regions, the region-based image-to-image similarity model in Eq. (21)

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**Figure 13.** The analysis of retrieval results using the conventional region-based relevance feedback approach. Top row: dominant color distributions and percentage of area *Poa* for each region in initial query image, "cucumber" and

The retrieval performance can be improved by automatically determining the user's query perception. In the following, we would like to evaluate the advantages of our proposed rele‐ vance feedback approach. For the second query, the integrated region-based relevance feed‐ back contains not only the salient-region information, but also the "specified-region" information based on relevant images set *Rs* . The retrieval results based on our integrated

could cause improper ranks in visualization.

"lemon" images. Bottom row: the corresponding segmented images.


**Figure 12.** The retrieval results based on new pseudo query image *I* <sup>+</sup> for the first iteration.


box) of the image "cucumber". In addition, the percentages of area (0.393911, 0.316813, 0.289276) of initial image "gorilla" are similar to the percentage of area (region#2, 0.264008) of the image "cucumber". The other similarity comparisons between "gorilla" and "cucumber" image are not presented here because the maximum similarity be‐ tween two regions in Eq. (14) is very small. In brief, without considering the exclusion of irrelevant regions, the region-based image-to-image similarity model in Eq. (21) could cause improper ranks in visualization.

*Rs* ={*I*, *<sup>I</sup>* <sup>1</sup>

, *I* <sup>2</sup>

terest region sets. For the next query, *I* <sup>+</sup>

rect matches as shown in Fig. 12.

**1.** The pseudo query image *I* <sup>+</sup>

66 Search Algorithms for Engineering Optimization

**2.** Using the pseudo image *I* <sup>+</sup>

but fifth, as shown in Fig. 12.

shown in Fig. 12. The following are discussions.

**Figure 12.** The retrieval results based on new pseudo query image *I* <sup>+</sup>

which ranks are 7th, 8th and 12th, respectively.

, ..., *I* 5} . The proposed similarity matrix model is able to find the region-of-in‐

composed of some salient regions. The retrieval results based on the new query image *I* <sup>+</sup>

sidering the Boolean model in Eq. (21), the similarity measure by Eq. (21) returns 16 cor‐

could be regarded as a new query image which is

is capable to reflect user's query perception. Without con‐

as query image, the initial query image is not ranked first

for the first iteration.

**3.** The retrieval results return three dissimilar images (marked by red rectangle boxes),

**4.** To analyze the improper result, the dominant color vectors and percentage of area of "cucumber" and "lemon" are listed. See Fig. 13. We can see that each of the images "go‐ rilla", "cucumber" and "lemon" contains three segmented regions. For each region, the number of the dominant colors, percentage of area and BV value are listed and colored red. For similarity matching, the dominant colors (i.e. region#1, region#2 and region#3) of initial image "gorilla" are similar to the dominant color (marked by red rectangle

are


**Figure 13.** The analysis of retrieval results using the conventional region-based relevance feedback approach. Top row: dominant color distributions and percentage of area *Poa* for each region in initial query image, "cucumber" and "lemon" images. Bottom row: the corresponding segmented images.

The retrieval performance can be improved by automatically determining the user's query perception. In the following, we would like to evaluate the advantages of our proposed rele‐ vance feedback approach. For the second query, the integrated region-based relevance feed‐ back contains not only the salient-region information, but also the "specified-region" information based on relevant images set *Rs* . The retrieval results based on our integrated region-based relevance feedback are shown in Fig. 14. Observations and discussions are de‐ scribed as follows.

In Fig. 15-17, further examples are tested to evaluate the performance of the integrated re‐ gion-based relevance feedback for nature images. In Fig. 15, the contents of the query image include a red car on country road by the side of grasslands. If the user is only interested in the red car, four positive images marked by red boxes will be selected as shown in Fig. 15 (b). In this case, retrieval results (RR=0.25, NMRR=0.7841) are far from satisfactory perform‐

(b)

and relevant images set *Rs* based on user's

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**Figure 15.** The initial query image and positive images. (a) Query image. (b) The 4 positive images marked by red box‐

feedback information, the first feedback retrieval returns 10 images containing "red car" as shown in Fig. 16. For this example, the first feedback retrieval achieves an ARR improve‐ ment of 28.6%. More precise results can be achieved by increasing of the number of regionof-interest sets and relevant image set based for the second feedback retrieval as shown in Fig. 17. The retrieval results for the second feedback retrieval returns 11 images containing "red car", and achieve an NMRR improvement of 35% compared to the initial query. Fur‐

thermore, the rank order in Fig. 17 is more reasonable than that in Fig. 16.

ance for the initial query.

(a)

es which are selected by user.

After the submission of pseudo query image *I* <sup>+</sup>


**Figure 14.** The retrieval results based on our integrated region-based relevance feedback.

In Fig. 15-17, further examples are tested to evaluate the performance of the integrated re‐ gion-based relevance feedback for nature images. In Fig. 15, the contents of the query image include a red car on country road by the side of grasslands. If the user is only interested in the red car, four positive images marked by red boxes will be selected as shown in Fig. 15 (b). In this case, retrieval results (RR=0.25, NMRR=0.7841) are far from satisfactory perform‐ ance for the initial query.

region-based relevance feedback are shown in Fig. 14. Observations and discussions are de‐

**2.** In Fig. 13, region#1 and region#3 in query image are two grass-like regions, which are labeled as inner region, i.e., *BV* =1 . On the other hand, the region#2 in image "cucum‐ ber" is a green region that is similar to the grass-like regions in query image. In our method, this problem can be solved by examining the BV value in Eq. (21). As we can see, none of the three incorrect images including "cucumber", "lemon" and "carrot" in

**3.** In contrast, it is possible that the grass-like regions are parts of the user's aspect. In this case, the three feature vectors including entire image, foreground and background can be used to compensate the loss of generality. In Fig. 14 retrieval results indicate that the

**4.** Our proposed relevance feedback approach can capture the query concept effectively. In Fig. 14, it can be seen that most of the retrieval results are considered to be highly correlated. In this example, 90% of top 20 images are correct images. In general, the fea‐ tures in all retrieval results look similar to gorilla or grass. The results reveal that the

proposed method improves the performance of the region-based image retrieval.

**1.** The system returns 18 correct matches as shown in Fig. 14.

Fig. 12 appears in the top 20 images in Fig. 14.

high performance is achieved by using these features.

**Figure 14.** The retrieval results based on our integrated region-based relevance feedback.

scribed as follows.

68 Search Algorithms for Engineering Optimization

**Figure 15.** The initial query image and positive images. (a) Query image. (b) The 4 positive images marked by red box‐ es which are selected by user.

After the submission of pseudo query image *I* <sup>+</sup> and relevant images set *Rs* based on user's feedback information, the first feedback retrieval returns 10 images containing "red car" as shown in Fig. 16. For this example, the first feedback retrieval achieves an ARR improve‐ ment of 28.6%. More precise results can be achieved by increasing of the number of regionof-interest sets and relevant image set based for the second feedback retrieval as shown in Fig. 17. The retrieval results for the second feedback retrieval returns 11 images containing "red car", and achieve an NMRR improvement of 35% compared to the initial query. Fur‐ thermore, the rank order in Fig. 17 is more reasonable than that in Fig. 16.

To show the effectiveness of our proposed region-based relevance feedback approach, the quantitative results for individual class and average performance (ARR, ANMRR) are listed in Table 2 and 3, which show the comparison of the performance for each query. It can be seen that the performance of retrieving precision and rank are relatively poor for the initial query. Through the adding positive examples by user, feedback information could have more potential in finding the user's query concept by means of optimal pseudo query image *I* + and relevant images set *Rs* as described in Section 5.4. In summary, the first feedback query improves 30.8% of ARR gain and 28% of ANMRR gain, and the second feedback query further improves 10.6% of ARR gain and 11% of ANMRR gain as compared with first feedback query. Although the improvement of retrieval efficiency is decreases progressively after two or three feedback queries, the proposed technique is able to provide satisfactory retrieval results in that few feedback queries.

**Figure 17.** The retrieval results by our integrated region-based relevance feedback for the second iteration.

The conventional existing region-based relevance feedback approaches work well in some specified applications; however, their performances depend on the accuracy of segmenta‐ tion techniques. To solve this problem, we have introduced a novel region-based relevance feedback for image retrieval with the modified dominant color descriptor. The term "speci‐ fied area", which combines main objects and irrelevant regions in image, has been defined for compensating the inaccuracy of segmentation algorithm. In order to manipulate the opti‐ mal query, we have proposed the similarity matrix model to form the salient region sets. Our integrated region-based relevance feedback approach contains relevance information

the user's query perception. Experimental results indicate that the proposed technique ach‐

and relevant images set *Rs* , which are capable to reflect

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71

**7. Conclusion**

including pseudo query image *I* <sup>+</sup>

ieves precise results in general-purpose image database.

**Figure 16.** The retrieval results by our integrated region-based relevance feedback for the first iteration.

**Figure 17.** The retrieval results by our integrated region-based relevance feedback for the second iteration.

## **7. Conclusion**

To show the effectiveness of our proposed region-based relevance feedback approach, the quantitative results for individual class and average performance (ARR, ANMRR) are listed in Table 2 and 3, which show the comparison of the performance for each query. It can be seen that the performance of retrieving precision and rank are relatively poor for the initial query. Through the adding positive examples by user, feedback information could have more potential in finding the user's query concept by means of optimal pseudo query image

 and relevant images set *Rs* as described in Section 5.4. In summary, the first feedback query improves 30.8% of ARR gain and 28% of ANMRR gain, and the second feedback query further improves 10.6% of ARR gain and 11% of ANMRR gain as compared with first feedback query. Although the improvement of retrieval efficiency is decreases progressively after two or three feedback queries, the proposed technique is able to provide satisfactory

**Figure 16.** The retrieval results by our integrated region-based relevance feedback for the first iteration.

*I* +

retrieval results in that few feedback queries.

70 Search Algorithms for Engineering Optimization

The conventional existing region-based relevance feedback approaches work well in some specified applications; however, their performances depend on the accuracy of segmenta‐ tion techniques. To solve this problem, we have introduced a novel region-based relevance feedback for image retrieval with the modified dominant color descriptor. The term "speci‐ fied area", which combines main objects and irrelevant regions in image, has been defined for compensating the inaccuracy of segmentation algorithm. In order to manipulate the opti‐ mal query, we have proposed the similarity matrix model to form the salient region sets. Our integrated region-based relevance feedback approach contains relevance information including pseudo query image *I* <sup>+</sup> and relevant images set *Rs* , which are capable to reflect the user's query perception. Experimental results indicate that the proposed technique ach‐ ieves precise results in general-purpose image database.


**Class Initial query The 1st feedback query The 2nd feedback query**

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**Table 3.** Comparisons of ANMRR performance with different iterations by our proposed integrated region-based

relevance feedback approach.

 0.735 0.399 0.306 0.624 0.395 0.326 0.741 0.519 0.503 0.246 0.135 0.118 0.745 0.694 0.643 0.744 0.643 0.581 0.783 0.721 0.633 0.762 0.578 0.537 0.215 0.155 0.132 0.745 0.571 0.553 0.794 0.619 0.557 0.331 0.156 0.144 0.683 0.591 0.517 0.807 0.728 0.709 0.514 0.256 0.161 0.687 0.559 0.416 0.712 0.579 0.554 0.836 0.81 0.798 0.763 0.512 0.438 0.699 0.548 0.488 0.716 0.311 0.293 0.805 0.664 0.581 0.851 0.809 0.797 0.725 0.691 0.556 0.782 0.645 0.623 0.699 0.587 0.503 0.791 0.688 0.628 0.642 0.613 0.561 0.851 0.687 0.649 0.662 0.321 0.287 0.779 0.587 0.514 Avg. 0.692548 0.541 0.48729

**Table 2.** Comparisons of ARR performance with different iterations by our proposed integrated region-based relevance feedback approach.


**Class Initial query The 1st feedback query The 2nd feedback query**

**Table 2.** Comparisons of ARR performance with different iterations by our proposed integrated region-based

relevance feedback approach.

 0.28 0.465 0.635 0.56 0.785 0.845 0.31 0.53 0.535 0.8375 0.85 0.9 0.19 0.275 0.32 0.255 0.355 0.385 0.2 0.29 0.3 0.165 0.235 0.245 0.73 0.985 1 0.345 0.525 0.625 0.23 0.345 0.4 0.835 1 1 0.33 0.52 0.63 0.235 0.38 0.4 0.655 0.885 0.98 0.435 0.625 0.705 0.365 0.465 0.515 0.235 0.275 0.275 0.32 0.505 0.59 0.34 0.59 0.635 0.37 0.76 0.865 0.22 0.355 0.495 0.15 0.21 0.225 0.31 0.46 0.565 0.25 0.43 0.465 0.38 0.515 0.61 0.245 0.34 0.395 0.385 0.415 0.46 0.195 0.325 0.41 0.4125 0.8 0.8875 0.3 0.51 0.61 Avg. 0.357097 0.51629 0.577661

Search Algorithms for Engineering Optimization

**Table 3.** Comparisons of ANMRR performance with different iterations by our proposed integrated region-based relevance feedback approach.

## **Acknowledgement**

This work was supported by the National Science Counsel of Republic of China Granted NSC. 97-2221-E-214-053-.

**Author details**

Nai-Chung Yang1

siung, Taiwan R.O.C.

**References**

R.O.C.

, Chung-Ming Kuo1

and Wei-Han Chang2

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http://dx.doi.org/10.5772/45841

75

1 Department of Information Engineering, I-Shou University Tahsu, Kaohsiung, Taiwan

2 Department of Information Management, Fortune Institute of Technology, Daliao, Kaoh‐

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## **Appendix**

*BV* : Boolean model, which is used to determine whether the segmented region *R* belongs to the background or foreground.

*F* : dominant color descriptor

*D* <sup>2</sup> : similarity measure (dominant color descriptor)

*I <sup>R</sup> <sup>i</sup>* : the ith non-overlaping region in I

*RDCD* : dominate color descriptor (DCD) of a segmented region R

*RLBP*\_*<sup>h</sup> <sup>K</sup>* : the value of kth bin in LBP histogram

*R* \_*S* : region-based color similarity

*R* \_*Sc* : the maximum similarity between two regions in similar color percentage

*R*\_*Spoa* : similarity of the area percentage

*R* \_*ST* : region based texture similarity

*Rbackground* : defined background based on foreground assumption

*Rforeground* : defined foreground based on foreground assumption

*Rpoa* : the percentage of area for region R in the image

*Rs* : relevant image set

*Rtexture* : texture feature of region R

*ai*, *j* : similarity coefficient between two color clusters (dominant color descriptor)

*ci* : dominant color vector (dominant color descriptor)

*di*, *<sup>j</sup>* : Euclidean distance between two color clusters (dominant color descriptor)

*pi* : percentage of each dominant color (dominant color descriptor)

## **Author details**

**Acknowledgement**

74 Search Algorithms for Engineering Optimization

NSC. 97-2221-E-214-053-.

the background or foreground.

*F* : dominant color descriptor

*<sup>R</sup> <sup>i</sup>* : the ith non-overlaping region in I

*R* \_*S* : region-based color similarity

*R*\_*Spoa* : similarity of the area percentage

*R* \_*ST* : region based texture similarity

*Rs* : relevant image set

*ai*, *j*

*ci*

*di*, *<sup>j</sup>*

*pi*

*Rtexture* : texture feature of region R

: similarity measure (dominant color descriptor)

: the value of kth bin in LBP histogram

*RDCD* : dominate color descriptor (DCD) of a segmented region R

*Rbackground* : defined background based on foreground assumption

*Rforeground* : defined foreground based on foreground assumption

*Rpoa* : the percentage of area for region R in the image

: dominant color vector (dominant color descriptor)

*R* \_*Sc* : the maximum similarity between two regions in similar color percentage

: similarity coefficient between two color clusters (dominant color descriptor)

: Euclidean distance between two color clusters (dominant color descriptor)

: percentage of each dominant color (dominant color descriptor)

**Appendix**

*D* <sup>2</sup>

*RLBP*\_*<sup>h</sup> <sup>K</sup>*

*I*

This work was supported by the National Science Counsel of Republic of China Granted

*BV* : Boolean model, which is used to determine whether the segmented region *R* belongs to

Nai-Chung Yang1 , Chung-Ming Kuo1 and Wei-Han Chang2

1 Department of Information Engineering, I-Shou University Tahsu, Kaohsiung, Taiwan R.O.C.

2 Department of Information Management, Fortune Institute of Technology, Daliao, Kaoh‐ siung, Taiwan R.O.C.

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**Section 2**

**Telecommunication Applications**

**Telecommunication Applications**

**Chapter 4**

**Provisional chapter**

**Multidimensional Optimization-Based Heuristics**

**Applied to Wireless Communication Systems**

**Multidimensional Optimization-Based Heuristics**

In the last two decades, the mobile communications technologies and the Internet have grown almost exponentially, reaching a significant numbers of subscribers around the world. The mobile cellular service got a very large growth of users along with the increase of mobile data services. On the other hand, the Internet provides a great opportunity for users to access the

In this scenario, stands out the spread spectrum communication techniques that until the mid-80 were restricted to military applications and is currently in a final technological consolidation phase through the cellular mobile communication systems of third and fourth

Such multiple access-based systems use a matched filter bank to detect the interest signal, being however unable to recover the signal in an optimal way, regardless is affected by additive white Gaussian noise (AWGN), flat fading or selective fading channels, since the direct sequence code division multiple access (DS/CDMA) signal is corrupted by multiple access interference (MAI) and severely affected by the near-far effect, resulting in a system whose capacity may remain remarkably below the channel capacity [2] if specific techniques are not introduced to mitigate these effects, such as multiuser detection (MuD) [3], diversity

Thus, one of the biggest challenges in the multiuser communication systems development is the interference mitigation. This challenge becomes obvious to the modern and current wireless networks like cellular networks, wireless local area network (WLAN) and wireless metropolitan area network (WMAN), due to the high spectral efficiency need, requiring

The third and fourth generations of cellular mobile systems and wireless networks were designed to support many services through the use of multirate transmission schemes, different quality of service (QoS) requirements and multidimensional diversity (time,

> ©2012 Ciriaco et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Ciriaco et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 Ciriaco et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

advanced techniques for frequency reuse and interference mitigation.

**Applied to Wireless Communication Systems**

Fernando Ciriaco, Taufik Abrão and

Taufik Abrão and Paul Jean E. Jeszensky

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

information via fixed and/or wireless networks.

generations used throughout the world [1].

exploration [4, 5] and so forth.

Paul Jean E. Jeszensky

Fernando Ciriaco,

**1. Introduction**

http://dx.doi.org/10.5772/54152

**Provisional chapter**

## **Multidimensional Optimization-Based Heuristics Applied to Wireless Communication Systems Multidimensional Optimization-Based Heuristics Applied to Wireless Communication Systems**

Fernando Ciriaco, Taufik Abrão and Paul Jean E. Jeszensky Fernando Ciriaco, Taufik Abrão and Paul Jean E. Jeszensky

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/54152

## **1. Introduction**

In the last two decades, the mobile communications technologies and the Internet have grown almost exponentially, reaching a significant numbers of subscribers around the world. The mobile cellular service got a very large growth of users along with the increase of mobile data services. On the other hand, the Internet provides a great opportunity for users to access the information via fixed and/or wireless networks.

In this scenario, stands out the spread spectrum communication techniques that until the mid-80 were restricted to military applications and is currently in a final technological consolidation phase through the cellular mobile communication systems of third and fourth generations used throughout the world [1].

Such multiple access-based systems use a matched filter bank to detect the interest signal, being however unable to recover the signal in an optimal way, regardless is affected by additive white Gaussian noise (AWGN), flat fading or selective fading channels, since the direct sequence code division multiple access (DS/CDMA) signal is corrupted by multiple access interference (MAI) and severely affected by the near-far effect, resulting in a system whose capacity may remain remarkably below the channel capacity [2] if specific techniques are not introduced to mitigate these effects, such as multiuser detection (MuD) [3], diversity exploration [4, 5] and so forth.

Thus, one of the biggest challenges in the multiuser communication systems development is the interference mitigation. This challenge becomes obvious to the modern and current wireless networks like cellular networks, wireless local area network (WLAN) and wireless metropolitan area network (WMAN), due to the high spectral efficiency need, requiring advanced techniques for frequency reuse and interference mitigation.

The third and fourth generations of cellular mobile systems and wireless networks were designed to support many services through the use of multirate transmission schemes, different quality of service (QoS) requirements and multidimensional diversity (time,

©2012 Ciriaco et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative

frequency and space). Thus, modern systems must accept users transmitting simultaneously in different rates in asymmetric traffic channels (uplink and downlink may be required to work at different rates), and also ensure the minimum specifications of QoS for each offered service.

**1.2. Heuristics applied to communication systems**

quality-of-services, and multicarrier CDMA systems.

with growing scarceness of spectrum and energy.

in relation to conventional topologies.

performance.

optimization [26–28].

In the last decade, the literature has been collecting sub-optimal solutions proposals based on iterative algorithms and heuristics, particularly evolutionary and local search, applied to inherent multiple access communication systems problems, among which we could cite the following heuristic solutions: optimal multiuser detection [12–19]; spreading sequences selection [20, 21]; parameter estimation, particularly the channel coefficients estimation, delay and power users [19, 22, 23]; power control problem [24, 25]; and resource allocation

Multidimensional Optimization-Based Heuristics Applied to Wireless Communication Systems

http://dx.doi.org/10.5772/54152

83

However, in recent years, the multiuser detection optimization problem in a single DS/CDMA system have been changed for others more complex applications, such as systems with multiple transmit and receive antennas, multirate coded systems with different

Differently from the most results reported in the literature, this chapter considers a multidimensional approach which aims to recover optimally (or very close to the optimal point) all the information of all users within the same processing window, considering multipath channels with different power-delay profiles, data rates, time or space-time coding, multiple antenna and multicarrier. In dealing with this sort of system, it will possible to provide various high-rate user's services including voice, data and video under a scenario

Moreover, to establish quality criteria that meet the acceptance requirements of the scientific community and telecommunications industry standards, this work analyzes convergence and performance aspects of a wide representative heuristic algorithms, considering metrics such as stability, capacity, control and implementation aspects, as well as the algorithm complexity

However, aiming at the multidimensional optimization analysis for high performance systems, various techniques based on heuristic algorithms are deployed in this work. Heuristic algorithms have been applied in several optimization problems, showing excellent results in large combination problems for practical cases. Still, there is an inherent difficulty in selecting and setting up the algorithm steps, since correct choices will result in good performances, and contrary, a poorly calibrated parameters may result in a disastrous

Therefore, the manipulation of several variables associated with each heuristic algorithm requires knowledge of the problem to be optimized, experience and keen perception of the algorithm behavior when selecting the parameters. Often, the parameters used are appropriate only for very restrictive settings, and frequently there are no consensus on the (sub-)optimal input parameters to be adopted or even the more conducive internal strategies to adjust those input parameters. Thus, parameters are chosen by past accumulated experience in dealing with other optimization problems or even through non-exhaustive trial tests. This scenario has resulted in a somewhat distrust level of such alternatives application

Thus, the motivation in pursuit of heuristic algorithms to ensure optimal performance is the core of this chapter. To do so, it will be analyzed in a systematic way the main meta-heuristic and hyper-heuristic algorithms deployed in wireless systems, which may be mentioned the genetic algorithm, evolutionary programming, local search (*k*-optimum),

in optimization problems that commonly arise in communications systems.

Hence, current industry standards for wireless networks use a combination of the following techniques to improve the frequency spectrum efficiency: multicarrier, spread spectrum, multiple antennas, spatial multiplexing and coding, reinforcing researches in order to improve the capacity of these systems, considering efficient transmission schemes, multiple diversity combination, multiuser detection methods, among others.

## **1.1. Multiuser detection**

One way to reduce substantially the interference and increase spread spectrum system capacity consists in modifying the detection strategy, using the information of other interfering users signals for detection process of interest user information. This strategy is called multiuser detection (MuD) [3, 6, 7].

In MuD strategy, active user information in the system are used together in order to better detect each individual user, increasing the system performance and/or capacity.

From the 1986 pioneering Verdu's work [3, 6] on optimum multiuser detector (OMuD) to a wide variety of multiuser detectors aiming to improve the performance obtained with the conventional detector in multiple access systems, a remarkable advance in the field has been achieved in the last twenty years. However, given the exponential complexity of the optimum detector, the research efforts has been focused on the development of sub-optimal or near-optimal multiuser detectors with lower complexity.

Alternatives to OMuD include the classical linear multiuser detectors such as Decorrelator [3] and the minimum mean square error (MMSE) [8], the nonlinear MuDs, such as interference cancellation (IC) [9, 10] and zero forcing decision feedback (ZF-DFE) detector [11], and heuristics-based multiuser detectors [12, 13].

However, both classical linear MMSE and Decorrelator multiuser detector algorithms presents two drawbacks; a) for practical system scenarios, both MuD result in performance degradation regarding OMuD; b) they need to perform a correlation matrix inversion, which implies in a high complexity for practical wireless systems with a high number of active users and/or systems with real-time detection in with the active number of users randomly and quickly changes along the time.

The operation principle for the non-linear classical IC and ZF-DF multiuser detectors is the reconstruction of MAI estimates, followed by cancellation (subtraction) for the interest user signal. The operations of MAI reconstruction and cancellation can be repeated in a multistage structure, resulting in more reliable signals canceling each new stage when estimates can be obtained with relative accuracy. The complexity of these detectors increases with the number of necessary stages for demodulation and after a certain number of stages there is no significant performance gain due to the propagation of interference estimation error. This limits the performance of these algorithms. Although the advantage of lower complexity regarding the MMSE and Decorrelator, performance achieved by the non-linear subtractive MuD detectors remain below the MMSE detector for almost all practical interest scenarios.

## **1.2. Heuristics applied to communication systems**

2 Search Algorithms

service.

**1.1. Multiuser detection**

is called multiuser detection (MuD) [3, 6, 7].

heuristics-based multiuser detectors [12, 13].

and quickly changes along the time.

or near-optimal multiuser detectors with lower complexity.

frequency and space). Thus, modern systems must accept users transmitting simultaneously in different rates in asymmetric traffic channels (uplink and downlink may be required to work at different rates), and also ensure the minimum specifications of QoS for each offered

Hence, current industry standards for wireless networks use a combination of the following techniques to improve the frequency spectrum efficiency: multicarrier, spread spectrum, multiple antennas, spatial multiplexing and coding, reinforcing researches in order to improve the capacity of these systems, considering efficient transmission schemes, multiple

One way to reduce substantially the interference and increase spread spectrum system capacity consists in modifying the detection strategy, using the information of other interfering users signals for detection process of interest user information. This strategy

In MuD strategy, active user information in the system are used together in order to better

From the 1986 pioneering Verdu's work [3, 6] on optimum multiuser detector (OMuD) to a wide variety of multiuser detectors aiming to improve the performance obtained with the conventional detector in multiple access systems, a remarkable advance in the field has been achieved in the last twenty years. However, given the exponential complexity of the optimum detector, the research efforts has been focused on the development of sub-optimal

Alternatives to OMuD include the classical linear multiuser detectors such as Decorrelator [3] and the minimum mean square error (MMSE) [8], the nonlinear MuDs, such as interference cancellation (IC) [9, 10] and zero forcing decision feedback (ZF-DFE) detector [11], and

However, both classical linear MMSE and Decorrelator multiuser detector algorithms presents two drawbacks; a) for practical system scenarios, both MuD result in performance degradation regarding OMuD; b) they need to perform a correlation matrix inversion, which implies in a high complexity for practical wireless systems with a high number of active users and/or systems with real-time detection in with the active number of users randomly

The operation principle for the non-linear classical IC and ZF-DF multiuser detectors is the reconstruction of MAI estimates, followed by cancellation (subtraction) for the interest user signal. The operations of MAI reconstruction and cancellation can be repeated in a multistage structure, resulting in more reliable signals canceling each new stage when estimates can be obtained with relative accuracy. The complexity of these detectors increases with the number of necessary stages for demodulation and after a certain number of stages there is no significant performance gain due to the propagation of interference estimation error. This limits the performance of these algorithms. Although the advantage of lower complexity regarding the MMSE and Decorrelator, performance achieved by the non-linear subtractive MuD detectors remain below the MMSE detector for almost all practical interest scenarios.

detect each individual user, increasing the system performance and/or capacity.

diversity combination, multiuser detection methods, among others.

In the last decade, the literature has been collecting sub-optimal solutions proposals based on iterative algorithms and heuristics, particularly evolutionary and local search, applied to inherent multiple access communication systems problems, among which we could cite the following heuristic solutions: optimal multiuser detection [12–19]; spreading sequences selection [20, 21]; parameter estimation, particularly the channel coefficients estimation, delay and power users [19, 22, 23]; power control problem [24, 25]; and resource allocation optimization [26–28].

However, in recent years, the multiuser detection optimization problem in a single DS/CDMA system have been changed for others more complex applications, such as systems with multiple transmit and receive antennas, multirate coded systems with different quality-of-services, and multicarrier CDMA systems.

Differently from the most results reported in the literature, this chapter considers a multidimensional approach which aims to recover optimally (or very close to the optimal point) all the information of all users within the same processing window, considering multipath channels with different power-delay profiles, data rates, time or space-time coding, multiple antenna and multicarrier. In dealing with this sort of system, it will possible to provide various high-rate user's services including voice, data and video under a scenario with growing scarceness of spectrum and energy.

Moreover, to establish quality criteria that meet the acceptance requirements of the scientific community and telecommunications industry standards, this work analyzes convergence and performance aspects of a wide representative heuristic algorithms, considering metrics such as stability, capacity, control and implementation aspects, as well as the algorithm complexity in relation to conventional topologies.

However, aiming at the multidimensional optimization analysis for high performance systems, various techniques based on heuristic algorithms are deployed in this work. Heuristic algorithms have been applied in several optimization problems, showing excellent results in large combination problems for practical cases. Still, there is an inherent difficulty in selecting and setting up the algorithm steps, since correct choices will result in good performances, and contrary, a poorly calibrated parameters may result in a disastrous performance.

Therefore, the manipulation of several variables associated with each heuristic algorithm requires knowledge of the problem to be optimized, experience and keen perception of the algorithm behavior when selecting the parameters. Often, the parameters used are appropriate only for very restrictive settings, and frequently there are no consensus on the (sub-)optimal input parameters to be adopted or even the more conducive internal strategies to adjust those input parameters. Thus, parameters are chosen by past accumulated experience in dealing with other optimization problems or even through non-exhaustive trial tests. This scenario has resulted in a somewhat distrust level of such alternatives application in optimization problems that commonly arise in communications systems.

Thus, the motivation in pursuit of heuristic algorithms to ensure optimal performance is the core of this chapter. To do so, it will be analyzed in a systematic way the main meta-heuristic and hyper-heuristic algorithms deployed in wireless systems, which may be mentioned the genetic algorithm, evolutionary programming, local search (*k*-optimum), simulated annealing, heuristic algorithm based on Tabu list and a hyper-heuristic-basis selection.

**2.1. Received signal in multipath MIMO channels**

in one of the antennas is:

*rnRx* (*t*) <sup>=</sup> *<sup>I</sup>*−<sup>1</sup>

the *I* bits transmitted base rate, **x**

frequency spread, respectively, *<sup>τ</sup>*(*g*)

(*g*)

*<sup>k</sup>*,ℓ,*ge <sup>j</sup>φ*(*i*)

*<sup>k</sup>*,ℓ,*<sup>g</sup>* have an uniform distribution *<sup>φ</sup>*(*i*)

transmission, *d*

group, resulting in:

can be written as:

(*i*) *<sup>k</sup>*,ℓ,*<sup>g</sup>* <sup>=</sup> *<sup>β</sup>*(*i*)

where *c*

for ∀*k*, *g*, *i*.

of *c* (*i*)

∑ *i*=0 *K*(*g*) ∑ *k*=1

*G* ∑ *g*=1

·*s* (*g*) *Fk*,*m*cos

 *<sup>M</sup>* ∑ *m*=1

*m*(*g*) ∑ *j*=1 *A*′ *<sup>k</sup>*,*g***x** (*i*) *<sup>k</sup>*(*g*) [*j*]*<sup>s</sup>*

(*i*)

*<sup>k</sup>* , as well as the propagation delay, <sup>∆</sup>(*g*)

*h* (*i*) *<sup>k</sup>*,*<sup>g</sup>* (*t*) =

the small-scale fading envelope with a Rayleigh distribution.

Additionally, we considered normalized channel gain for all users, i.e., **E**

*<sup>τ</sup>*(*g*) *<sup>k</sup>*,<sup>ℓ</sup> <sup>=</sup> <sup>∆</sup>(*g*)

> *L* ∑ ℓ=1 *c* (*i*) *<sup>k</sup>*,ℓ,*gδ <sup>t</sup>* − <sup>∆</sup>(*g*)

<sup>2</sup>*<sup>π</sup> fmt* <sup>+</sup> *<sup>φ</sup>*(*g*)

*k*,*m* <sup>∗</sup> *<sup>h</sup>* (*i*)

where *K*(*g*) is the number of physical users belonging to *g*-th multirate group being *K* = *K*(1) + *K*(2) + ... + *K*(*g*) + ... + *K*(*G*) the total number of active users in the physical system, divided into *g* user groups of same rate, *t* ∈ [0, *T*], *M* represents the number of subcarriers, the amplitude *Ak*,*g*′ is the amplitude of the received *k*-th user of *g*-th multirate group, including the effects of path loss and shadowing channel, and assumed constant over

to the *i* th symbol interval; *sCk*, *sk* and *sFk*,*<sup>m</sup>* represent the sequences of channeling, time and

user; *fm* represents the respective subcarriers frequencies; *hk*,*<sup>g</sup>* is the impulse response of the channel and the term *η*(*t*) is the AWGN with bilateral power spectral density equal to *N*0/2.

The *k*-th user delay of *g*-th multirate group takes into account the nature of the asynchronous

The channel impulse response to the *k*-th user of *g*-th multirate group in the range of *i*-th bit

multirate group, ℓ-th path and *δ*(*t*) is the unit impulse function. It is assumed that the phase

*<sup>k</sup>*,<sup>ℓ</sup> + *d*

(*g*)

*<sup>k</sup>*,<sup>ℓ</sup> <sup>−</sup> *iT*

*<sup>k</sup>*,ℓ,*<sup>g</sup>* indicates the complex channel coefficient for the *k*-th user of *g*-th

*<sup>k</sup>*,ℓ,*<sup>g</sup>* <sup>∈</sup> [0, 2*π*) and the module channel *<sup>β</sup>*(*i*)

*<sup>k</sup>*,<sup>ℓ</sup> is the random delay, *<sup>φ</sup>*(*g*)

Considering the reverse link and assuming a set of bits transmitted (frame) consisting of *I* bit for each multirate user, the resulting signal propagates through *G* independent Rayleigh fading paths. Thus, the equivalent baseband received signal (assuming ideal low-pass filter)

(*g*)

*Ck* (*<sup>t</sup>* <sup>−</sup> *jT*)*<sup>s</sup>*

Multidimensional Optimization-Based Heuristics Applied to Wireless Communication Systems

(*g*) *k <sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*(*g*)

*<sup>k</sup>*,*<sup>g</sup>* (*t*) + *η* (*t*)

*<sup>k</sup>*(*g*) [*j*] <sup>∈</sup> {±1} is the symbol of coded information passed

*<sup>k</sup>*,<sup>ℓ</sup> <sup>−</sup> *iT*

http://dx.doi.org/10.5772/54152

*<sup>k</sup>*,*<sup>m</sup>* corresponds to the initial *k*-th

*<sup>k</sup>*,<sup>ℓ</sup> for *<sup>k</sup>*-th user, <sup>ℓ</sup>-th path, *<sup>g</sup>*-th multirate

*<sup>k</sup>* (2)

·

(1)

85

(3)

*<sup>k</sup>*,ℓ,*<sup>g</sup>* represents


 *L* ∑ ℓ=1

## **2. System characteristics**

Considering the provision of various services with high quality, we opted a transmission/reception scheme that adds several dimensions in order to explore diversity. Figure 1 shows the transmitter and receiver topologies deployed in this work. Hence, lets consider the *k*-th user transmitter and noting that the channel coding stage is necessary to correct the received signal in the presence of errors through the use of redundancy (code diversity). The multirate modulation block aims to ensure the provision of various services to users at different data rates, ensuring the possibility of optimum resource management strategies. The time-spreading code guarantees a rejection level of multiple access interference, and the identification of each DS/CDMA user as well, acting as a kind of time-diversity. The frequency spreading block implements frequency-diversity through the information transmission on different sub-carriers. Finally, the multiple-input-multiple-output (MIMO) antennas block deploys techniques that provide spatial-diversity, either through simple arrangements with various antennas or even by space-time block code (STBC) or trellis code.

**Figure 1.** Communication system overview with use of space, time, frequency and coding diversities.

The transmitted signal of the *k*-th user propagates through a channel whose model includes attenuation of small and large scale, i.e., path loss, shadowing and multipath effects.

The *k*-th user signals at the receiver input are demodulated via an antenna array in order to exploit spatial diversity. Structures can be used with several receiving antennas physically separated by a sufficient distance to avoid overlapping signals and block-basis or trellis-basis signal processing techniques. Subsequently, the signals are despread in frequency and time ensuring the channel rejection and multiple access interference rejection, respectively. At this point, it is evident the frequency- and time-diversity exploitation. Thus, the demodulated signals are reassembled considering the *k*-th user transmission rates. This receptor is known as Rake receiver. Finally, the signals are decoded by means of particular techniques, resulting in a type of diversity code.

## **2.1. Received signal in multipath MIMO channels**

4 Search Algorithms

selection.

User k Data

Estimated User k Data

**2. System characteristics**

space-time block code (STBC) or trellis code.

Multirate Modulation

Multirate Demodulation

Heuristic Algorithms

Time Spreading

Transmitter

Time Despreading

Receiver

**Figure 1.** Communication system overview with use of space, time, frequency and coding diversities.

The transmitted signal of the *k*-th user propagates through a channel whose model includes

The *k*-th user signals at the receiver input are demodulated via an antenna array in order to exploit spatial diversity. Structures can be used with several receiving antennas physically separated by a sufficient distance to avoid overlapping signals and block-basis or trellis-basis signal processing techniques. Subsequently, the signals are despread in frequency and time ensuring the channel rejection and multiple access interference rejection, respectively. At this point, it is evident the frequency- and time-diversity exploitation. Thus, the demodulated signals are reassembled considering the *k*-th user transmission rates. This receptor is known as Rake receiver. Finally, the signals are decoded by means of particular techniques, resulting

attenuation of small and large scale, i.e., path loss, shadowing and multipath effects.

Frequency Spreading

Frequency Despreading

Multiple Antennas

> Multiple Antennas

MIMO Channel

ATx

AR x

Channel Encoder

Channel Decoder

in a type of diversity code.

simulated annealing, heuristic algorithm based on Tabu list and a hyper-heuristic-basis

Considering the provision of various services with high quality, we opted a transmission/reception scheme that adds several dimensions in order to explore diversity. Figure 1 shows the transmitter and receiver topologies deployed in this work. Hence, lets consider the *k*-th user transmitter and noting that the channel coding stage is necessary to correct the received signal in the presence of errors through the use of redundancy (code diversity). The multirate modulation block aims to ensure the provision of various services to users at different data rates, ensuring the possibility of optimum resource management strategies. The time-spreading code guarantees a rejection level of multiple access interference, and the identification of each DS/CDMA user as well, acting as a kind of time-diversity. The frequency spreading block implements frequency-diversity through the information transmission on different sub-carriers. Finally, the multiple-input-multiple-output (MIMO) antennas block deploys techniques that provide spatial-diversity, either through simple arrangements with various antennas or even by Considering the reverse link and assuming a set of bits transmitted (frame) consisting of *I* bit for each multirate user, the resulting signal propagates through *G* independent Rayleigh fading paths. Thus, the equivalent baseband received signal (assuming ideal low-pass filter) in one of the antennas is:

$$r\_{\eta\_{\rm Kr}}(t) = \sum\_{i=0}^{I-1} \sum\_{k=1}^{K^{(g)}} \sum\_{g=1}^{G} \left[ \sum\_{m=1}^{M} \sum\_{j=1}^{m^{(g)}} A\_{k,g}^{'} \mathbf{x}\_{k^{(g)}}^{(i)} \left[ j \right] \mathbf{s}\_{\rm CK}^{(g)} \left( t - jT \right) \mathbf{s}\_{k}^{(g)} \left( t - \tau\_{k,\ell}^{(g)} - iT \right) \cdot \right. \tag{1}$$
 
$$\therefore \mathbf{s}\_{\rm FK,m}^{(g)} \cos \left( 2\pi f\_{\rm m} t + \phi\_{k,m}^{(g)} \right) \Big| \ast h\_{k,g}^{(i)} \left( t \right) + \eta \left( t \right)$$

where *K*(*g*) is the number of physical users belonging to *g*-th multirate group being *K* = *K*(1) + *K*(2) + ... + *K*(*g*) + ... + *K*(*G*) the total number of active users in the physical system, divided into *g* user groups of same rate, *t* ∈ [0, *T*], *M* represents the number of subcarriers, the amplitude *Ak*,*g*′ is the amplitude of the received *k*-th user of *g*-th multirate group, including the effects of path loss and shadowing channel, and assumed constant over the *I* bits transmitted base rate, **x** (*i*) *<sup>k</sup>*(*g*) [*j*] <sup>∈</sup> {±1} is the symbol of coded information passed to the *i* th symbol interval; *sCk*, *sk* and *sFk*,*<sup>m</sup>* represent the sequences of channeling, time and frequency spread, respectively, *<sup>τ</sup>*(*g*) *<sup>k</sup>*,<sup>ℓ</sup> is the random delay, *<sup>φ</sup>*(*g*) *<sup>k</sup>*,*<sup>m</sup>* corresponds to the initial *k*-th user; *fm* represents the respective subcarriers frequencies; *hk*,*<sup>g</sup>* is the impulse response of the channel and the term *η*(*t*) is the AWGN with bilateral power spectral density equal to *N*0/2.

The *k*-th user delay of *g*-th multirate group takes into account the nature of the asynchronous transmission, *d* (*g*) *<sup>k</sup>* , as well as the propagation delay, <sup>∆</sup>(*g*) *<sup>k</sup>*,<sup>ℓ</sup> for *<sup>k</sup>*-th user, <sup>ℓ</sup>-th path, *<sup>g</sup>*-th multirate group, resulting in:

$$
\pi\_{k,\ell}^{(g)} = \Delta\_{k,\ell}^{(g)} + d\_k^{(g)} \tag{2}
$$

The channel impulse response to the *k*-th user of *g*-th multirate group in the range of *i*-th bit can be written as:

$$h\_{k,g}^{(i)}\left(t\right) = \sum\_{\ell=1}^{L} c\_{k,\ell,g}^{(i)} \delta\left(t - \Delta\_{k,\ell}^{(g)} - iT\right) \tag{3}$$

where *c* (*i*) *<sup>k</sup>*,ℓ,*<sup>g</sup>* <sup>=</sup> *<sup>β</sup>*(*i*) *<sup>k</sup>*,ℓ,*ge <sup>j</sup>φ*(*i*) *<sup>k</sup>*,ℓ,*<sup>g</sup>* indicates the complex channel coefficient for the *k*-th user of *g*-th multirate group, ℓ-th path and *δ*(*t*) is the unit impulse function. It is assumed that the phase of *c* (*i*) *<sup>k</sup>*,ℓ,*<sup>g</sup>* have an uniform distribution *<sup>φ</sup>*(*i*) *<sup>k</sup>*,ℓ,*<sup>g</sup>* <sup>∈</sup> [0, 2*π*) and the module channel *<sup>β</sup>*(*i*) *<sup>k</sup>*,ℓ,*<sup>g</sup>* represents the small-scale fading envelope with a Rayleigh distribution.

Additionally, we considered normalized channel gain for all users, i.e., **E** *L* ∑ ℓ=1 |*ck*,ℓ,*g*| 2 = 1 for ∀*k*, *g*, *i*.

Therefore, we can rewrite the received signal in each *ARx* antennas replacing the eq.(3) into eq. (1), resulting in:

$$r\_{\mathbb{H}\mathbf{x}}\left(t\right) = \sum\_{i=0}^{I-1} \sum\_{k=1}^{K^{(\mathfrak{z})}} \sum\_{g=1}^{G} \sum\_{m=1}^{M^{(\mathfrak{z})}} \sum\_{\ell=1}^{L} A\_{k,g,\ell}^{'} \mathbf{x}\_{k^{(\mathfrak{z})}}^{(i)} \left[ \left[ \mathbf{s}\_{\mathbb{C}\mathbf{x}}^{(g)} \left(t - jT\right) \mathbf{s}\_{k}^{(g)} \left(t - \tau\_{k,\ell}^{(g)} - iT\right) \cdot \right. \tag{4} \right. \tag{4}$$
 
$$\cdot \mathbf{s}\_{Fk,m}^{(g)} \cos \left(2\pi f\_{m} t + \phi\_{k,m}^{(g)} \right) c\_{k,\ell,g}^{(i)} \delta \left(t - \Delta\_{k,\ell}^{(g)} - iT\right) + \eta\_{\mathbb{H}\mathbf{x}} \tag{5}$$

where *j* = 1 : *m*(*g*). The first term corresponds to the desired signal, the second term to the self-interference (SI), the third to the MAI on the ℓ-th multipath component of the *k*-th user of *g*-th multirate group, *m*-th subcarrier and *nRx*-th antenna, as well the last term corresponds

In this case, the Rake receiver combines the outputs of the matched filters bank available for each user (fingers)2 and weighted by the respective channel gains [29]. The Maximal Ratio

*k*,ℓ,*g*,*nRx*,*mβ*ˆ(*i*)

*<sup>k</sup>*,ℓ,*g*,*nRx*,*<sup>m</sup>* and *<sup>φ</sup>*ˆ(*i*)

estimates of the channel coefficients, respectively, for the *i*-th processing interval for the *<sup>k</sup>*-th user, ℓ-th path, *<sup>g</sup>*-th multirate group, *nRx*-th antenna and *<sup>m</sup>*-th subcarrier. Again, the performance is degraded proportionally when there are errors in the channel estimates.

Finally, the estimates for the *m*(*g*) information symbols of *k*-th user of *g*-th multirate group

Therefore, the estimated symbol frame for all users in the range of *i*-th bit with *DKv* × 1

The performance obtained with the MRC Rake receiver will be deteriorated considerably when the number of users sharing the same channel grow3 and/or when the interfering

The best performance among the multiuser detectors is achieved with OMuD, where the goal is to maximize the maximum likelihood function [3]. Given the conditional probability:

> = *e* − <sup>1</sup> 2*σ*2 *<sup>I</sup>*−<sup>1</sup>

, *i* ∈ [0, *I* − 1]

<sup>2</sup> In addition to multipath effects, multiple subcarriers and multiple receiving antennas.

(*i*) *<sup>K</sup>*(1),1 ... **ˆx**

**ˆy** (*i*) *k*,*g* 

> (*i*) 1,*<sup>G</sup>* **ˆx** (*i*) 2,*<sup>G</sup>* ... **ˆx**

*<sup>k</sup>*,ℓ,*g*,*nRx*,*me*

Multidimensional Optimization-Based Heuristics Applied to Wireless Communication Systems

−*jφ*ˆ(*i*) *<sup>k</sup>*,ℓ,*g*,*nRx*,*<sup>m</sup>* 

> (*i*) *K*(*G*),*G T*

*<sup>i</sup>*=<sup>0</sup> [**y**(*i*)−**S***t*(**ˆx**(*i*)

)]2 *dt*

*<sup>k</sup>*,ℓ,*g*,*nRx*,*<sup>m</sup>* are the magnitude and phase

http://dx.doi.org/10.5772/54152

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87

(9)

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(11)

Combiner (MRC) combines the signals from the *D* correlators in coherent way:

**ˆx** (*i*) *<sup>k</sup>*,*<sup>g</sup>* <sup>=</sup> *sgn*

to the filtered AWGN.

dimension is given by:

users power increase.

**2.3. Optimum multiuser detector**

*Pr* **y**(*i*) |**ˆx**(*i*)

<sup>3</sup> Increasing the MAI, third term of eq. (7).

**ˆy** (*i*) *<sup>k</sup>*,*<sup>g</sup>* =

where ℜ{.} is the real part operator, *<sup>β</sup>*ˆ(*i*)

are obtained through an abrupt decision rule:

**ˆx**(*i*) = **ˆx** (*i*) 1,1 **ˆx** (*i*) 2,1 ... **ˆx**

*D* ∑ ℓ=1

*ARx* ∑ *nRx*=1

*M* ∑ *m*=1 ℜ **<sup>y</sup>**(*i*)

For simplicity and without generality loss, we consider ordered random delays, i.e.:

$$0 = \tau\_{1,1}^{(1)} \le \tau\_{1,2}^{(1)} \le \dots \le \tau\_{1,L}^{(1)} \le \tau\_{2,1}^{(1)} \le \dots \le \tau\_{K^{(1)},L}^{(1)} \le \dots \le \tau\_{K^{(G)},L}^{(G)} < T \tag{5}$$

#### **2.2. Conventional SIMO detection systems**

Considering the system with only one transmission antenna, one can rewrite eq. (4) considering just the *nRx*-th receiving antenna and *m*-th subcarrier as:

$$\begin{split} \boldsymbol{\sigma}\_{\boldsymbol{\eta}\_{\rm{Rx}},\rm{\rm{m}}}(t) = \sum\_{i=0}^{I-1} \sum\_{k=1}^{K^{(\boldsymbol{g})}} \sum\_{g=1}^{G} \sum\_{i=1}^{m^{(\boldsymbol{g})}} \sum\_{\ell=1}^{L} A\_{k,\rm{g},\ell} \mathbf{x}\_{k^{(\boldsymbol{g})}}^{(i)} \left[ \boldsymbol{j} \right] \mathbf{s}\_{\rm{CK}}^{(g)} \left( t - \boldsymbol{j} \boldsymbol{T} \right) \mathbf{s}\_{k}^{(g)} \left( t - \tau\_{k,\ell}^{(g)} \boldsymbol{i} \boldsymbol{T} \right) \mathbf{s}\_{\rm{Fk},\rm{m}}^{(g)} \cdot \\ \boldsymbol{\cdot} \boldsymbol{\beta}\_{k,\ell,g,\rm{m}}^{(i)} \boldsymbol{e}^{l(\boldsymbol{\omega}\_{\rm{m}} t + \boldsymbol{\varphi}\_{k,\ell,g,\rm{m}}^{(i)})} + \eta\_{\rm{Hz}}(t) \end{split} \tag{6}$$

where *ηq*(*t*) corresponds to the Additive White Gaussian Noise (AWGN) for *nRx*-th receiving antenna.

For multipath fading channels, multirate and multicarrier scheme, the receiver for each subcarrier demodulation use the Rake receiver consisting of a bank of *KD* matched filters to the multirate physical users spread sequences with path diversity order<sup>1</sup> *D* ≤ *L*, followed by the second despreading (channeling) aiming recovering *m*(*g*) simultaneously transmitted bits in parallel channels. To be able achieve a perfect synchronism (maximum auto-correlation) of spread sequence at the receiver must use delay accurate estimates for the ℓ-th path of the *k*-th user of *g*-th multirate group, *τ*ˆ (*g*) *<sup>k</sup>*,<sup>ℓ</sup> . Performance is degraded proportionally when there are errors in the delays estimates.

Thus, the *m*(*g*) matched filter outputs for the *k*-th physical user, *g*-th multirate group and corresponding to ℓ-th multipath component, *<sup>m</sup>*-th subcarrier and *nRx*-th antenna, sampled at the end of basic information period *T* of *i*-th interval symbol can be expressed as:

$$\begin{split} \mathcal{Y}\_{k,\ell,\operatorname{g},\operatorname{yn},\operatorname{m}}^{(i)}[j] &= \frac{1}{\sqrt{\mathcal{N}\_{\operatorname{\mathsf{C}}}}} \int\_{0}^{\mathsf{T}} r\_{\operatorname{\mathsf{u}},\operatorname{\mathsf{s}},\operatorname{\mathsf{m}}}(t) \operatorname{s}\_{\operatorname{\mathsf{k}}}^{\operatorname{\mathsf{(g)}}} \left(t - \tau\_{\operatorname{k},\operatorname{\mathsf{e}}}^{\operatorname{\mathsf{(g)}}} - i\mathcal{T}\right) \operatorname{s}\_{\operatorname{\mathsf{C}k}}^{\operatorname{\mathsf{(g)}}}(t - j\mathcal{T}) \operatorname{s}\_{\operatorname{\mathsf{F}},\operatorname{\mathsf{m}}}^{\operatorname{\mathsf{(g)}}} e^{\left(-j\omega\_{\operatorname{\mathsf{m}}}t\right)} dt \\ &= \underbrace{\mathcal{A}\_{\operatorname{\mathsf{k}},\operatorname{\mathsf{g}}}^{\operatorname{\mathsf{T}}} \operatorname{\mathsf{Tc}}\_{\operatorname{\mathsf{k}},\operatorname{\mathsf{\mathsf{f}}},\operatorname{\mathsf{ym}}}^{\operatorname{\mathsf{(i)}}}(j)}\_{(1)} + \underbrace{\mathcal{S} \operatorname{I}\_{\operatorname{\mathsf{k}},\operatorname{\mathsf{\mathsf{E}}},\operatorname{\mathsf{vec}}}^{\operatorname{\mathsf{(i)}}} + \underbrace{\mathcal{M} \operatorname{AI}\_{\operatorname{\mathsf{k}},\operatorname{\mathsf{\mathsf{E}}},\operatorname{\mathsf{y}},\operatorname{\mathsf{u}}}^{\operatorname{\mathsf{(j)}}}}\_{(111)} + \underbrace{\mathcal{n}\_{\operatorname{\mathsf{k}},\operatorname{\mathsf{\mathsf{E}}},\operatorname{\mathsf{y}}}^{\operatorname{\mathsf{(i)}}}}\_{(1l)} + \underbrace{\mathcal{n}\_{\operatorname{\mathsf{k}},\operatorname{\mathsf{\mathsf{E}}},\operatorname{\mathsf{y}}}^{\operatorname{\mathsf{(i)}}}}\_{(1l)} \end{split} \tag{7}$$

<sup>1</sup> If *D* < *L*, in each Rake receiver the matched filters to spread sequences are synchronized to *D* energy major paths.

where *j* = 1 : *m*(*g*). The first term corresponds to the desired signal, the second term to the self-interference (SI), the third to the MAI on the ℓ-th multipath component of the *k*-th user of *g*-th multirate group, *m*-th subcarrier and *nRx*-th antenna, as well the last term corresponds to the filtered AWGN.

In this case, the Rake receiver combines the outputs of the matched filters bank available for each user (fingers)2 and weighted by the respective channel gains [29]. The Maximal Ratio Combiner (MRC) combines the signals from the *D* correlators in coherent way:

$$\mathfrak{F}\_{k,\emptyset}^{(i)} = \sum\_{\ell=1}^{D} \sum\_{n\_{\mathcal{K}}=1}^{A\_{\mathcal{K}}} \sum\_{m=1}^{M} \mathfrak{R} \left\{ \mathbf{y}\_{k,\ell,\emptyset,n\_{\mathcal{K}},m}^{(i)} \hat{\boldsymbol{\beta}}\_{k,\ell,\emptyset,n\_{\mathcal{K}},m}^{(i)} e^{-j\hat{\boldsymbol{\phi}}\_{k,\ell,\emptyset,n\_{\mathcal{K}},m}^{(i)}} \right\} \tag{8}$$

where ℜ{.} is the real part operator, *<sup>β</sup>*ˆ(*i*) *<sup>k</sup>*,ℓ,*g*,*nRx*,*<sup>m</sup>* and *<sup>φ</sup>*ˆ(*i*) *<sup>k</sup>*,ℓ,*g*,*nRx*,*<sup>m</sup>* are the magnitude and phase estimates of the channel coefficients, respectively, for the *i*-th processing interval for the *<sup>k</sup>*-th user, ℓ-th path, *<sup>g</sup>*-th multirate group, *nRx*-th antenna and *<sup>m</sup>*-th subcarrier. Again, the performance is degraded proportionally when there are errors in the channel estimates.

Finally, the estimates for the *m*(*g*) information symbols of *k*-th user of *g*-th multirate group are obtained through an abrupt decision rule:

$$\mathfrak{A}\_{k,\mathfrak{g}}^{(i)} = \text{sgn}\left(\mathfrak{F}\_{k,\mathfrak{g}}^{(i)}\right) \tag{9}$$

Therefore, the estimated symbol frame for all users in the range of *i*-th bit with *DKv* × 1 dimension is given by:

$$\mathbf{x}^{(i)} = \begin{bmatrix} \mathbf{x}\_{1,1}^{(i)} \ \mathbf{x}\_{2,1}^{(i)} \dots \ \mathbf{x}\_{K^{(1)},1}^{(i)} \dots \ \mathbf{x}\_{1,G}^{(i)} \ \mathbf{x}\_{2,G}^{(i)} \dots \ \mathbf{x}\_{K^{(G)},G}^{(i)} \end{bmatrix}^T \tag{10}$$

The performance obtained with the MRC Rake receiver will be deteriorated considerably when the number of users sharing the same channel grow3 and/or when the interfering users power increase.

#### **2.3. Optimum multiuser detector**

6 Search Algorithms

eq. (1), resulting in:

*rnRx* (*t*) <sup>=</sup> *<sup>I</sup>*−<sup>1</sup>

∑ *i*=0

> ·*s* (*g*) *Fk*,*m*cos

<sup>0</sup> <sup>=</sup> *<sup>τ</sup>*(1)

*rnRx*,*m*(*t*) =

*k*-th user of *g*-th multirate group, *τ*ˆ

are errors in the delays estimates.

*<sup>k</sup>*,ℓ,*g*,*nRx*,*m*[*j*] = <sup>1</sup>

<sup>√</sup>*NC T*

<sup>=</sup> *<sup>A</sup>*′

0

*<sup>k</sup>*,ℓ,*g*,*<sup>m</sup>* **x**

 (*I*)

*k*,*gTc*(*i*)

antenna.

*y* (*i*) *K*(*g*) ∑ *k*=1

1,1 <sup>≤</sup> *<sup>τ</sup>*(1)

**2.2. Conventional SIMO detection systems**

*K*(*g*) ∑ *k*=1

*G* ∑ *g*=1

*I*−1 ∑ *i*=0

*G* ∑ *g*=1

*M* ∑ *m*=1

1,2 ≤··· *<sup>τ</sup>*(1)

considering just the *nRx*-th receiving antenna and *m*-th subcarrier as:

*L* ∑ ℓ=1

(*g*)

*rnRx*,*<sup>m</sup>* (*t*)*s*

(*i*) *<sup>k</sup>*(*g*) [*j*]

*m*(*g*) ∑ *j*=1

*m*(*g*) ∑ *j*=1

<sup>2</sup>*<sup>π</sup> fmt* <sup>+</sup> *<sup>φ</sup>*(*g*)

*L* ∑ ℓ=1 *A*′ *<sup>k</sup>*,*g*,ℓ**x** (*i*) *<sup>k</sup>*(*g*) [*j*]*<sup>s</sup>*

> *k*,*m c* (*i*)

For simplicity and without generality loss, we consider ordered random delays, i.e.:

1,*<sup>L</sup>* <sup>≤</sup> *<sup>τ</sup>*(1)

*Ak*,*g*,<sup>ℓ</sup> **x** (*i*) *<sup>k</sup>*(*g*) [*j*]*<sup>s</sup>*

Therefore, we can rewrite the received signal in each *ARx* antennas replacing the eq.(3) into

(*g*)

*<sup>k</sup>*,ℓ,*gδ*(*<sup>t</sup>* <sup>−</sup> <sup>∆</sup>(*g*)

2,1 ≤···≤ *<sup>τ</sup>*(1)

(*g*)

*Ck* (*<sup>t</sup>* <sup>−</sup> *jT*) *<sup>s</sup>*

· *<sup>β</sup>*(*i*) *<sup>k</sup>*,ℓ,*g*,*<sup>m</sup> e*

Considering the system with only one transmission antenna, one can rewrite eq. (4)

where *ηq*(*t*) corresponds to the Additive White Gaussian Noise (AWGN) for *nRx*-th receiving

For multipath fading channels, multirate and multicarrier scheme, the receiver for each subcarrier demodulation use the Rake receiver consisting of a bank of *KD* matched filters to the multirate physical users spread sequences with path diversity order<sup>1</sup> *D* ≤ *L*, followed by the second despreading (channeling) aiming recovering *m*(*g*) simultaneously transmitted bits in parallel channels. To be able achieve a perfect synchronism (maximum auto-correlation) of spread sequence at the receiver must use delay accurate estimates for the ℓ-th path of the

Thus, the *m*(*g*) matched filter outputs for the *k*-th physical user, *g*-th multirate group and corresponding to ℓ-th multipath component, *<sup>m</sup>*-th subcarrier and *nRx*-th antenna, sampled at

*<sup>k</sup>*,<sup>ℓ</sup> <sup>−</sup> *iT*

*k*,ℓ,*g*,*nRx*,*m* (*I I*)

 *s* (*g*)

+ *MAI*(*i*)

the end of basic information period *T* of *i*-th interval symbol can be expressed as:

(*g*) *k <sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*(*g*)

+ *SI*(*i*)

<sup>1</sup> If *D* < *L*, in each Rake receiver the matched filters to spread sequences are synchronized to *D* energy major paths.

*Ck* (*<sup>t</sup>* <sup>−</sup> *jT*)*<sup>s</sup>*

(*g*) *k <sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*(*g*)

*<sup>k</sup>*,<sup>ℓ</sup> <sup>−</sup> *iT*) + *<sup>η</sup>nRx* (*t*)

*<sup>K</sup>*(1),*<sup>L</sup>* <sup>≤</sup> ... *<sup>τ</sup>*(*G*)

(*g*) *<sup>k</sup>* (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*(*g*)

*<sup>k</sup>*,<sup>ℓ</sup> . Performance is degraded proportionally when there

*Ck* (*<sup>t</sup>* <sup>−</sup> *jT*) *<sup>s</sup>*

*k*,ℓ,*g*,*nRx*,*m* (*III*)

(*g*) *Fk*,*me*

+ *<sup>n</sup>*(*i*)

(−*ωmt*) *dt*

*k*,ℓ,*g*,*nRx*,*m* (*IV*)

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(*ωmt*+*ϕ*(*i*)

*<sup>k</sup>*,<sup>ℓ</sup> <sup>−</sup> *iT*

 ·

*<sup>K</sup>*(*G*),*<sup>L</sup>* <sup>&</sup>lt; *<sup>T</sup>* (5)

(*g*) *Fk*,*<sup>m</sup>* ·

*<sup>k</sup>*,<sup>ℓ</sup> *iT*)*s*

*<sup>k</sup>*,ℓ,*g*,*<sup>m</sup>*) <sup>+</sup> *<sup>η</sup>nRx* (*t*)

(4)

(6)

The best performance among the multiuser detectors is achieved with OMuD, where the goal is to maximize the maximum likelihood function [3]. Given the conditional probability:

$$P\_r\left(\mathbf{y}^{(i)}|\mathbf{\hat{x}}^{(i)}, i \in [0, I-1]\right) = e^{\left\{-\frac{1}{2\sigma^2} \int\_{l=0}^{l=1} \left[\mathbf{y}^{(i)} - \mathbf{S}\_l(\mathbf{\hat{x}}^{(l)})\right]^2 dt\right\}}\tag{11}$$

<sup>2</sup> In addition to multipath effects, multiple subcarriers and multiple receiving antennas.

<sup>3</sup> Increasing the MAI, third term of eq. (7).

where the total received signal, reconstructed from the estimated parameters and known at the receiver is:

$$\begin{split} \mathbf{S}\_{l} \left( \hat{\mathbf{b}}^{(i)} \right) = \sum\_{k=1}^{K^{(i)}} \sum\_{g=1}^{G} \sum\_{m=1}^{M} \sum\_{j=1}^{M^{(g)}} \sum\_{\ell=1}^{L} A\_{k,g,\ell,\eta\_{Rx}}^{'} \mathbf{s}\_{k,g}^{(i)} \mathbf{s}\_{\complement\hspace{0.0cm}k}^{(i)} (t-jT) \mathbf{s}\_{k}^{(g)} \left( t - \tau\_{k,\ell,\eta\_{Rx}}^{(g)} - iT \right) \cdot \\ \cdot \mathbf{s}\_{\text{F},\text{m}}^{(g)} \hat{\boldsymbol{\beta}}\_{k,\ell,\eta\_{\text{F}Rx}}^{(i)} \mathbf{e}^{j \left( \omega\_{\text{f},\ell} + \hat{\mathbf{b}}\_{\ell,\ell,g\_{Rx}}^{(i)} \right)}\_{\text{F}} \end{split} \tag{12}$$

In this context, the maximum likelihood vector that must be found by OMuD has *DKv* × 1 dimension, and by:

$$\hat{\mathbf{x}} = \begin{bmatrix} \hat{\mathbf{x}}^{(0)^T} \ \hat{\mathbf{x}}^{(1)^T} \ \hat{\mathbf{x}}^{(2)^T} \dots \ \hat{\mathbf{x}}^{(I-1)^T} \end{bmatrix}^T \tag{13}$$

Finally, the block Toeplitz tridiagonal correlation matrix **R**, dimension *D*K*<sup>v</sup> I* × *D*K*<sup>v</sup> I*, is

**R** [0] **R***<sup>T</sup>* [1] **0** ··· **0 0 R** [1] **R** [0] **R***<sup>T</sup>* [1] ··· **0 0 0 R** [1] **<sup>R</sup>** [0] ... **0 0**

. ... ... ... .

**00 0** ··· **R** [1] **R** [0]

Therefore, the complete frame for the *I* estimated symbols from all *Kv* users can be obtained

max B∈{±1}K*<sup>v</sup> <sup>I</sup>*

The OMuD consists in finding the best data symbols vector in a set with all the possibilities, i.e., it is a NP-complete combination problem [30], which the traditional algorithms are inefficient. Most of these result in exponential complexity growth when one or more of the following factors: number of users, frame, number of receiver antennas, number of paths,

Therefore, the use of heuristic methods for this class of problems shows up attractive, since it is possible to obtain optimal solutions (or near-optimal) using reduced search spaces. Thus, the proposed strategy in this Chapter aiming for maximize the LLF by testing different candidates symbols vectors at each new iteration/generation of heuristic algorithms. Such attempts seek to maximize the system average performance, approaching or even equaling that obtained by OMuD, but with remarkable reduction in the computational complexity.

This section presents a brief review of heuristic algorithms, specifically local and evolutionary search, describing variants and required parameters. Such variants include encoding (mapping) problem, initialization algorithms step (parameters choice), cost function evaluation, search space scanning step, and replacement candidates step. For the analysis, 1-opt and *k*-opt local search, simulated annealing, short-term and reactive Tabu search, genetic, as well evolutionary programming algorithms have been considered in this work.

The encoding for MuD problem is inherently binary, because the data vector is naturally binary. Therefore, following the Keep it Simple (KIS) principle as much as possible, it is not necessary to perform an encoding (mapping) of candidate solutions differently to binary form. Thus, these candidates vectors will be directly represented by the information bits that will be tested by the cost function, considering only polarized binary encoding, i.e., for each candidate position of the vector Y is able to assume just only one value in the set {±1}.

. ... **<sup>R</sup>** [0] **<sup>R</sup>***<sup>T</sup>* [1]

Multidimensional Optimization-Based Heuristics Applied to Wireless Communication Systems

[<sup>Ω</sup> (B)]�

. .  

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89

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**R** =

by optimizing eq. (14), resulting in:

**3. Heuristic algorithms**

**3.1. Encoding problem**

number of subcarriers, among others increase.

 

. . . . .

. . . . . . . .

�**<sup>x</sup>** <sup>=</sup> arg �

defined as [3]:

Note that the minimum square difference exists in eq. (11) ensures the maximization of the maximum likelihood function. Expanding the quadratic difference in eq. (11), based on the output of the matched filter, vector **y**(*i*), find the maximum likelihood vector **ˆx** is equivalent to selecting the bit vector B, with same size, which maximizes the Log Likelihood Function (LLF) [3]:

$$\Omega\left(\mathcal{B}\right) = 2\sum\_{n\_{\mathcal{R}\mathcal{I}}=1}^{A\_{\mathcal{R}\mathcal{I}}} \Re\left\{\mathcal{B}^T \mathcal{C}^H \mathcal{A} \mathcal{V}\right\} - \mathcal{B}^T \mathcal{C} \mathcal{A} \mathbf{R} \mathcal{A} \mathcal{C}^H \mathcal{B} \tag{14}$$

where each matrix must be determined for each receiving antenna and (·) *<sup>H</sup>* refers to the Hermitian transpose operator.

The diagonal channel coefficients and amplitudes matrices4 for *nRx*-th receiving antenna of *DKv I* dimension are defined, respectively, by:

$$\mathcal{C} = \text{diag}\left[\mathbf{C}^{(0)} \; \mathbf{C}^{(1)} \; \mathbf{C}^{(2)} \dots \mathbf{C}^{(l-1)}\right] \tag{15}$$

$$\mathcal{A} = \text{diag}\left[\mathbf{A}^{(0)} \; \mathbf{A}^{(1)} \; \mathbf{A}^{(2)} \dots \; \mathbf{A}^{(I-1)}\right] \tag{16}$$

The output vector of the matched filter (MFB), composed by *I* vectors **y**(*i*) with dimension *DKv I* × 1 is given by:

$$\mathcal{V} = \begin{bmatrix} \mathbf{y}^{(0)} \ \mathbf{y}^{(1)} \ \mathbf{y}^{(2)} \cdots \ \mathbf{y}^{(I-1)} \end{bmatrix}^T \tag{17}$$

In general, the MFB output vector Y is deployed as initial guess in the LLF cost function, eq. (14) when a heuristic multiuser detection is performed. The general rule should be ensure the maximization of the cost function, considering the same Y for all receiving antennas.

<sup>4</sup> To simplify the notation, hereafter we have omitted the matrix index *nRx*.

Finally, the block Toeplitz tridiagonal correlation matrix **R**, dimension *D*K*<sup>v</sup> I* × *D*K*<sup>v</sup> I*, is defined as [3]:

$$\mathbf{R} = \begin{bmatrix} \mathbf{R} \begin{bmatrix} \mathbf{0} \end{bmatrix} \cdot \mathbf{R}^T \begin{bmatrix} 1 \end{bmatrix} & \mathbf{0} & \cdots & \mathbf{0} & \mathbf{0} \\ \mathbf{R} \begin{bmatrix} 1 \end{bmatrix} \cdot \mathbf{R} \begin{bmatrix} 0 \end{bmatrix} \cdot \mathbf{R}^T \begin{bmatrix} 1 \end{bmatrix} \cdot \cdots & \mathbf{0} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{R} \begin{bmatrix} 1 \end{bmatrix} \cdot \mathbf{R} \begin{bmatrix} 0 \end{bmatrix} \cdot \cdots & \mathbf{0} & \mathbf{0} \\\\ \vdots & \vdots & \ddots & \ddots & \vdots \\\\ \vdots & \vdots & \vdots & \ddots & \mathbf{R} \begin{bmatrix} 0 \end{bmatrix} \cdot \mathbf{R}^T \begin{bmatrix} 1 \end{bmatrix} \end{bmatrix} \tag{18}$$

Therefore, the complete frame for the *I* estimated symbols from all *Kv* users can be obtained by optimizing eq. (14), resulting in:

$$\widehat{\mathbf{x}} = \arg \left\{ \max\_{\mathcal{B} \in \{\pm 1\}^{\mathcal{E}\_{\mathcal{P}}}} [\Omega \, (\mathcal{B})] \right\} \tag{19}$$

The OMuD consists in finding the best data symbols vector in a set with all the possibilities, i.e., it is a NP-complete combination problem [30], which the traditional algorithms are inefficient. Most of these result in exponential complexity growth when one or more of the following factors: number of users, frame, number of receiver antennas, number of paths, number of subcarriers, among others increase.

Therefore, the use of heuristic methods for this class of problems shows up attractive, since it is possible to obtain optimal solutions (or near-optimal) using reduced search spaces. Thus, the proposed strategy in this Chapter aiming for maximize the LLF by testing different candidates symbols vectors at each new iteration/generation of heuristic algorithms. Such attempts seek to maximize the system average performance, approaching or even equaling that obtained by OMuD, but with remarkable reduction in the computational complexity.

## **3. Heuristic algorithms**

8 Search Algorithms

the receiver is:

dimension, and by:

*G* ∑ *g*=1

*M* ∑ *m*=1 *m*(*g*) ∑ *j*=1

*ARx* ∑ *nRx*=1

**ˆx** = **ˆx**(0) *T* **ˆx**(1) *T* **ˆx**(2) *T*

Ω (B) = 2

*DKv I* dimension are defined, respectively, by:

*ARx* ∑ *nRx*=1

C = diag

A = diag

Y = 

<sup>4</sup> To simplify the notation, hereafter we have omitted the matrix index *nRx*.

ℜ 

where each matrix must be determined for each receiving antenna and (·)

*L* ∑ ℓ=1 *A*′ *<sup>k</sup>*,*g*,ℓ,*nRx* **ˆx**

**S***t* **bˆ** (*i*) = *K*(*g*) ∑ *k*=1

(LLF) [3]:

Hermitian transpose operator.

*DKv I* × 1 is given by:

where the total received signal, reconstructed from the estimated parameters and known at

In this context, the maximum likelihood vector that must be found by OMuD has *DKv* × 1

Note that the minimum square difference exists in eq. (11) ensures the maximization of the maximum likelihood function. Expanding the quadratic difference in eq. (11), based on the output of the matched filter, vector **y**(*i*), find the maximum likelihood vector **ˆx** is equivalent to selecting the bit vector B, with same size, which maximizes the Log Likelihood Function

B*T*C*H*AY

The diagonal channel coefficients and amplitudes matrices4 for *nRx*-th receiving antenna of

The output vector of the matched filter (MFB), composed by *I* vectors **y**(*i*) with dimension

**<sup>y</sup>**(0) **<sup>y</sup>**(1) **<sup>y</sup>**(2) ... **<sup>y</sup>**(*I*−1)

In general, the MFB output vector Y is deployed as initial guess in the LLF cost function, eq. (14) when a heuristic multiuser detection is performed. The general rule should be ensure the maximization of the cost function, considering the same Y for all receiving antennas.

**<sup>C</sup>**(0) **<sup>C</sup>**(1) **<sup>C</sup>**(2) ... **<sup>C</sup>**(*I*−1)

**<sup>A</sup>**(0) **<sup>A</sup>**(1) **<sup>A</sup>**(2) ... **<sup>A</sup>**(*I*−1)

(*i*) *<sup>k</sup>*,*g***s** (*g*) *Ck* (*<sup>t</sup>* <sup>−</sup> *jT*)*<sup>s</sup>*

(*g*) *k <sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*(*g*)

*<sup>k</sup>*,ℓ,*g*,*nRx e j <sup>ω</sup>mt*+*φ*ˆ(*i*) *<sup>k</sup>*,ℓ,*g*,*nRx* 

− B*T*CA**R**AC*H*B (14)

·*s* (*g*) *Fk*,*mβ*ˆ(*i*)

... **ˆx**(*I*−1)

*<sup>T</sup> <sup>T</sup>*

*T*

*<sup>k</sup>*,ℓ,*nRx* <sup>−</sup> *iT*

 ·

(12)

(13)

*<sup>H</sup>* refers to the

(15)

(16)

(17)

This section presents a brief review of heuristic algorithms, specifically local and evolutionary search, describing variants and required parameters. Such variants include encoding (mapping) problem, initialization algorithms step (parameters choice), cost function evaluation, search space scanning step, and replacement candidates step. For the analysis, 1-opt and *k*-opt local search, simulated annealing, short-term and reactive Tabu search, genetic, as well evolutionary programming algorithms have been considered in this work.

## **3.1. Encoding problem**

The encoding for MuD problem is inherently binary, because the data vector is naturally binary. Therefore, following the Keep it Simple (KIS) principle as much as possible, it is not necessary to perform an encoding (mapping) of candidate solutions differently to binary form. Thus, these candidates vectors will be directly represented by the information bits that will be tested by the cost function, considering only polarized binary encoding, i.e., for each candidate position of the vector Y is able to assume just only one value in the set {±1}.

## **3.2. Search space definition**

After the encoding step, we must define the problem search space, in which case the MuD problem is characterized by all possible combinations bits that users can transmit. In this case, for *Kv* virtual users transmiting *I* bits through a multipath channel with *L* paths and *D* processing signal branches at the receiver, the total search universe, considering optimizing the output of matched filter and signals combination from multirate users, is a binary set of dimension:

$$\Theta\left(\mathcal{K}\_{\mathcal{V}}, I, D\right) = 2^{D\mathcal{K}\_{\mathcal{V}}I}, \qquad \text{with} \quad 1 \le D \le L. \tag{20}$$

**3.4. Local Search (LS) algorithms**

few problems have a guide or direction.

search space as large as brute force methods [36].

**3.5. Simulated Annealing (SA) algorithms**

to form a homogeneous structure with lowest energy [37, 39].

solutions, promoting diversification in the search.

combinatorial problems.

equation.

**3.6. Tabu search**

areas and fields.

The Local Search (LS) strategy is based on the better established existing principle for combinatorial optimization methods: trial and error. This is a natural and simple idea, but in fact, surprised by the success degree that this method has the most varied types of

Multidimensional Optimization-Based Heuristics Applied to Wireless Communication Systems

http://dx.doi.org/10.5772/54152

91

The only parameters to be selected corresponds to the search starting point and the neighborhood size. The choice of starting point is usually done by intuition, because very

For neighborhood definition, it should be pointed out that small neighborhood set leads in a low complexity algorithm, since the search space consists of few alternatives. On the other hand, the reduced size of the neighborhood set may not provide good solutions due to local minimum or maximum in this reduced region. Large neighborhood sets, on the other hand, provide good solutions but bring much greater complexity, since these sets may result in

The simulated annealing (SA) algorithm proposed by Kirkpatrick [37] was inspired by the annealing process of physical systems. This was based on the algorithm originally proposed by Metropolis [38] as a strategy for determining equilibrium states (or configurations) of a collection of atoms at a given temperature. The basic idea comes from the statistical thermodynamics, which is a physics branch responsible for theoretical predictions about the behavior of macroscopic systems based on the laws that govern their atoms. Using analogies, the SA algorithm was proposed based on the similarity between the annealing procedure implemented by the Metropolis algorithm and combinatorial optimization processes.

Thus, the concept of the SA algorithm is associated with the principle of thermodynamics, in which a solid heated to a very high temperature and then cooled gradually tends to solidify

This way, the SA algorithm must be started with one strategy and three parameters: initial temperature, *T*(0), cooling step, *ǫ*, size range (plateau) *LSA*, and acceptance probability

Tabu search algorithm was originally proposed in 1977 with the pioneering Glover's work [40] and later described in its current form in 1986 [41], being used in various knowledge

The short-term Tabu search (STTS) algorithm is based on the deterministic mode of memory operation. The memory is implemented by recording characteristics of displacement of previously visited solutions. This is described by the Tabu list, which is formed by the recent past search, being called the short-term memory effect. These displacement characteristics are prohibited by Tabu list for a number of iterations. This helps prevent returns to local

It is evident that for the proposed MuD problem, a total search universe should result smaller than 2*DKv <sup>I</sup>* , since each transmitted bit must be detected in a way that results in a same estimated bit value for all *D* processing branches, namely:

$$
\mathfrak{b}\_{k,1,\emptyset}^{(i)} = \mathfrak{b}\_{k,2,\emptyset}^{(i)} = \dots = \mathfrak{b}\_{k,D,\emptyset}^{(i)} \in \{+1, -1\} \tag{21}
$$

This implies that the search universe covered by the heuristic MuD algorithm should be independent of the number of paths, resulting in:

$$\Theta\left(\mathcal{K}\_{\mathcal{U}}\right) = \mathcal{Z}^{\mathcal{K}\_{\mathcal{U}}\mathcal{I}} \tag{22}$$

As a result, the universe of possible solutions is then formed by all vectors candidates that satisfy (21).

The other possibilities belong to the so-called forbidden universe, composing the non-tested candidates set into a heuristic methodology. This guarantees the final solution quality in MuD problem with multipath diversity, because it enable a correct estimate for all paths of the same transmitted bit could be made.

#### **3.3. Evolutionary Programming (EP) algorithms**

The evolutionary heuristic algorithms methods are non-deterministic search mechanisms based on natural selection and evolution mechanisms from Darwin's theory [31]5. This theory explains the life history by the physical processes action and genetic operators in populations or species. These processes are known for reproduction, disturbance, competition and selection.

Considering the computational implementation aspects, the parameters and strategies such as population size, mating pool size, selection strategy, crossover type and rate, mutation type and rate and replacement strategy should be chosen carefully by the user for each class of optimization problem, allowing numerous plausible combinations [33–35].

<sup>5</sup> Because [31] is a rare and difficult access reference, we consider newer editions of the Darwin's work, for instance [32].

## **3.4. Local Search (LS) algorithms**

10 Search Algorithms

dimension:

than 2*DKv <sup>I</sup>*

satisfy (21).

[32].

**3.2. Search space definition**

After the encoding step, we must define the problem search space, in which case the MuD problem is characterized by all possible combinations bits that users can transmit. In this case, for *Kv* virtual users transmiting *I* bits through a multipath channel with *L* paths and *D* processing signal branches at the receiver, the total search universe, considering optimizing the output of matched filter and signals combination from multirate users, is a binary set of

It is evident that for the proposed MuD problem, a total search universe should result smaller

*b* (*i*)

This implies that the search universe covered by the heuristic MuD algorithm should be

As a result, the universe of possible solutions is then formed by all vectors candidates that

The other possibilities belong to the so-called forbidden universe, composing the non-tested candidates set into a heuristic methodology. This guarantees the final solution quality in MuD problem with multipath diversity, because it enable a correct estimate for all paths of

The evolutionary heuristic algorithms methods are non-deterministic search mechanisms based on natural selection and evolution mechanisms from Darwin's theory [31]5. This theory explains the life history by the physical processes action and genetic operators in populations or species. These processes are known for reproduction, disturbance,

Considering the computational implementation aspects, the parameters and strategies such as population size, mating pool size, selection strategy, crossover type and rate, mutation type and rate and replacement strategy should be chosen carefully by the user for each class

<sup>5</sup> Because [31] is a rare and difficult access reference, we consider newer editions of the Darwin's work, for instance

of optimization problem, allowing numerous plausible combinations [33–35].

*<sup>k</sup>*,2,*<sup>g</sup>* <sup>=</sup> ... <sup>=</sup> <sup>ˆ</sup>

, since each transmitted bit must be detected in a way that results in a same

, with 1 ≤ *D* ≤ *L*. (20)

*<sup>k</sup>*,*D*,*<sup>g</sup>* <sup>∈</sup> {+1, <sup>−</sup>1} (21)

Θ (*Kv*, *I*) = 2*Kv <sup>I</sup>* (22)

Θ (*Kv*, *I*, *D*) = 2*DKv <sup>I</sup>*

estimated bit value for all *D* processing branches, namely:

ˆ *b* (*i*) *<sup>k</sup>*,1,*<sup>g</sup>* <sup>=</sup> <sup>ˆ</sup> *b* (*i*)

independent of the number of paths, resulting in:

the same transmitted bit could be made.

competition and selection.

**3.3. Evolutionary Programming (EP) algorithms**

The Local Search (LS) strategy is based on the better established existing principle for combinatorial optimization methods: trial and error. This is a natural and simple idea, but in fact, surprised by the success degree that this method has the most varied types of combinatorial problems.

The only parameters to be selected corresponds to the search starting point and the neighborhood size. The choice of starting point is usually done by intuition, because very few problems have a guide or direction.

For neighborhood definition, it should be pointed out that small neighborhood set leads in a low complexity algorithm, since the search space consists of few alternatives. On the other hand, the reduced size of the neighborhood set may not provide good solutions due to local minimum or maximum in this reduced region. Large neighborhood sets, on the other hand, provide good solutions but bring much greater complexity, since these sets may result in search space as large as brute force methods [36].

## **3.5. Simulated Annealing (SA) algorithms**

The simulated annealing (SA) algorithm proposed by Kirkpatrick [37] was inspired by the annealing process of physical systems. This was based on the algorithm originally proposed by Metropolis [38] as a strategy for determining equilibrium states (or configurations) of a collection of atoms at a given temperature. The basic idea comes from the statistical thermodynamics, which is a physics branch responsible for theoretical predictions about the behavior of macroscopic systems based on the laws that govern their atoms. Using analogies, the SA algorithm was proposed based on the similarity between the annealing procedure implemented by the Metropolis algorithm and combinatorial optimization processes.

Thus, the concept of the SA algorithm is associated with the principle of thermodynamics, in which a solid heated to a very high temperature and then cooled gradually tends to solidify to form a homogeneous structure with lowest energy [37, 39].

This way, the SA algorithm must be started with one strategy and three parameters: initial temperature, *T*(0), cooling step, *ǫ*, size range (plateau) *LSA*, and acceptance probability equation.

## **3.6. Tabu search**

Tabu search algorithm was originally proposed in 1977 with the pioneering Glover's work [40] and later described in its current form in 1986 [41], being used in various knowledge areas and fields.

The short-term Tabu search (STTS) algorithm is based on the deterministic mode of memory operation. The memory is implemented by recording characteristics of displacement of previously visited solutions. This is described by the Tabu list, which is formed by the recent past search, being called the short-term memory effect. These displacement characteristics are prohibited by Tabu list for a number of iterations. This helps prevent returns to local solutions, promoting diversification in the search.

The reactive Tabu search (RTS) version combines the short-term memory effect with another memory effect to avoid the local maximum returns and ensure efficient search. This effect is known as long-term memory, which alternates between intensification and diversification phases, adapting the prohibition period during the search, provide that the prohibition period takes different values for each iteration [42].

found by the heuristic algorithm and may take values milder or stricter6. But this quality analysis should consider the acceptance limits of the solution, i.e., what are the thresholds of deviation from the desired value that can still be accepted as a solution for the optimization

Multidimensional Optimization-Based Heuristics Applied to Wireless Communication Systems

One of the metrics widely used to evaluate the stability and capacity of the search algorithms is the so-called algorithm stability index (ASI), which corresponds to an ability measure of the algorithm to produce consistent results, described by the ratio between the dispersion of allowed solutions and the dispersion of current solution. The other metric, namely algorithm capability index (ACI) is a measure of how far from the specified limits the solution propitiates by the algorithm is, in terms of quality of the solutions obtained. The ASI and

> *ASI* <sup>=</sup> (*USL* <sup>−</sup> *LSL*) 6*σX*¯

In the literature, this methodology is also known as "*Six Sigma*" methodology [47]. The ACI metric measures how close the algorithm's solution is from its purpose, as well as the consistency around their average performance. An algorithm may have a minimal variation, but if it is away from the objective value for one of the specification limits, it will result in a lower ACI value, whereas the ASI metric may still be high. On the other hand, an algorithm could result, on average, in solutions exactly equal to the purpose, but presents a large variation in performance. In this case, ACI is still small and ASI can still be large. Thus, the ACI metric just will be large if and only if it reaches the vicinity of the desirable

Note that for practical reasons, it has been considered a good criterion to ensure *ASI* > 2

In the next section, the stability and capacity indexes have been considered in the input parameters optimization step of the heuristic algorithms, since the adopted benchmark functions have well-defined values for the global minimum, as described in the following.

Considering the quality metrics discussed previously, it was decided to hold a *Decathlon* marathon type [43, 49] in order to evaluate the efficiency, stability an convergence capacity of the proposed heuristic MuD algorithms. For this purpose, ten benchmark functions, which correspond to the ten races of marathon, have been deployed aiming to define performance thresholds, as well as parameters determination that provide good solutions for all heuristic

<sup>6</sup> A usual way is to take *USL* = *X*¯ + 3*σ<sup>X</sup>* and *LSL* = *X*¯ − 3*σX*, where *σ<sup>X</sup>* is the standard deviation of the process *X*.

and *ACI* > 1.33 for most of engineering applications with practical interest [46, 48].

or *ACI* <sup>=</sup> (*X*¯ <sup>−</sup> *LSL*)

3*σX*¯

*ACI* <sup>=</sup> (*USL* <sup>−</sup> *<sup>X</sup>*¯)

objective value consistently and with minimal variation.

**4.2. Input parameter optimization**

algorithms considered.

3*σX*¯

(23)

93

http://dx.doi.org/10.5772/54152

(24)

problem.

ACI can be calculated as:

**4.1. Algorithm stability and capacity indexes**

## **3.7. Hyperheuristic strategies**

Over the past 50 years, the well-known meta-heuristic algorithms have been used as optimization tool for a wide range of optimization problems. The ability of the meta-heuristic algorithms to avoid local optimum-solutions offer us the ability to adapt this class of optimization strategy to solve various problems with the robustness and easiness of implementation, contributing to various optimization fields, mainly in those problems where deterministic or traditional optimization methods become inefficient or highly complex.

However, it is not easy or even possible to predict which of the many existing heuristic algorithms is the best choice for a specific optimization problem and be able to produce the same result given the same input parameters. The difficulty of choosing is associated to the performance unpredictability, which constitutes the major factor limiting their use by the scientific community and industry.

Furthermore, for a large optimization problems variety, the input parameter values should be controlled as the search evolves. In this context, recently, the idea of working with a higher level of automation in a heuristic design have been resulted in the development of the so-called hyper-heuristic (HH) strategy [43, 44].

Thus, the HH algorithms consist in applying a high-level methodology to control the selection or generation of generic search strategies using a specific number of different low-level heuristics.

It is worth noting that meta-heuristics are quality techniques to solve complex optimization problems, but efficient implementations of these methods usually require many specific knowledge about the problem being treated. Thus, the HH methodologies have been proposed aiming to build robust optimization algorithms, allowing the use of meta-heuristics methods with minor adaptations.

For HHs based on heuristics selection, one should choose the suitable number of iterations for the HH, the selection strategy, as well as the acceptance strategy.

## **4. Performance metrics for heuristic algorithms evaluation**

Aiming to quantify the performance of the heuristic algorithms in terms of stability and convergence guarantee, it is necessary to know the specified limits for the tolerances calculation, the so-called upper specification limit (USL) and lower specification limit (LSL) [45, 46].

The USL and LSL are simply an upper and lower bounds to measure the algorithm's performance. Thus, as in the case of control charts, it is desired that the algorithm behaves within these two limits. These parameters are often set by the need for quality of solutions found by the heuristic algorithm and may take values milder or stricter6. But this quality analysis should consider the acceptance limits of the solution, i.e., what are the thresholds of deviation from the desired value that can still be accepted as a solution for the optimization problem.

## **4.1. Algorithm stability and capacity indexes**

12 Search Algorithms

The reactive Tabu search (RTS) version combines the short-term memory effect with another memory effect to avoid the local maximum returns and ensure efficient search. This effect is known as long-term memory, which alternates between intensification and diversification phases, adapting the prohibition period during the search, provide that the prohibition

Over the past 50 years, the well-known meta-heuristic algorithms have been used as optimization tool for a wide range of optimization problems. The ability of the meta-heuristic algorithms to avoid local optimum-solutions offer us the ability to adapt this class of optimization strategy to solve various problems with the robustness and easiness of implementation, contributing to various optimization fields, mainly in those problems where deterministic or traditional optimization methods become inefficient or highly complex.

However, it is not easy or even possible to predict which of the many existing heuristic algorithms is the best choice for a specific optimization problem and be able to produce the same result given the same input parameters. The difficulty of choosing is associated to the performance unpredictability, which constitutes the major factor limiting their use by the

Furthermore, for a large optimization problems variety, the input parameter values should be controlled as the search evolves. In this context, recently, the idea of working with a higher level of automation in a heuristic design have been resulted in the development of the

Thus, the HH algorithms consist in applying a high-level methodology to control the selection or generation of generic search strategies using a specific number of different

It is worth noting that meta-heuristics are quality techniques to solve complex optimization problems, but efficient implementations of these methods usually require many specific knowledge about the problem being treated. Thus, the HH methodologies have been proposed aiming to build robust optimization algorithms, allowing the use of meta-heuristics

For HHs based on heuristics selection, one should choose the suitable number of iterations

Aiming to quantify the performance of the heuristic algorithms in terms of stability and convergence guarantee, it is necessary to know the specified limits for the tolerances calculation, the so-called upper specification limit (USL) and lower specification limit (LSL)

The USL and LSL are simply an upper and lower bounds to measure the algorithm's performance. Thus, as in the case of control charts, it is desired that the algorithm behaves within these two limits. These parameters are often set by the need for quality of solutions

for the HH, the selection strategy, as well as the acceptance strategy.

**4. Performance metrics for heuristic algorithms evaluation**

period takes different values for each iteration [42].

**3.7. Hyperheuristic strategies**

scientific community and industry.

methods with minor adaptations.

low-level heuristics.

[45, 46].

so-called hyper-heuristic (HH) strategy [43, 44].

One of the metrics widely used to evaluate the stability and capacity of the search algorithms is the so-called algorithm stability index (ASI), which corresponds to an ability measure of the algorithm to produce consistent results, described by the ratio between the dispersion of allowed solutions and the dispersion of current solution. The other metric, namely algorithm capability index (ACI) is a measure of how far from the specified limits the solution propitiates by the algorithm is, in terms of quality of the solutions obtained. The ASI and ACI can be calculated as:

$$ASI = \frac{\left(USL - LSL\right)}{6\sigma\_{\mathcal{R}}}\tag{23}$$

$$ACI = \frac{(LSL - \bar{X})}{3\sigma\_{\bar{X}}} \quad \text{or} \quad ACI = \frac{(\bar{X} - LSL)}{3\sigma\_{\bar{X}}} \tag{24}$$

In the literature, this methodology is also known as "*Six Sigma*" methodology [47]. The ACI metric measures how close the algorithm's solution is from its purpose, as well as the consistency around their average performance. An algorithm may have a minimal variation, but if it is away from the objective value for one of the specification limits, it will result in a lower ACI value, whereas the ASI metric may still be high. On the other hand, an algorithm could result, on average, in solutions exactly equal to the purpose, but presents a large variation in performance. In this case, ACI is still small and ASI can still be large. Thus, the ACI metric just will be large if and only if it reaches the vicinity of the desirable objective value consistently and with minimal variation.

Note that for practical reasons, it has been considered a good criterion to ensure *ASI* > 2 and *ACI* > 1.33 for most of engineering applications with practical interest [46, 48].

In the next section, the stability and capacity indexes have been considered in the input parameters optimization step of the heuristic algorithms, since the adopted benchmark functions have well-defined values for the global minimum, as described in the following.

## **4.2. Input parameter optimization**

Considering the quality metrics discussed previously, it was decided to hold a *Decathlon* marathon type [43, 49] in order to evaluate the efficiency, stability an convergence capacity of the proposed heuristic MuD algorithms. For this purpose, ten benchmark functions, which correspond to the ten races of marathon, have been deployed aiming to define performance thresholds, as well as parameters determination that provide good solutions for all heuristic algorithms considered.

<sup>6</sup> A usual way is to take *USL* = *X*¯ + 3*σ<sup>X</sup>* and *LSL* = *X*¯ − 3*σX*, where *σ<sup>X</sup>* is the standard deviation of the process *X*.

Hence, in order to optimize the input parameters of each heuristic algorithm, ten benchmarks (test) functions described in Table 1 have been deployed, considering functions commonly used in the literature [34, 44, 50–52], but with different characteristics in terms of local optima and dimensionality. In this study the first three functions of De Jong's work [50] have been considered, and in order to guarantee diversity in the characteristics, a set of seven additional test functions have been chosen.

GA

EP-C

SA

HH

*k*-opt

algorithms.

iteration/generation necessary for convergence.

*k* ∑ *i*=1  *Qindiv i*

Population Size: *p* = 10 ·

Mating Pool Size: *T* = 0.7*p* Selection Strategy: *p*-sort

Population Size: *p* = 10 ·

Cloning Rate: *Ic* = 20% Selection Strategy: *p*-sort

Initial temperature: *T*(0) = ln(*It*) Step Size (Plateau): *Lsa* = 2 Cooling Step: *ε* =

Acceptance Probability: *<sup>x</sup>* (*i*) <sup>=</sup> exp <sup>|</sup>∆*e*<sup>|</sup>

Number of HH iteration: *It*(*HH*) = 10 Selection Strategy: Simply random

**Table 2.** A summary for the optimized input parameters and strategies adopted in all considered heuristic algorithms

GA *pGt* (O*FC* + 11, 7*Qindiv* + 3 log (*Qindiv*)) *pGt*O*FC* EP-C *pGt* (O*FC* + 6, 1*Qindiv* + 3 log (*Qindiv*)) *pGt*O*FC* 1-opt *QindivGt* (O*FC* + 2*Qindiv* + 2) *QindivGt*O*FC*

*Gt* (O*FC* + 2*Qindiv* + 2)

SA *QindivGt* (O*FC* + 3*Qindiv* + 5) *QindivGt*O*FC* STTS *QindivGt* (O*FC* + 3, 5*Qindiv* + log (*Qindiv*) + 6) *QindivGt*O*FC* RTS *QindivGt* (O*FC* + 3, 5*Qindiv* + log (*Qindiv*) + 7) *QindivGt*O*FC* HH (0, 6*Qindiv* + 0, 4*p*) *Gt*O*FC*+ (0, 6*Qindiv* + 0, 4*p*) *Gt*O*FC*

**Table 3.** Average complexity in terms of number of operations (Flops) and predominant term, for all considered heuristic

+0, 2*QindivGt* (10*Qindiv* + 18*p* + 28)

of operations relevant to the cost function calculation, and *Gt* is the number of

STTS Prohibition Period: *P* = *Qindiv*/2 RTS Initial Prohibition Period: *<sup>P</sup>*(0) = *Qindiv*/2 Reduction/Increase Rate: *x* = 50%

Acceptance Strategy: Naive

Crossover Type/Rate: Uniform / *pc* = 50% Mutation Type/Rate: Gaussian / *pm* = 10% Replacement Strategy: *µ* + *λ* (with *µ* = *p*)

Mutation Type/Rate: Gaussian / *pm* = 15% Replacement Strategy: *µ* + *λ* (with *µ* = *p*) *k*-opt Neighborhood Search Choose neighborhood size (*k*)

Multidimensional Optimization-Based Heuristics Applied to Wireless Communication Systems

 2 ln(*i*)

*T*(*i*) − 1

**Flops Foremost Term**

*k* ∑ *i*=1  *Qindiv i*

 *Gt*O*FC*

0.345 *<sup>π</sup>* (*<sup>l</sup>* <sup>−</sup> <sup>1</sup>) <sup>+</sup> <sup>2</sup>

0.345 *<sup>π</sup>* (*<sup>l</sup>* <sup>−</sup> <sup>1</sup>) <sup>+</sup> <sup>2</sup>

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**Table 1.** Benchmark functions deployed for the heuristic input parameters optimization.

In order to eliminate eventual bias in the analysis, a large number of simulations have been considered. Hence, in all numerical results presented in this work an average over at least 1000 realization for each numerical parameter determination of each algorithm and for each function have been carried out, aiming to determine means and respective standard deviations, as well as for the calculation of the ASI and ACI quality measures. Thus, the numerical results show confidence intervals that provide consistent analyzes.

As a result of these analyzes, Table 2 presents a summary for the input parameters optimization and adopted strategies (in order to guarantee diversity on the search space) for each (hyper)-heuristic algorithm analyzed in this work. *Qindiv* indicates the length of each individual-candidate solution which of course is a function of the problem dimensionality.

## **4.3. Computational complexity**

Table 3 presents the generic complexity of the heuristic algorithms for subsequent determination of the heuristic multiuser detectors (MuD) complexity operating under different telecommunications systems scenarios in addition to the presentation of quantitative computational complexity of algorithms for application in Decathlon proof considered in this work (F1 to F10 functions). Notation: O*FC* represents the number


14 Search Algorithms

test functions have been chosen.

Hence, in order to optimize the input parameters of each heuristic algorithm, ten benchmarks (test) functions described in Table 1 have been deployed, considering functions commonly used in the literature [34, 44, 50–52], but with different characteristics in terms of local optima and dimensionality. In this study the first three functions of De Jong's work [50] have been considered, and in order to guarantee diversity in the characteristics, a set of seven additional

> ∑ *i*=1

*x*2 <sup>1</sup> <sup>+</sup> *<sup>x</sup>*<sup>2</sup> 2 0,25

<sup>1</sup> <sup>+</sup> <sup>2</sup>*x*<sup>2</sup>

4 − 2, 1*x*<sup>2</sup>

In order to eliminate eventual bias in the analysis, a large number of simulations have been considered. Hence, in all numerical results presented in this work an average over at least 1000 realization for each numerical parameter determination of each algorithm and for each function have been carried out, aiming to determine means and respective standard deviations, as well as for the calculation of the ASI and ACI quality measures. Thus, the

As a result of these analyzes, Table 2 presents a summary for the input parameters optimization and adopted strategies (in order to guarantee diversity on the search space) for each (hyper)-heuristic algorithm analyzed in this work. *Qindiv* indicates the length of each individual-candidate solution which of course is a function of the problem dimensionality.

Table 3 presents the generic complexity of the heuristic algorithms for subsequent determination of the heuristic multiuser detectors (MuD) complexity operating under different telecommunications systems scenarios in addition to the presentation of quantitative computational complexity of algorithms for application in Decathlon proof considered in this work (F1 to F10 functions). Notation: O*FC* represents the number

100

∑ *i*=1

∑ *i*=1 *i* · *x*<sup>2</sup> *i*

*xi*−<sup>1</sup> − *x*<sup>2</sup> *i*

> ∑ *i*=1 ⌊*xi*⌋

> > sin *ix*<sup>2</sup> *i π*

<sup>2</sup> <sup>−</sup> 0, 3 cos (3*πx*1) <sup>−</sup> 0, 4 cos (4*πx*2) <sup>+</sup> 0, 7

<sup>1</sup> <sup>+</sup> *<sup>x</sup>*1*x*<sup>2</sup> <sup>+</sup>

*j* cos [(*j* + 1) *xi* + *j*]

*<sup>i</sup>* <sup>−</sup> *<sup>A</sup>* cos (2*πxi*)


sin (*xi*)

sin<sup>2</sup> 50 *x*2 <sup>1</sup> <sup>+</sup> *<sup>x</sup>*<sup>2</sup> 2 0,1 + 1 

∑ *i*=1 *x*2

∑ *i*=1 −*xi* sin

<sup>1</sup> <sup>+</sup> *<sup>x</sup>*<sup>4</sup> 1 3 *x*2

 *<sup>m</sup>* ∑ *j*=1

∏ *i*=1 <sup>2</sup> <sup>+</sup> (<sup>1</sup> <sup>−</sup> *xi*)

<sup>2</sup>*<sup>m</sup>*

−4 + 4*x*<sup>2</sup> 2 *x*2 2

2

**Name Definition** De Jong [50] *<sup>F</sup>*<sup>1</sup> (**x**) <sup>=</sup> *<sup>n</sup>*

De Jong [50] *<sup>F</sup>*<sup>3</sup> (**x**) <sup>=</sup> *<sup>n</sup>*

De Jong [50] *<sup>F</sup>*<sup>2</sup> (**x**) <sup>=</sup> *<sup>n</sup>*−<sup>1</sup>

Michalewicz [52] *<sup>F</sup>*<sup>4</sup> (**x**) <sup>=</sup> <sup>−</sup> *<sup>n</sup>*

Rastrigin [55] *<sup>F</sup>*<sup>7</sup> (**x**) <sup>=</sup> *An* <sup>+</sup> *<sup>n</sup>*

Schwefel [56] *<sup>F</sup>*<sup>8</sup> (**x**) <sup>=</sup> *An* <sup>+</sup> *<sup>n</sup>*

Shubert [58] *<sup>F</sup>*<sup>10</sup> (**x**) <sup>=</sup> *<sup>n</sup>*

**Table 1.** Benchmark functions deployed for the heuristic input parameters optimization.

numerical results show confidence intervals that provide consistent analyzes.

Schaffer [53] *F*5 (**x**) =

Ackley [54] *F*6 (**x**) = *x*<sup>2</sup>

6-Hump Camelback [57] *F*9 (**x**) =

**4.3. Computational complexity**


of operations relevant to the cost function calculation, and *Gt* is the number of iteration/generation necessary for convergence.


**Table 3.** Average complexity in terms of number of operations (Flops) and predominant term, for all considered heuristic algorithms.

## **5. Numerical results for DS/CDMA systems with multidimensional diversity**

This section discuss representative numerical results for multiuser detection obtained under various types of diversity scenarios. First, we present results for a SIMO MC-CDMA systems, i.e. systems in which frequency and space diversities have been deployed jointly, due to the use of multicarrier and multiple receiving antennas. In a second step, a scenario with code and spatial diversity using multiple receive and/or transmission antennas have been analyzed.

performance improvement (due to the spatial diversity) for a medium loading system and

Multidimensional Optimization-Based Heuristics Applied to Wireless Communication Systems

10−6

**Figure 2.** Convergence speed for the conventional and various heuristic algorithm detectors under SIMO MC-CDMA scenarios

Figure 3 shows the BER performance for SIMO MC-CDMA with different number of receiving antennas (*ARx* = 1, 2, . . . 5) and medium loading system (*K* = 20). Accordingly, a *BER* <sup>=</sup> <sup>10</sup>−<sup>5</sup> for a moderate number of *ARx* antennas and SNR has been achieved. Thus, there is an expressive performance gain with heuristic MuD strategies regarding the Conventional detector when the number of antennas is increased for signal-to-noise ratio in the range

Systems with multiple-input-multiple-output (MIMO) and space-time block code (STBC) represent a promising solution often incorporated in commercial standards such as Wimax. Furthermore, a better performance × complexity trade-off can be obtained through the use of low density parity check codes (LDPC). The choice of STBC topology should take into account performance criteria, such as coding gain, diversity gain, multiplexing gain, and obviously the decoder complexity. However, these topics are not the focus of the this work

The considered MIMO system is formed by *ATx* = 4 transmit antennas and *ARx* ≥ 1 receiving antennas, with 4 symbols transmitted simultaneously. Furthermore, the following parameters have been adopted (see Table 4): *ATxARx* flat fading statistically independent channels, *M*−QAM modulation, quasi-orthogonal STBC (QO-STBC) scheme with rate 1 [4], short LDPC(204,102) code, perfect channel state information knowledge at receiver, random

and, therefore, more information can be found in the references [4, 60].

sequences of length *N* = 32 and two scenarios with loading system L = *<sup>K</sup>*

10−5

10−4

Conventional GA EP−C SA, STTS e RTS

A = 1

HH

10−3

0 5 10 15 20 25

Iterations / Generations

Conventional GA EP−C SA, STTS e RTS

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A = 3

A = 4

*<sup>N</sup>* <sup>=</sup> <sup>1</sup>

<sup>32</sup> and 1.

HH

A = 3 A = 4

low signal-to-noise ratio (SNR).

10−3

[0; 18] dB.

with *Eb*/*N*<sup>0</sup> = 9dB and *K* = 20 users.

10−2

BER

10−1

0 5 10 15 20 25

A = 2

Iterations / Generations

The main parameters of the system and channel coefficients are presented in Table 4. In all simulations results, random spreading sequences and slow Rayleigh channel model have been considered. Furthermore, it was assumed that the channel parameters are perfectly known at the receiver side, as well each subcarrier of the MC-CDMA system is subjected to flat frequency fading. Besides, low (BPSK) and high order modulation (M-QAM) format, LDPC and Turbo coding, as well as different spreading codes length, ranging from *N* = 8 to *N* = 64 have been deployed in this section.


**Table 4.** Adopted channel and multicarrier multiple-antenna system parameters.

Figures 2 and 3 consider systems with space and frequency diversity, in SIMO MC-CDMA scenarios. Monte Carlo simulation results indicate that the GA, SA, RTS, STTS, EP-C and HH multiuser detectors result in the same near-optimal performance in terms of solution quality after convergence, but with different complexities. However, the local search algorithm (1-LS and 3-LS) presented ACI and ASI measures below desirable thresholds and are not suitable for applications in scenarios of multiuser detection with multidimensional diversity.

Figure 2 shows the convergence behavior as a function of the number of iterations for all heuristic MuD algorithms considered. Note the equality of BER performance achieved after total convergence for all heuristic techniques. Specifically for convergence evaluation, different initial solutions were considered, while all achieving performances significantly superior to the Conventional detector. However, the number of generations/iterations for convergence proved to be different, which will be analyzed in details on Section 5.1 and 5.2. Note that increasing the number of receiving antennas implies in a significant performance improvement (due to the spatial diversity) for a medium loading system and low signal-to-noise ratio (SNR).

16 Search Algorithms

**diversity**

analyzed.

*N* = 64 have been deployed in this section.

**Table 4.** Adopted channel and multicarrier multiple-antenna system parameters.

**5. Numerical results for DS/CDMA systems with multidimensional**

This section discuss representative numerical results for multiuser detection obtained under various types of diversity scenarios. First, we present results for a SIMO MC-CDMA systems, i.e. systems in which frequency and space diversities have been deployed jointly, due to the use of multicarrier and multiple receiving antennas. In a second step, a scenario with code and spatial diversity using multiple receive and/or transmission antennas have been

The main parameters of the system and channel coefficients are presented in Table 4. In all simulations results, random spreading sequences and slow Rayleigh channel model have been considered. Furthermore, it was assumed that the channel parameters are perfectly known at the receiver side, as well each subcarrier of the MC-CDMA system is subjected to flat frequency fading. Besides, low (BPSK) and high order modulation (M-QAM) format, LDPC and Turbo coding, as well as different spreading codes length, ranging from *N* = 8 to

**Parameters Fig.2 Fig.3 Fig.4 Fig.5 Fig. 6** # Users 20 20 1 and 32 1 and 32 64 # Antenas Tx 1 1 4 4 1 and 2 # Antenas Rx 1 to 4 1 to 5 1 4 1 and 2 Modulation BPSK BPSK M-QAM M-QAM BPSK Spread Sequence *N* = 8 *N* = 8 *N* = 32 *N* = 32 *N* = 64 Subcarriers *M* = 4 *M* = 4- - *M* = 64 SNR (*γ*) 9dB 0 to 18dB -2 to 32dB -10 to 24dB 0.5 to 5dB Max. Doppler Freq. 100Hz 100Hz 20Hz 20Hz 30Hz Channel - - short LDPC short LDPC Turbo Coding - - (204,102) [59] (204,102) [59] (*R* = 1/2) Channel - - Belief Belief Turbo Decoding - - Propag. Propag. (MAP) Space-Time - - Rate 1 Rate 1 Rate 1 Coding - - *RSTBC* = 1 [4] *RSTBC* = 1 [4] *RSTBC* = 1 [60]

Figures 2 and 3 consider systems with space and frequency diversity, in SIMO MC-CDMA scenarios. Monte Carlo simulation results indicate that the GA, SA, RTS, STTS, EP-C and HH multiuser detectors result in the same near-optimal performance in terms of solution quality after convergence, but with different complexities. However, the local search algorithm (1-LS and 3-LS) presented ACI and ASI measures below desirable thresholds and are not suitable

Figure 2 shows the convergence behavior as a function of the number of iterations for all heuristic MuD algorithms considered. Note the equality of BER performance achieved after total convergence for all heuristic techniques. Specifically for convergence evaluation, different initial solutions were considered, while all achieving performances significantly superior to the Conventional detector. However, the number of generations/iterations for convergence proved to be different, which will be analyzed in details on Section 5.1 and 5.2. Note that increasing the number of receiving antennas implies in a significant

for applications in scenarios of multiuser detection with multidimensional diversity.

**Figure 2.** Convergence speed for the conventional and various heuristic algorithm detectors under SIMO MC-CDMA scenarios with *Eb*/*N*<sup>0</sup> = 9dB and *K* = 20 users.

Figure 3 shows the BER performance for SIMO MC-CDMA with different number of receiving antennas (*ARx* = 1, 2, . . . 5) and medium loading system (*K* = 20). Accordingly, a *BER* <sup>=</sup> <sup>10</sup>−<sup>5</sup> for a moderate number of *ARx* antennas and SNR has been achieved. Thus, there is an expressive performance gain with heuristic MuD strategies regarding the Conventional detector when the number of antennas is increased for signal-to-noise ratio in the range [0; 18] dB.

Systems with multiple-input-multiple-output (MIMO) and space-time block code (STBC) represent a promising solution often incorporated in commercial standards such as Wimax. Furthermore, a better performance × complexity trade-off can be obtained through the use of low density parity check codes (LDPC). The choice of STBC topology should take into account performance criteria, such as coding gain, diversity gain, multiplexing gain, and obviously the decoder complexity. However, these topics are not the focus of the this work and, therefore, more information can be found in the references [4, 60].

The considered MIMO system is formed by *ATx* = 4 transmit antennas and *ARx* ≥ 1 receiving antennas, with 4 symbols transmitted simultaneously. Furthermore, the following parameters have been adopted (see Table 4): *ATxARx* flat fading statistically independent channels, *M*−QAM modulation, quasi-orthogonal STBC (QO-STBC) scheme with rate 1 [4], short LDPC(204,102) code, perfect channel state information knowledge at receiver, random sequences of length *N* = 32 and two scenarios with loading system L = *<sup>K</sup> <sup>N</sup>* <sup>=</sup> <sup>1</sup> <sup>32</sup> and 1.

**Figure 3.** BER performance for heuristic algorithms. SIMO MC-CDMA system with *K* = 20 users.

Figure 4 depicts the BER performance *versus* SNR at the receiver input for different modulation constellations with *ATx* = 4 and *ARx* = 1 antennas. As expected, the single-user performance achieves very low BER7 under smaller SNR than that necessary with high loading system (with *K* = 32 users). However, it is observed that with an increment of 2–3dB in SNR for systems with high loading it is possible to obtain very lower BER, especially for 4-QAM modulation. However, higher order modulations such as 256-QAM enable the transmission of more bits per symbol period, which result in higher throughput systems if more power/energy is available at transmitter. Furthermore, the performance loss by increasing the loading proves be small, enabling the deployment of heuristic MuD algorithms in coded CDMA systems.

0 5 10 15 20 25 30

γ¯ (dB)

System with Heuristics and ATx=4 x ARx=4

**Figure 4.** BER performance of the heuristic decoders for QO-STBC MIMO systems with short LDPC(204,102), *ATx* = 4, *ARx* = 1

−10 −5 0 5 10 15 20

**Figure 5.** BER performance of the heuristic decoders for QO-STBC MIMO systems with short coding LDPC(204,102), *ATx* = 4,

γ¯ (dB)

System with Heuristics and ATx=4 x ARx=1

Multidimensional Optimization-Based Heuristics Applied to Wireless Communication Systems

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10−4

10−5

10−4

10−3

BER

10−2

10−1

antennas and *K* = 1 or *K* = 32 users.

10−3

4x1−4QAM (K = 1) 4x1−16QAM (K = 1) 4x1−64QAM (K = 1) 4x1−256QAM (K = 1) 4x1−4QAM (K = 32) 4x1−16QAM (K = 32) 4x1−64QAM (K = 32) 4x1−256QAM (K = 32)

4x4−4QAM (K = 1) 4x4−16QAM (K = 1) 4x4−64QAM (K = 1) 4x4−256QAM (K = 1) 4x4−4QAM (K = 32) 4x4−16QAM (K = 32) 4x4−64QAM (K = 32) 4x4−256QAM (K = 32)

*ARx* = 4 antennas and *K* = 1 or *K* = 32 users.

BER

10−2

10−1

Figure 5 compares the heuristic MuD performance in terms of BER *versus* SNR for high order modulation constellations and considering *ATx* = 4 and *ARx* = 4 antennas. We observe the same behavior shown in Figure 4. But in this case the number of receiving antennas has been increased to *ARx* = 4, resulting in significant performance improvement with reduction of ≈ 10dB SNR requirement, in order to obtain similar BERs. Again, the performance loss under total loading system is marginal; as a result, all heuristic algorithms discussed herein can be considered suitable for multiuser detection in coded CDMA systems. It is noteworthy that all heuristics algorithms showed the same level of BER performance after 40 generations for GA and EP-C, and 45 iterations for the SA, STTS, RTS and HH algorithms.

Figure 6 shows the BER performance of a multicarrier DS/CDMA (MC-CDMA) system with various types of diversity, considering multiple-input-multiple-output (MIMO) STBC coding, encoding and decoding turbo. This topology<sup>8</sup> was adopted in order to represent a transmission-reception topology with a great diversity order, making possible to obtain

<sup>7</sup> Beyond a certain SNR value, the performance improves sharply.

<sup>8</sup> Several other topologies can be considered for multidimensional analysis. For details, please see [61].

18 Search Algorithms

10−7

in coded CDMA systems.

10−6

10−5

10−4

BER

10−3

10−2

10−1

0 2 4 6 8 10 12 14 16 18

Conventional Heuristics A = 1 A = 2 A = 3 A = 4 A = 5

γ¯ (dB)

Figure 4 depicts the BER performance *versus* SNR at the receiver input for different modulation constellations with *ATx* = 4 and *ARx* = 1 antennas. As expected, the single-user performance achieves very low BER7 under smaller SNR than that necessary with high loading system (with *K* = 32 users). However, it is observed that with an increment of 2–3dB in SNR for systems with high loading it is possible to obtain very lower BER, especially for 4-QAM modulation. However, higher order modulations such as 256-QAM enable the transmission of more bits per symbol period, which result in higher throughput systems if more power/energy is available at transmitter. Furthermore, the performance loss by increasing the loading proves be small, enabling the deployment of heuristic MuD algorithms

Figure 5 compares the heuristic MuD performance in terms of BER *versus* SNR for high order modulation constellations and considering *ATx* = 4 and *ARx* = 4 antennas. We observe the same behavior shown in Figure 4. But in this case the number of receiving antennas has been increased to *ARx* = 4, resulting in significant performance improvement with reduction of ≈ 10dB SNR requirement, in order to obtain similar BERs. Again, the performance loss under total loading system is marginal; as a result, all heuristic algorithms discussed herein can be considered suitable for multiuser detection in coded CDMA systems. It is noteworthy that all heuristics algorithms showed the same level of BER performance after 40 generations

Figure 6 shows the BER performance of a multicarrier DS/CDMA (MC-CDMA) system with various types of diversity, considering multiple-input-multiple-output (MIMO) STBC coding, encoding and decoding turbo. This topology<sup>8</sup> was adopted in order to represent a transmission-reception topology with a great diversity order, making possible to obtain

for GA and EP-C, and 45 iterations for the SA, STTS, RTS and HH algorithms.

<sup>8</sup> Several other topologies can be considered for multidimensional analysis. For details, please see [61].

<sup>7</sup> Beyond a certain SNR value, the performance improves sharply.

**Figure 3.** BER performance for heuristic algorithms. SIMO MC-CDMA system with *K* = 20 users.

**Figure 4.** BER performance of the heuristic decoders for QO-STBC MIMO systems with short LDPC(204,102), *ATx* = 4, *ARx* = 1 antennas and *K* = 1 or *K* = 32 users.

**Figure 5.** BER performance of the heuristic decoders for QO-STBC MIMO systems with short coding LDPC(204,102), *ATx* = 4, *ARx* = 4 antennas and *K* = 1 or *K* = 32 users.

excellent BER performance even for low SNR region. Of course, this topology is promising for adoption as a commercial standard.

The presented results considering multiuser detection with different level of diversity exploitation in (non)coded telecommunication systems have demonstrated the effective applicability of the proposed heuristics MuD techniques, due to significant improvement in BER with reduced complexity regarding the optimal multiuser detector (OMuD). Complexity

Multidimensional Optimization-Based Heuristics Applied to Wireless Communication Systems

As the complexity of the algorithms in terms of number of operations has been determined, one can determine the complexity of each telecommunication scenario considering the

Both terms of the cost function, defined by F<sup>1</sup> = C*H*AY and F<sup>2</sup> = CA**R**AC*<sup>H</sup>* in eq. (14) can be obtained before the loop optimization in each MuD heuristic algorithm. Thus, for each candidate-solution evaluation, B*T*F<sup>1</sup> and B*T*F2B are computed, which in terms of

bit generations of order *KID*, as well as 2*K I* cost function calculations for the simultaneous detection of a frame consisting of *I* bits of *K* users on a system with multiuser detection

needed. It is noteworthy that 42 different scenarios have been analyzed9. Besides, scores to

Thus, the computational complexity of each proposed heuristic MuD algorithm, in terms of number of operations, has been obtained under different operation scenarios. Table 5 presents such complexities. Strategy with the lowest complexity using the adopted scoring system, for each analyzed scenario has been indicated with bold numbers. It is noteworthy that the scores were normalized considering the higher value with score of 100. For sake of

**Scenario GA EP-C SA STTS RTS HH OMuD** 1 – Fig.2 (×106) 4,597 **4,328** 5,572 5,606 5,607 5,144 8, 22.10<sup>21</sup> 2 – Fig.3 (×108) 2,322 **2,303** 3,333 3,336 3,336 2,926 1, 52.10<sup>45</sup> 3 – Fig.4 (×109) 4,390 **4,382** 5,794 5,795 5,795 5,451 1, 03.10<sup>57</sup> 4 – Fig.5 (×1013) **2,569 2,569** 18,877 18,878 18,878 12,483 > 10300 5 – Fig.6 (×1013) 1,028 **1,027** 22,277 22,278 22,278 13,860 > 10300 Score 87 100 83 72 56 85 – Position 2nd 1st 4th 5th 6th 3rd –

Furthermore, the complexity in terms of computational time has been determined for each telecommunication scenario. As a result, the computational time for calculating one cost function according each specific scenario has been quantified. We have deployed a personal

<sup>2</sup> + 4*KID* operations. For OMuD detector, the number

2*K I*(*KID*)<sup>2</sup>

<sup>2</sup> + 4*KID* operations are

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101

. It takes 2*K I*

aspects of the proposed heuristic MuDs are discussed in the next section.

of operations grows exponentially with the number of users, i.e., O

Therefore, in order to calculate the cost function, O*FC* = (*KID*)

define the best strategy have been considered this metric in different scenarios.

comparison, the optimal MuD complexity is presented in the last column.

<sup>9</sup> But not shown herein due to the lack of space. Additional results can be checked in [61].

**Table 5.** Necessary number of operations in all optimized Scenarios.

*5.1.1. Computational Time*

**5.1. Overall systems complexity**

operations is equivalent to (*KID*)

operating on fading channels.

complexity for the cost function calculation.

Specifically, in the performance of Figure 6, a Turbo encoding and decoding of rate 1/2, which result in a spectral efficiency of 0.5 bps/Hz have been adopted. Again, there is a remarkable performance gain increasing when the transmit-receive antenna array becomes larger for both MMSE and heuristic multiuser detectors. Another important aspect to be mentioned is the extremely low BERs for *γ*¯ ≤ 3 dB achieved with heuristics MuD topologies even under 1 × 1 antenna array configuration. Increasing the antenna array to 2 × 2 and adopting Alamouti code [60] it was possible to obtain a performance of *BER* ≤ 10−<sup>4</sup> even for *γ*¯ ≤ 2 dB. Thus, the performance achieved by MuD heuristics topology approaches the single-user bound (SuB) demonstrating the huge potential applicability in commercial communication systems and standards, specially those ones with high-performance and reliability requirements.

**Figure 6.** BER performance against SNR for a system with turbo channel coding.

Moreover, different topologies of the chosen one can be analyzed by considering, for example, channel coding (Convolutional or LDPC codes) and spatial diversity with other settings. However, the purpose of this section is to validate the potential of application of heuristics in MuD scenarios with multidimensional diversity and not compare topologies and system settings.

It is noteworthy that all heuristic MuD algorithms showed the same BER performance level after 100 generations for GA and EP-C algorithms, and 120 iterations for the SA, STTS, RTS and HH algorithms.

The presented results considering multiuser detection with different level of diversity exploitation in (non)coded telecommunication systems have demonstrated the effective applicability of the proposed heuristics MuD techniques, due to significant improvement in BER with reduced complexity regarding the optimal multiuser detector (OMuD). Complexity aspects of the proposed heuristic MuDs are discussed in the next section.

## **5.1. Overall systems complexity**

20 Search Algorithms

for adoption as a commercial standard.

reliability requirements.

10−1

10−4

settings.

and HH algorithms.

10−3

10−2

BER

excellent BER performance even for low SNR region. Of course, this topology is promising

Specifically, in the performance of Figure 6, a Turbo encoding and decoding of rate 1/2, which result in a spectral efficiency of 0.5 bps/Hz have been adopted. Again, there is a remarkable performance gain increasing when the transmit-receive antenna array becomes larger for both MMSE and heuristic multiuser detectors. Another important aspect to be mentioned is the extremely low BERs for *γ*¯ ≤ 3 dB achieved with heuristics MuD topologies even under 1 × 1 antenna array configuration. Increasing the antenna array to 2 × 2 and adopting Alamouti code [60] it was possible to obtain a performance of *BER* ≤ 10−<sup>4</sup> even for *γ*¯ ≤ 2 dB. Thus, the performance achieved by MuD heuristics topology approaches the single-user bound (SuB) demonstrating the huge potential applicability in commercial communication systems and standards, specially those ones with high-performance and

> MMSE (1x1) MMSE (2x2) Heuristics (1x1) Heuristics (2x2) SuB (BPSK)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

γ¯ (dB)

Moreover, different topologies of the chosen one can be analyzed by considering, for example, channel coding (Convolutional or LDPC codes) and spatial diversity with other settings. However, the purpose of this section is to validate the potential of application of heuristics in MuD scenarios with multidimensional diversity and not compare topologies and system

It is noteworthy that all heuristic MuD algorithms showed the same BER performance level after 100 generations for GA and EP-C algorithms, and 120 iterations for the SA, STTS, RTS

**Figure 6.** BER performance against SNR for a system with turbo channel coding.

As the complexity of the algorithms in terms of number of operations has been determined, one can determine the complexity of each telecommunication scenario considering the complexity for the cost function calculation.

Both terms of the cost function, defined by F<sup>1</sup> = C*H*AY and F<sup>2</sup> = CA**R**AC*<sup>H</sup>* in eq. (14) can be obtained before the loop optimization in each MuD heuristic algorithm. Thus, for each candidate-solution evaluation, B*T*F<sup>1</sup> and B*T*F2B are computed, which in terms of operations is equivalent to (*KID*) <sup>2</sup> + 4*KID* operations. For OMuD detector, the number of operations grows exponentially with the number of users, i.e., O 2*K I*(*KID*)<sup>2</sup> . It takes 2*K I* bit generations of order *KID*, as well as 2*K I* cost function calculations for the simultaneous detection of a frame consisting of *I* bits of *K* users on a system with multiuser detection operating on fading channels.

Therefore, in order to calculate the cost function, O*FC* = (*KID*) <sup>2</sup> + 4*KID* operations are needed. It is noteworthy that 42 different scenarios have been analyzed9. Besides, scores to define the best strategy have been considered this metric in different scenarios.

Thus, the computational complexity of each proposed heuristic MuD algorithm, in terms of number of operations, has been obtained under different operation scenarios. Table 5 presents such complexities. Strategy with the lowest complexity using the adopted scoring system, for each analyzed scenario has been indicated with bold numbers. It is noteworthy that the scores were normalized considering the higher value with score of 100. For sake of comparison, the optimal MuD complexity is presented in the last column.



## *5.1.1. Computational Time*

Furthermore, the complexity in terms of computational time has been determined for each telecommunication scenario. As a result, the computational time for calculating one cost function according each specific scenario has been quantified. We have deployed a personal

<sup>9</sup> But not shown herein due to the lack of space. Additional results can be checked in [61].

computer with the following configuration: Motherboard ASUS P8H67-M EVO, Intel I7-2600 with 3.4 GHz clock and 8MB cache; Memory 8GB Corsair Dominator DDR3 1333MHz and a Video board Radeon HD6950 2GB DDR5.

iterations10. In order to evaluate the stability of the algorithms after convergence, some tests have been conducted for a large number of generations/iterations. For all algorithms, the average performance level and number of generations/iterations for total convergence hold

Multidimensional Optimization-Based Heuristics Applied to Wireless Communication Systems

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103

Therefore, for the multiuser detection problems presented in this work it is concluded that the proposed heuristic algorithms are able to find the optimum solution or very near-optimum solution with a reduced number of cost function tests, approaching the MuD performance since an adequate and sufficient number of generations/iterations has been available. In this context we are interest in analyze the variation of the solutions when decreasing the number of generations/iterations; as a result, the heuristic MuD algorithms were also classified

In addition, for the complexity analysis, we chose to average between scores (number of operations and computational time), thus avoiding an unfair analysis among optimization

Table 7 presents the classification for the six heuristic MuD with the adoption of equal weight features for the final score calculation. In conclusion, under equal weight metric for computational complexity and quality/stability of the solutions found, the following choice

**Score GA EP-C SA STTS RTS HH**

Number of Operation 87 100 83 72 56 85 Computational Time 88 100 73 66 56 80 Complexity (Average) 88 100 78 69 56 83

Quality and Stability 80 67 78 64 62 100

Final Score (Average) 84 83 78 66 59 91

Several heuristic techniques applied to multiuser detection problems under different channel and system configuration scenarios, as well as diversity dimensionality, such as time, frequency, space and coding have been analyzed in this work. The main purpose in

<sup>10</sup> This result was obtained when the initial inputs guess for all algorithms considered are the same, indicating that all algorithms were able to achieve convergence after a certain number of generations/iterations, except the LS

**Table 7.** Scores for all considered heuristic algorithms applicable to wireless communication Scenarios 1 – 5.

Final Position 2nd 3rd 4th 5th 6th 1st

following the criterion of the smaller deviation values of solutions.

under the same boundaries.

strategies or possible bias.

**6. Main conclusions**

criteria for MuD problem can be established:

**Criterion Algorithm Choice**

Best BER performance: HH Lowest Complexity: EP-C Performance-Complexity trade-off: HH (score: 91)

algorithm, which did not show enough stability for adoption.

**Figure 7.** Average time for cost function calculation and respective polynomial approximation.

Figure 7 depicts the time to calculate a cost function as the size of the individual increases, while Table 6 shows the average time required for optimization algorithms in each scenario considered, as well as the respective scores and classification.


**Table 6.** Average time (in seconds) spend by the optimized heuristic algorithms under Scenarios 1 – 5.

In conclusion, again the algorithm EP-C presents the lowest complexity in terms of computational time. Note that all heuristic algorithms result in a computational time, as well as number of operation very close each other. Thus, we can adopt any topologies without significant loss in terms of performance × complexity trade-off metric, validating the deployment of heuristic MuD approach with optimized input parameters as shown in Table 2.

## **5.2. Quality, stability and topology choice**

For all scenarios considered, convergence curves in terms of BER have been resulted in the same level achieved for all algorithms, but after a different number of generations or iterations10. In order to evaluate the stability of the algorithms after convergence, some tests have been conducted for a large number of generations/iterations. For all algorithms, the average performance level and number of generations/iterations for total convergence hold under the same boundaries.

Therefore, for the multiuser detection problems presented in this work it is concluded that the proposed heuristic algorithms are able to find the optimum solution or very near-optimum solution with a reduced number of cost function tests, approaching the MuD performance since an adequate and sufficient number of generations/iterations has been available. In this context we are interest in analyze the variation of the solutions when decreasing the number of generations/iterations; as a result, the heuristic MuD algorithms were also classified following the criterion of the smaller deviation values of solutions.

In addition, for the complexity analysis, we chose to average between scores (number of operations and computational time), thus avoiding an unfair analysis among optimization strategies or possible bias.

Table 7 presents the classification for the six heuristic MuD with the adoption of equal weight features for the final score calculation. In conclusion, under equal weight metric for computational complexity and quality/stability of the solutions found, the following choice criteria for MuD problem can be established:


**Table 7.** Scores for all considered heuristic algorithms applicable to wireless communication Scenarios 1 – 5.

## **6. Main conclusions**

22 Search Algorithms

Table 2.

Video board Radeon HD6950 2GB DDR5.

2.5 x 10−4

Cost Function Time Polynomial Regression

**Figure 7.** Average time for cost function calculation and respective polynomial approximation.

**Table 6.** Average time (in seconds) spend by the optimized heuristic algorithms under Scenarios 1 – 5.

considered, as well as the respective scores and classification.

**5.2. Quality, stability and topology choice**

0.5

1

1.5

Time (s)

2

computer with the following configuration: Motherboard ASUS P8H67-M EVO, Intel I7-2600 with 3.4 GHz clock and 8MB cache; Memory 8GB Corsair Dominator DDR3 1333MHz and a

Processing Time to Multiuser Detection Cost Function

<sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>500</sup> <sup>600</sup> <sup>700</sup> <sup>800</sup> <sup>900</sup> <sup>1000</sup> <sup>0</sup>

Individual Size

Figure 7 depicts the time to calculate a cost function as the size of the individual increases, while Table 6 shows the average time required for optimization algorithms in each scenario

> **Scenario GA EP-C SA STTS RTS HH** 1 – Fig.2 0,039 **0,0378** 0,0474 0,0477 0,0477 0,0436 2 – Fig.3 0,297 **0,2892** 0,7509 0,7532 0,7538 0,5948 3 – Fig.4 0,4638 **0,4515** 1,1704 1,1741 1,1749 0,9414 4 – Fig.5 128,126 **128,001** 914,891 914,876 914,906 609,049 5 – Fig.6 5,2576 **5,1283** 24,0933 24,1408 24,1608 18,2711 Score 88 100 73 66 56 80 Position 2nd 1st 4th 5th 6th 3rd

In conclusion, again the algorithm EP-C presents the lowest complexity in terms of computational time. Note that all heuristic algorithms result in a computational time, as well as number of operation very close each other. Thus, we can adopt any topologies without significant loss in terms of performance × complexity trade-off metric, validating the deployment of heuristic MuD approach with optimized input parameters as shown in

For all scenarios considered, convergence curves in terms of BER have been resulted in the same level achieved for all algorithms, but after a different number of generations or Several heuristic techniques applied to multiuser detection problems under different channel and system configuration scenarios, as well as diversity dimensionality, such as time, frequency, space and coding have been analyzed in this work. The main purpose in

<sup>10</sup> This result was obtained when the initial inputs guess for all algorithms considered are the same, indicating that all algorithms were able to achieve convergence after a certain number of generations/iterations, except the LS algorithm, which did not show enough stability for adoption.

combining different types of diversity with heuristic detection is to provide system capacity increasing and/or reliability improvement.

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Near single-user bound performance has been achieved by all MuD heuristic algorithms analyzed in this work, considering different system and channel configurations, while offer a dramatic complexity reduction regarding the OMuD with marginal performance loss, even in aggressive fading channels and high loading systems conditions.

Among the analyzed detectors, the best MuD heuristic algorithm choice must take into account that one which offer either smallest computational complexity or the best BER performance, i.e. EP-C or HH multiuser detectors, respectively. Hence, the criteria for topology ranking established in this work allow us to quantify the parameter optimization level, reflecting on the quality and stability of the solutions obtained.

The heuristic input parameter optimization, as well as the proposed methodology for the heuristic MuD topology choice represent the main contribution of this work. Under optimized input parameters condition of all heuristic MuD algorithms, the quality and stability analyses have been carried out deploying ten benchmark functions. The numerical results for the MuD problem confirmed the near-optimal performance achieved by the heuristic algorithms for a wide channel and system configurations, corroborating the methodology adopted for the ranking topology.

## **Acknowledgements**

This work was supported in part by the National Council for Scientific and Technological Development (CNPq) of Brazil under Grants 202340/2011-2, 303426/2009-8 and in part by Londrina State University - Paraná State Government (UEL).

## **Author details**

Fernando Ciriaco1,⋆, Taufik Abrão<sup>1</sup> and Paul Jean E. Jeszensky<sup>2</sup>

<sup>⋆</sup> Address all correspondence to: fciriaco@uel.br; abrao@ieee.org; pjj@lcs.poli.usp.br

1 Electrical Engineering Department, State University of Londrina (DEEL-UEL), Londrina, Paraná, Brazil

2 Polytechnic School of the University of Sao Paulo (EPUSP), Sao Paulo, Brazil

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Near single-user bound performance has been achieved by all MuD heuristic algorithms analyzed in this work, considering different system and channel configurations, while offer a dramatic complexity reduction regarding the OMuD with marginal performance loss, even

Among the analyzed detectors, the best MuD heuristic algorithm choice must take into account that one which offer either smallest computational complexity or the best BER performance, i.e. EP-C or HH multiuser detectors, respectively. Hence, the criteria for topology ranking established in this work allow us to quantify the parameter optimization

The heuristic input parameter optimization, as well as the proposed methodology for the heuristic MuD topology choice represent the main contribution of this work. Under optimized input parameters condition of all heuristic MuD algorithms, the quality and stability analyses have been carried out deploying ten benchmark functions. The numerical results for the MuD problem confirmed the near-optimal performance achieved by the heuristic algorithms for a wide channel and system configurations, corroborating the

This work was supported in part by the National Council for Scientific and Technological Development (CNPq) of Brazil under Grants 202340/2011-2, 303426/2009-8 and in part by

<sup>⋆</sup> Address all correspondence to: fciriaco@uel.br; abrao@ieee.org; pjj@lcs.poli.usp.br

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**Chapter 5**

**Provisional chapter**

**Ant Colony Optimization for Resource Allocation and**

**and Anomaly Detection in Communication Networks**

Due to the paramount importance of (wireless) communication systems and computer networks, in the last decade both resource allocation (RA) and anomaly detection (AD) problems addressed herein have been intensively studied and numerous solutions have been proposed in the specialized literature [1]. The RA specially in wireless multiple access networks and the AD in computer networks are problems of complex nature and an engineering compromise solution has much appeal in terms of practical and effective

Resource allocation problems in wireless communication networks include power consumption minimization, information rate and network capacity maximization, battery lifetime maximization, energy-efficient and bandwidth-efficient optimal design among others. In computer networks, anomaly detection system (ADS) consists of a set of techniques aiming to detect anomalies in network operation, helping the administrator to decide which action need to be performed in each situation. Anomaly detection is not an easy task and brings together a range of techniques in several areas, such as machine learning, signal processing techniques based on specification techniques, and data mining among others. Generally, for most scenarios of practical interest, these optimization formulations result in non-convex problems, which is hard to solve or even impossible using conventional convex

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> ©2012 Marques et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Sampaio et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 Sampaio et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Ant Colony Optimization for Resource Allocation**

**Anomaly Detection in Communication Networks**

Mateus de Paula Marques, Mário H. A. C. Adaniya,

Taufik Abrão and Paul Jean E. Jeszensky

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Lucas Hiera Dias Sampaio and Paul Jean E. Jeszensky

Lucas Hiera Dias Sampaio,

Mário H. A. C. Adaniya, Taufik Abrão,

Mateus de Paula Marques,

http://dx.doi.org/10.5772/53338

**1. Introduction**

deployment.


**Provisional chapter**

## **Ant Colony Optimization for Resource Allocation and Anomaly Detection in Communication Networks Ant Colony Optimization for Resource Allocation and Anomaly Detection in Communication Networks**

Lucas Hiera Dias Sampaio, Mateus de Paula Marques, Mário H. A. C. Adaniya, Taufik Abrão and Paul Jean E. Jeszensky Mateus de Paula Marques, Mário H. A. C. Adaniya, Taufik Abrão, Lucas Hiera Dias Sampaio and Paul Jean E. Jeszensky

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/53338

## **1. Introduction**

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*and Evolutionary Computation Conference (GECCO)*, pages 2205–2208, July 2009.

Due to the paramount importance of (wireless) communication systems and computer networks, in the last decade both resource allocation (RA) and anomaly detection (AD) problems addressed herein have been intensively studied and numerous solutions have been proposed in the specialized literature [1]. The RA specially in wireless multiple access networks and the AD in computer networks are problems of complex nature and an engineering compromise solution has much appeal in terms of practical and effective deployment.

Resource allocation problems in wireless communication networks include power consumption minimization, information rate and network capacity maximization, battery lifetime maximization, energy-efficient and bandwidth-efficient optimal design among others. In computer networks, anomaly detection system (ADS) consists of a set of techniques aiming to detect anomalies in network operation, helping the administrator to decide which action need to be performed in each situation. Anomaly detection is not an easy task and brings together a range of techniques in several areas, such as machine learning, signal processing techniques based on specification techniques, and data mining among others. Generally, for most scenarios of practical interest, these optimization formulations result in non-convex problems, which is hard to solve or even impossible using conventional convex optimization techniques, even after imposing relaxations to deal with RA problems.

In this sense, heuristics have been widely used to solve problems that deterministic optimization methods result in high computational complexity and therefore have no application in real systems. In this context, the ant colony optimization (ACO) algorithm [2] developed in recent years has attracted a lot of interest from so many professionals due

Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Sampaio et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Sampaio et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Marques et al., licensee InTech. This is an open access chapter distributed under the terms of the

to its robustness and great performance in deal with discrete (combinatorial) and continuous optimization problems.

**2. Resource Allocation in Wireless Multiple Access Networks**

resources in a simple<sup>1</sup> and optimized manner is pretty important.

recently done we enumerate some notorious in the next section.

**2.1. Related Work**

networks.

as a weighted SNIR's productory.

<sup>1</sup> Here, simple is used as a synonym for low computational complexity.

The optimized resource allocation in wireless multiple access networks, specially the power rate allocation, is a problem of great interest for telecommunications enterprises and users. It is well known that spectrum and power are valuable resources due to their scarcity, the first one is a natural non renewable resource and the second one is limited by the battery and device size. Therefore, proposing new techniques and algorithms that can allocate this

Ant Colony Optimization for Resource Allocation and Anomaly Detection in Communication Networks

http://dx.doi.org/10.5772/53338

111

In the last few decades many researchers have been working on this subject aiming to find a simple yet sturdy algorithm for resource allocation in wireless systems. Among many works

Among numerous solutions proposed to resource allocation in wireless multiple access networks we enumerate herein some of great importance works: Foschini and Miljanic [4] distributed power control algorithm (DPCA) stands as the main one. When it comes to metaheuristics, in [5] and [6] a genetic algorithm approach was used to propose the genetic algorithm for mobiles equilibrium, providing the joint power-rate control in CDMA multiple access networks. In [7], the particle swarm optimization (PSO) metaheuristic was used in order to establish a low-complexity power control algorithm. Finally, in [8] a power allocation approach was proposed to solve the parallel interference cancelation in multi-user detectors. Beyond the metaheuristic approaches, the work developed in [9] exploits an algorithm based on the dynamic cost assignment for downlink power allocation in CDMA networks. Besides, [10] addressed the uplink fairness maximization in a CDMA system with multiple processing gains. In [11], the Verhulst population model, firstly developed to describe the biological species growth with restrictions of space and food, was adapted to the distributed power control problem in a DS/CDMA network. It is noteworthy that this work was the first one to propose a Verhulst model adaptation to resource allocation problems in multiple access

Furthermore, in [12] an analytical approach was proposed for the weighted throughput maximization (WTM) problem, namely MAPEL. The algorithm performs the power control in the interference limited wireless networks, i.e., CDMA and MC/CDMA networks, through a monotonically increasing objective function that is not necessarily convex. This function was formulated as a multiplicative linear fractional programming (MLFP) problem, which is a special case of generalized linear fractional programming (GLFP). So, the GLFP problem presented in [12] was used in [13] in order to formulate a non-decreasing objective function

Finally, this section presents a heuristic approach through ant colony optimization in the continuous domains (ACO**R**) applicable to the power and rate allocation problems [13], and is organized as follows: subsection 2.2 describes aspects of the DS/CDMA networks and the power control problem on subsection 2.3 the power control problem and the cost function used with the ACO algorithm are presented; subsection 2.4 deals with the throughput maximization problem and how the ACO algorithm can be applied to solve this optimization

An important challenge for the future wireless communication systems has been how to acquire higher throughput with lower power consumption. Hence, in order to transfer the exponentially rising amount of available data to the user in an acceptable time, following the "Moore's Law", according to which both the processing power of CPUs and the capacity of mass storage devices doubles approximately every 18 months, the transmission rate in cellular network has been risen at the speed of nearly 10 times every 5 years. Meanwhile, the price paid for this enormous growth in data rates and market penetration is a rising power requirement of information and communication technologies (ICT) – although at a substantially lower speed than "Moore's Law" – the energy consumption doubles every 4-5 years [3].

In order to avoid the collapsing of communication systems and networks resources, an increasing interest and intensive researches in both energy and bandwidth efficient designs have mobilized enormous efforts of research's groups around the globe in the last decade. Against this background, the conventional efficient design of wireless networks mainly focuses on system capacity and spectral efficiency (SE). However, energy-efficient design in wireless networks is of paramount importance and is becoming an inevitable trend, since the deployment of multimedia wireless services and requirement of ubiquitous access have increased rapidly, and as a consequence, energy consumption at both the base station and mobile terminals side have experienced enormous increasing.

In order to achieve high throughput with low total transmit power, system resources such as spectrum (subcarriers, bandwidth), transmit power (energy, battery lifetime) and information rate (QoS requirements) in different multiple access wireless communication systems should be efficiently and appropriately allocated to the different active users. The first part of this chapter is dedicated to deal with the energy-efficient and spectral-efficient designs in DS/CDMA wireless communication systems through the appropriate heuristic optimization of energy and information rate resources.

The Internet has brought to our daily life easy and new ways to execute tasks as searching and gathering information, to communicate and spread ideas and others small gestures that are changing our lives. In order to prevent possible failures and loss of performance, the infrastructure providing theses services must be monitored, which unavoidably increases the responsibility and charge of the network administrator. The administrator is assisted by tools such as firewall, proxy, among others, including the anomaly detection system to help prevent abnormal network operation. Usually the anomaly behavior is a sudden increase or decrease into the network traffic. It can be caused by a simple programming error in some software to hardware failure, among many other causes that affect directly the network operation.

In the next sections of this Chapter the ACO methodology will be applied and analyzed regarding two distinct application of communication scenarios: the resource allocation in a direct sequence code division multiple access (DS/CDMA) systems, which is developed in Section 2 and the anomaly detection in computer networks is discussed in the Section 3. The conclusion remarks for both ACO-communication application problems are offered in Section 4.

## **2. Resource Allocation in Wireless Multiple Access Networks**

The optimized resource allocation in wireless multiple access networks, specially the power rate allocation, is a problem of great interest for telecommunications enterprises and users. It is well known that spectrum and power are valuable resources due to their scarcity, the first one is a natural non renewable resource and the second one is limited by the battery and device size. Therefore, proposing new techniques and algorithms that can allocate this resources in a simple<sup>1</sup> and optimized manner is pretty important.

In the last few decades many researchers have been working on this subject aiming to find a simple yet sturdy algorithm for resource allocation in wireless systems. Among many works recently done we enumerate some notorious in the next section.

## **2.1. Related Work**

2 Search Algorithms

years [3].

operation.

Section 4.

optimization problems.

to its robustness and great performance in deal with discrete (combinatorial) and continuous

An important challenge for the future wireless communication systems has been how to acquire higher throughput with lower power consumption. Hence, in order to transfer the exponentially rising amount of available data to the user in an acceptable time, following the "Moore's Law", according to which both the processing power of CPUs and the capacity of mass storage devices doubles approximately every 18 months, the transmission rate in cellular network has been risen at the speed of nearly 10 times every 5 years. Meanwhile, the price paid for this enormous growth in data rates and market penetration is a rising power requirement of information and communication technologies (ICT) – although at a substantially lower speed than "Moore's Law" – the energy consumption doubles every 4-5

In order to avoid the collapsing of communication systems and networks resources, an increasing interest and intensive researches in both energy and bandwidth efficient designs have mobilized enormous efforts of research's groups around the globe in the last decade. Against this background, the conventional efficient design of wireless networks mainly focuses on system capacity and spectral efficiency (SE). However, energy-efficient design in wireless networks is of paramount importance and is becoming an inevitable trend, since the deployment of multimedia wireless services and requirement of ubiquitous access have increased rapidly, and as a consequence, energy consumption at both the base station and

In order to achieve high throughput with low total transmit power, system resources such as spectrum (subcarriers, bandwidth), transmit power (energy, battery lifetime) and information rate (QoS requirements) in different multiple access wireless communication systems should be efficiently and appropriately allocated to the different active users. The first part of this chapter is dedicated to deal with the energy-efficient and spectral-efficient designs in DS/CDMA wireless communication systems through the appropriate heuristic optimization

The Internet has brought to our daily life easy and new ways to execute tasks as searching and gathering information, to communicate and spread ideas and others small gestures that are changing our lives. In order to prevent possible failures and loss of performance, the infrastructure providing theses services must be monitored, which unavoidably increases the responsibility and charge of the network administrator. The administrator is assisted by tools such as firewall, proxy, among others, including the anomaly detection system to help prevent abnormal network operation. Usually the anomaly behavior is a sudden increase or decrease into the network traffic. It can be caused by a simple programming error in some software to hardware failure, among many other causes that affect directly the network

In the next sections of this Chapter the ACO methodology will be applied and analyzed regarding two distinct application of communication scenarios: the resource allocation in a direct sequence code division multiple access (DS/CDMA) systems, which is developed in Section 2 and the anomaly detection in computer networks is discussed in the Section 3. The conclusion remarks for both ACO-communication application problems are offered in

mobile terminals side have experienced enormous increasing.

of energy and information rate resources.

Among numerous solutions proposed to resource allocation in wireless multiple access networks we enumerate herein some of great importance works: Foschini and Miljanic [4] distributed power control algorithm (DPCA) stands as the main one. When it comes to metaheuristics, in [5] and [6] a genetic algorithm approach was used to propose the genetic algorithm for mobiles equilibrium, providing the joint power-rate control in CDMA multiple access networks. In [7], the particle swarm optimization (PSO) metaheuristic was used in order to establish a low-complexity power control algorithm. Finally, in [8] a power allocation approach was proposed to solve the parallel interference cancelation in multi-user detectors.

Beyond the metaheuristic approaches, the work developed in [9] exploits an algorithm based on the dynamic cost assignment for downlink power allocation in CDMA networks. Besides, [10] addressed the uplink fairness maximization in a CDMA system with multiple processing gains. In [11], the Verhulst population model, firstly developed to describe the biological species growth with restrictions of space and food, was adapted to the distributed power control problem in a DS/CDMA network. It is noteworthy that this work was the first one to propose a Verhulst model adaptation to resource allocation problems in multiple access networks.

Furthermore, in [12] an analytical approach was proposed for the weighted throughput maximization (WTM) problem, namely MAPEL. The algorithm performs the power control in the interference limited wireless networks, i.e., CDMA and MC/CDMA networks, through a monotonically increasing objective function that is not necessarily convex. This function was formulated as a multiplicative linear fractional programming (MLFP) problem, which is a special case of generalized linear fractional programming (GLFP). So, the GLFP problem presented in [12] was used in [13] in order to formulate a non-decreasing objective function as a weighted SNIR's productory.

Finally, this section presents a heuristic approach through ant colony optimization in the continuous domains (ACO**R**) applicable to the power and rate allocation problems [13], and is organized as follows: subsection 2.2 describes aspects of the DS/CDMA networks and the power control problem on subsection 2.3 the power control problem and the cost function used with the ACO algorithm are presented; subsection 2.4 deals with the throughput maximization problem and how the ACO algorithm can be applied to solve this optimization

<sup>1</sup> Here, simple is used as a synonym for low computational complexity.

problem, while in subsection 2.5 the ACO algorithm itself is described. Finally, subsection 2.6 introduces the simulations scenarios, the numerical results and conclusions for the first part of this chapter.

#### **2.2. Resource Allocation in DS/CDMA Networks**

In DS/CDMA multirate networks, the Bit Error Rate (BER) is usually used as a QoS metric, since it is directly related to the Signal to Noise plus Interference Ratio (SNIR). Thus, the SNIR is associated to the Carrier to Interference Ratio as follows:

$$
\gamma\_{\dot{i}} = \frac{r\_c}{r\_{\dot{i}}} \times \Gamma\_{\dot{i}\prime} \qquad \qquad \dot{\imath} = 1, \ldots, \dot{\jmath}L \tag{1}
$$

Additionally, the theoretical Shannon channel capacity is defined as [15]:

[1]:

level.

can be defined as [1]:

where *BER* is the desired bit error rate (BER).

**2.3. Power Allocation Problem**

relation must be calculated as [16]:

QoS) that is acceptable to each user.

capacity model using the gap introduced in Eq (8):

respectively, and *<sup>γ</sup>*<sup>∗</sup>

be mathematically stated as:

where *C* is the channel capacity in *bits*/*s* and *γ* is the SNIR. It is worthy to note that since this is a theoretical bound a gap can be included, thus, the Shannon equation can be rewritten as

Ant Colony Optimization for Resource Allocation and Anomaly Detection in Communication Networks

where *θ* is the gap between the theoretical bound and the real information rate. Usually, *θ*

The power control objective is to find the minimal transmission power for each user that satisfy its QoS requirement, usually a minimum transmission rate. Since user rate is related to the user SNIR one may use it as a QoS measure. Thus, the power allocation problem may

min **p** = [*p*1, *p*2,..., *pU*]

0 ≤ *pi* ≤ *p*max

In order to enable the users to have minimum QoS warranty, the minimum CIR to SNIR

*i rc*

where Γ*i*,min and *Ri*,min are the minimum CIR and minimum information rate of each user,

This way, the minimum information rate can be mapped in the SNIR through the Shannon's

*<sup>i</sup>* is the minimum SNIR needed in order to obtain a minimum BER (or

s.t. *<sup>γ</sup><sup>i</sup>* <sup>≥</sup> *<sup>γ</sup>*<sup>∗</sup>

where *pi* and *<sup>γ</sup><sup>i</sup>* is the *<sup>i</sup>*th user power and SNIR, respectively, and *<sup>γ</sup>*<sup>∗</sup>

<sup>Γ</sup>*i*,min <sup>=</sup> *ri*,min*γ*<sup>∗</sup>

*<sup>θ</sup>* <sup>=</sup> <sup>−</sup> 1.5

*C* = *W* log(1 + *γ*) (6)

http://dx.doi.org/10.5772/53338

113

*C* = *W* log(1 + *θγ*) (7)

log(5*BER*) (8)

*<sup>i</sup>* (9)

, *i* = 1, ..., *U* (10)

*<sup>i</sup>* is the desired SNIR

where *γ<sup>i</sup>* is the *i*-th user's SNIR, *rc* is the chip rate, *ri* is the *i*-th user's information rate, *U* is the system load and Γ*<sup>i</sup>* is the *i*-th user's CIR defined as [11], [14]:

$$\Gamma\_{i} = \frac{p\_{i}|g\_{ii}|^{2}}{\sum\_{j=1, i \neq j}^{II} p\_{j}|g\_{ij}|^{2} + \sigma^{2}}, i = 1, \dots, II \tag{2}$$

where *pi* is the *i*-th user's power bounded by *pmax*, *U* the number of active users on the system, |*gii*| the channel gain of the *i*-th user, |*gij*| is the interfering signals gain and *σ*<sup>2</sup> the Additive White Gaussian Noise (AWGN) at the *i*-th receiver's input.

$$\mathbf{G}\_{\text{upl}} = \begin{bmatrix} \\$11 & \\$12 & \cdots & \\$1u \\ \\$21 & \\$22 & \cdots & \\$2u \\ \vdots & \vdots & \ddots & \vdots \\ \\$u & \\$u 2 & \cdots & \\$uu \end{bmatrix} \tag{3}$$

where the main diagonal (*gii*) shows the *i*-th user's channel attenuation, while the other values shows the interfering signals gain.

The path loss is inversely proportional to the distance between the mobile unit and the base station; the shadowing is obtained by a log-normal probability distribution random variable, and the multipath fading obtained assuming a Rayleigh probability distribution for cases without line of sight (LOS), and Rice distribution for cases with LOS.

In the DS/CDMA networks with multiple processing gains (MPG), where each user has a different processing gain *Fi* > 1, it is defined as a function of the chip rate:

$$F\_{\bar{l}} = \frac{r\_c}{r\_{\bar{l}}}, \qquad \bar{l} = 1, 2, \dots, \mathcal{U} \tag{4}$$

Therefore, from Eq. (1) and (4) follows:

$$
\gamma\_{\bar{\imath}} = F\_{\bar{\imath}} \times \Gamma\_{\bar{\imath}} \tag{5}
$$

Additionally, the theoretical Shannon channel capacity is defined as [15]:

$$\mathbb{C} = W \log(1 + \gamma) \tag{6}$$

where *C* is the channel capacity in *bits*/*s* and *γ* is the SNIR. It is worthy to note that since this is a theoretical bound a gap can be included, thus, the Shannon equation can be rewritten as [1]:

$$C = W \log(1 + \theta \gamma) \tag{7}$$

where *θ* is the gap between the theoretical bound and the real information rate. Usually, *θ* can be defined as [1]:

$$\theta = -\frac{1.5}{\log(5BER)}\tag{8}$$

where *BER* is the desired bit error rate (BER).

#### **2.3. Power Allocation Problem**

4 Search Algorithms

part of this chapter.

**2.2. Resource Allocation in DS/CDMA Networks**

SNIR is associated to the Carrier to Interference Ratio as follows:

*<sup>γ</sup><sup>i</sup>* <sup>=</sup> *rc ri*

the system load and Γ*<sup>i</sup>* is the *i*-th user's CIR defined as [11], [14]:

<sup>Γ</sup>*<sup>i</sup>* <sup>=</sup> *pi*|*gii*<sup>|</sup>

∑*<sup>U</sup>*

Additive White Gaussian Noise (AWGN) at the *i*-th receiver's input.

**G**upl =

without line of sight (LOS), and Rice distribution for cases with LOS.

*Fi* <sup>=</sup> *rc ri*

different processing gain *Fi* > 1, it is defined as a function of the chip rate:

values shows the interfering signals gain.

Therefore, from Eq. (1) and (4) follows:

 

. . . . . . ... . . .

where the main diagonal (*gii*) shows the *i*-th user's channel attenuation, while the other

The path loss is inversely proportional to the distance between the mobile unit and the base station; the shadowing is obtained by a log-normal probability distribution random variable, and the multipath fading obtained assuming a Rayleigh probability distribution for cases

In the DS/CDMA networks with multiple processing gains (MPG), where each user has a

problem, while in subsection 2.5 the ACO algorithm itself is described. Finally, subsection 2.6 introduces the simulations scenarios, the numerical results and conclusions for the first

In DS/CDMA multirate networks, the Bit Error Rate (BER) is usually used as a QoS metric, since it is directly related to the Signal to Noise plus Interference Ratio (SNIR). Thus, the

where *γ<sup>i</sup>* is the *i*-th user's SNIR, *rc* is the chip rate, *ri* is the *i*-th user's information rate, *U* is

2

where *pi* is the *i*-th user's power bounded by *pmax*, *U* the number of active users on the system, |*gii*| the channel gain of the *i*-th user, |*gij*| is the interfering signals gain and *σ*<sup>2</sup> the

> *g*<sup>11</sup> *g*<sup>12</sup> ... *g*1*<sup>U</sup> g*<sup>21</sup> *g*<sup>22</sup> ... *g*2*<sup>U</sup>*

*gU*<sup>1</sup> *gU*<sup>2</sup> ... *gUU*

× Γ*i*, *i* = 1, . . . , *U* (1)

*<sup>j</sup>*=1,*i*�=*<sup>j</sup> pj*|*gij*|<sup>2</sup> <sup>+</sup> *<sup>σ</sup>*<sup>2</sup> , *<sup>i</sup>* <sup>=</sup> 1, ..., *<sup>U</sup>* (2)

, *i* = 1, 2, . . . , *U* (4)

*γ<sup>i</sup>* = *Fi* × Γ*<sup>i</sup>* (5)

(3)

  The power control objective is to find the minimal transmission power for each user that satisfy its QoS requirement, usually a minimum transmission rate. Since user rate is related to the user SNIR one may use it as a QoS measure. Thus, the power allocation problem may be mathematically stated as:

$$\begin{aligned} \text{minim} \quad & \mathbf{p} = [p\_1, p\_2, \dots, p\_U] \\ \text{s.t.} \quad & \gamma\_i \ge \gamma\_i^\* \\ & 0 \le p\_i \le p\_{\text{max}} \end{aligned} \tag{9}$$

where *pi* and *<sup>γ</sup><sup>i</sup>* is the *<sup>i</sup>*th user power and SNIR, respectively, and *<sup>γ</sup>*<sup>∗</sup> *<sup>i</sup>* is the desired SNIR level.

In order to enable the users to have minimum QoS warranty, the minimum CIR to SNIR relation must be calculated as [16]:

$$\Gamma\_{i,\min} = \frac{r\_{i,\min} \gamma\_i^\*}{r\_c}, \quad i = 1, \ldots, U \tag{10}$$

where Γ*i*,min and *Ri*,min are the minimum CIR and minimum information rate of each user, respectively, and *<sup>γ</sup>*<sup>∗</sup> *<sup>i</sup>* is the minimum SNIR needed in order to obtain a minimum BER (or QoS) that is acceptable to each user.

This way, the minimum information rate can be mapped in the SNIR through the Shannon's capacity model using the gap introduced in Eq (8):

$$\mathcal{D}^{\frac{\gamma}{\gamma\_i}} = \max \left[ 1 + \theta\_i \gamma\_i \right] \\ = \max \left[ 1 + \frac{\theta\_i F\_{\bar{l}} \cdot p\_{\bar{i}} |g\_{\bar{i}\bar{l}}|^2}{\sum\_{i \neq j}^{\mathcal{U}} p\_j |g\_{\bar{i}j}|^2 + \sigma^2} \right] \tag{11}$$

where 2 *ri rc* is the normalized information rate for the *i*-th user, *θ<sup>i</sup>* is the inverse of the gap between the channel's theoretical capacity and the real information rate. Note that for the minimum SNIR *<sup>γ</sup>*<sup>∗</sup> *<sup>i</sup>* , Eq. 11 uses the minimum information rate established by the system, in order to guarantee QoS. Such that one obtain the condition needed for the minimum SNIR to be satisfied, given a minimum information rate:

$$\gamma\_i^\* = \frac{\mathfrak{D}^{r\_{i,\text{min}}} - 1}{\theta\_i} \tag{12}$$

function for power control problem in CDMA systems using genetic algorithms has been proposed. This function was later modified and used with swarm intelligence in [1] to solve the power control problem. Due to the good results obtained in [1] that cost function was

Ant Colony Optimization for Resource Allocation and Anomaly Detection in Communication Networks

*<sup>i</sup>* =

The increasing information traffic demand due to multimedia services on third generation networks (3G) and beyond, along with the need of telecommunications companies to improve their profits have motivated development on weighted throughput maximization (WTM)

0 ≤ *pi* ≤ *p*max

where *f*(**p**) is a cost function that describes the behaviour of information rate of each user regarding the allocated transmit power vector **p**; *ri* is the *i*-th user's information rate, *ri*,min the minimum rate needed to ensure QoS for user *i*, **p** is the power vector such that **p** = [*p*1, *p*2,..., *pU*], and *p*max is the maximum transmission power allowed in the system.

Therefore, we must incorporate the multirate criterion to the WTM problem subject to maximum power allowed per user. From this, the optimization problem is formulated as a special case of *generalized linear fractional programming* (GLFP) [18]. This way, the second RA

*vi*

where 2*ri*,min is the minimum information rate normalized by the bandwidth of the system (*rc*) of the *i*-th link, including null rate restrictions; *vi* > 0 is the priority of the *i*-th user to

*hi*(**p**) <sup>≥</sup> <sup>2</sup>*ri*,min , <sup>∀</sup>*<sup>i</sup>* <sup>=</sup> 1, . . . , *<sup>U</sup>*

problem, which aims to maximize the system throughput, been formulated as:

max*<sup>r</sup> <sup>f</sup>*(**p**) s.t. *ri* ≥ *ri*,min , ∀*i* = 1, 2, . . . , *U*

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115

(18)

(19)

0 ≤ *pi* ≤ *p*max (17)

*i* 0, otherwise

1, *<sup>γ</sup><sup>i</sup>* <sup>≥</sup> *<sup>γ</sup>*<sup>∗</sup>

used with the ACO algorithm and reproduced hereafter for convenience:

**F**th *i* · <sup>1</sup> <sup>−</sup> *pi <sup>p</sup>*max

*U* ∑ *i*=1

1 *U*

*i*

**2.4. Weighted Throughput Maximization (WTM) Problem**

*ri* = *ri*,min

*J*1(**p**) = max

where the threshold function is defined as **F**th

problem can be described as follows:

*J*2(**p**) = max

*U* ∏ *i*=1  *fi*(**p**) *hi*(**p**)

s.t. 0 < *pi* ≤ *pi*,max, *fi*(**p**)

s.t. *<sup>γ</sup><sup>i</sup>* <sup>≥</sup> *<sup>γ</sup>*<sup>∗</sup>

Consider the QoS normalized interference matrix **B** [1]:

$$B\_{ij} = \begin{cases} 0, & i = j; \\ \frac{\Gamma\_{l, \text{min} \mathcal{G} \mid l}}{\mathcal{G}^{\mu}}, & \text{otherwise.} \end{cases} \tag{13}$$

which Γ*i*,min can be obtained as follows:

$$
\Gamma\_{i, \text{min}} = \frac{r\_{i, \text{min}} \gamma\_i^\*}{r\_c}, \quad i = 1, \dots, U \tag{14}
$$

Now consider the information rate requirements for each user and the QoS normalized noise vector *u* = [*u*1, *u*2,..., *uk*] *<sup>T</sup>*, with elements:

$$
\mu\_i = \frac{\Gamma\_{i,\text{min}} \sigma\_i^2}{\mathcal{g}\_{ii}} \tag{15}
$$

The solution to the power control problem may be analytically obtained solving the following linear system:

$$\mathbf{p}^\* = (\mathbf{I} - \mathbf{B})^{-1}\mathbf{u} \tag{16}$$

where *IU*×*<sup>U</sup>* is the identity matrix. Note that that (**I** − **B**) is invertible only, and only if the maximum eigenvalue of *B* is smaller than one [17]. Only in this case, the power control problem will present a feasible solution. Nevertheless, due to the limited resources of mobile terminals, the use of this method is not feasible since its computational cost grows prohibitively when the number of users goes beyond some dozens due to a matrix inversion operation. Besides a totally distributed allocation scheme cannot be deployed using this analytical solution. To overcome this issues, this work proposes a metaheuristic approach for the optimum power-rate allocation problems.

In order to use the ACO algorithm to solve the power allocation problem one must map the problem objective into a mathematical function so-called cost function. In [5, 6] a new cost function for power control problem in CDMA systems using genetic algorithms has been proposed. This function was later modified and used with swarm intelligence in [1] to solve the power control problem. Due to the good results obtained in [1] that cost function was used with the ACO algorithm and reproduced hereafter for convenience:

$$J\_1(\mathbf{p}) = \max \, \frac{1}{U} \sum\_{i=1}^{U} \mathbb{F}\_i^{\mathbf{th}} \cdot \left(1 - \frac{p\_i}{p\_{\max}}\right), \qquad \forall i = 1, 2, \dots, U$$

$$\begin{array}{ll} \text{s.t.} & \gamma\_i \ge \gamma\_i^\* \\ & 0 \le p\_i \le p\_{\max} \\ & r\_i = r\_{i,\min} \end{array} \tag{17}$$

where the threshold function is defined as **F**th *<sup>i</sup>* = 1, *<sup>γ</sup><sup>i</sup>* <sup>≥</sup> *<sup>γ</sup>*<sup>∗</sup> *i* 0, otherwise

6 Search Algorithms

where 2 *ri*

minimum SNIR *<sup>γ</sup>*<sup>∗</sup>

2 *ri*

which Γ*i*,min can be obtained as follows:

the optimum power-rate allocation problems.

vector *u* = [*u*1, *u*2,..., *uk*]

linear system:

to be satisfied, given a minimum information rate:

Consider the QoS normalized interference matrix **B** [1]:

*rc* = max [1 + *θiγi*] = max

*γ*∗

<sup>Γ</sup>*i*,min <sup>=</sup> *ri*,min*γ*<sup>∗</sup>

*Bij* =

*<sup>T</sup>*, with elements:

 1 +

*rc* is the normalized information rate for the *i*-th user, *θ<sup>i</sup>* is the inverse of the gap

between the channel's theoretical capacity and the real information rate. Note that for the

order to guarantee QoS. Such that one obtain the condition needed for the minimum SNIR

*<sup>i</sup>* <sup>=</sup> <sup>2</sup>*ri*,min <sup>−</sup> <sup>1</sup> *θi*

 0, *i* = *j*; Γ*i*,min *gji*

> *i rc*

Now consider the information rate requirements for each user and the QoS normalized noise

*ui* <sup>=</sup> <sup>Γ</sup>*i*,min*σ*<sup>2</sup>

The solution to the power control problem may be analytically obtained solving the following

where *IU*×*<sup>U</sup>* is the identity matrix. Note that that (**I** − **B**) is invertible only, and only if the maximum eigenvalue of *B* is smaller than one [17]. Only in this case, the power control problem will present a feasible solution. Nevertheless, due to the limited resources of mobile terminals, the use of this method is not feasible since its computational cost grows prohibitively when the number of users goes beyond some dozens due to a matrix inversion operation. Besides a totally distributed allocation scheme cannot be deployed using this analytical solution. To overcome this issues, this work proposes a metaheuristic approach for

In order to use the ACO algorithm to solve the power allocation problem one must map the problem objective into a mathematical function so-called cost function. In [5, 6] a new cost

*i gii*

*θiFi* · *pi*|*gii*|

*<sup>i</sup>*�=*<sup>j</sup> pj*|*gij*|<sup>2</sup> <sup>+</sup> *<sup>σ</sup>*<sup>2</sup>

∑*<sup>U</sup>*

*<sup>i</sup>* , Eq. 11 uses the minimum information rate established by the system, in

2

*gii* , otherwise. (13)

, *i* = 1, ..., *U* (14)

**<sup>p</sup>**<sup>∗</sup> = (**<sup>I</sup>** − **<sup>B</sup>**)−1**<sup>u</sup>** (16)

(11)

(12)

(15)

## **2.4. Weighted Throughput Maximization (WTM) Problem**

The increasing information traffic demand due to multimedia services on third generation networks (3G) and beyond, along with the need of telecommunications companies to improve their profits have motivated development on weighted throughput maximization (WTM) problem, which aims to maximize the system throughput, been formulated as:

$$\begin{aligned} \max\_{r} & \quad f(\mathbf{p})\\ \text{s.t.} & \quad r\_{i} \ge r\_{i,\min} \\ & \quad 0 \le p\_{i} \le p\_{\max} \end{aligned} \tag{18}$$

where *f*(**p**) is a cost function that describes the behaviour of information rate of each user regarding the allocated transmit power vector **p**; *ri* is the *i*-th user's information rate, *ri*,min the minimum rate needed to ensure QoS for user *i*, **p** is the power vector such that **p** = [*p*1, *p*2,..., *pU*], and *p*max is the maximum transmission power allowed in the system.

Therefore, we must incorporate the multirate criterion to the WTM problem subject to maximum power allowed per user. From this, the optimization problem is formulated as a special case of *generalized linear fractional programming* (GLFP) [18]. This way, the second RA problem can be described as follows:

$$\begin{aligned} \underline{f\_2}(\mathbf{p}) = \max \quad & \prod\_{i=1}^{U} \left[ \frac{f\_i(\mathbf{p})}{h\_i(\mathbf{p})} \right]^{v\_i} \\ \text{s.t.} \quad & 0 < p\_i \le p\_{i,\max} \\ & \frac{f\_i(\mathbf{p})}{h\_i(\mathbf{p})} \ge 2^{r\_{\text{train}}} \end{aligned} \tag{19}$$

where 2*ri*,min is the minimum information rate normalized by the bandwidth of the system (*rc*) of the *i*-th link, including null rate restrictions; *vi* > 0 is the priority of the *i*-th user to transmit with satisfied QoS requirements, assumed normalized, such that ∑*<sup>U</sup> <sup>i</sup>*=<sup>1</sup> *vi* = 1. It is noteworthy that the second restriction in Eq. (19) is easily obtained from Eqs. (11) and (12), where the minimum information rate given can be transformed in the minimum SNIR through the Shannon capacity equation, considering a maximum tolerable BER for each user or service class. Hence, functions *fi*(**p**) and *hi*(**p**) can be readily defined as:

*Gi* (*x*) =

<sup>1</sup>, *<sup>µ</sup><sup>i</sup>*

Gaussian PDFs (*g<sup>i</sup>*

*l*

<sup>2</sup>,..., *<sup>µ</sup><sup>i</sup>*


input parameters is sketched in Fig. 1.

layer (in depth) shows the solution file in each iteration from *n* = 1 to *N*.

(*x*)) in the *i*-th kernel (*G<sup>i</sup>*


µ*<sup>i</sup>* = [*µ<sup>i</sup>*

set, |ω| = |µ*<sup>i</sup>*

*Fs* ∑ *l*=1 *ωlg<sup>i</sup> l* (*x*) =

*Fs* ∑ *l*=1 *ωl*

1 *σi l* √2*π*

*Fs*] is the vector of means and <sup>σ</sup>*<sup>i</sup>* = [*σ<sup>i</sup>*

exp

where *i* is the Gaussian kernel indexer, with *U* being the number of dimensions of the problem; ω = [*ω*1, *ω*2,..., *ωFs*] is the weight vector associated to each Gaussian in the kernel;

deviations. Hence, the cardinality of both vectors is equal to the number of Gaussians in the

For discrete combinatorial optimization problems, the pheromone informations are kept in a table. This is not possible when we need to deal with continuous problems, since there are an infinite number of points to keep, and as a consequence, an infinite ways to evolve. Thus, a solution file is deployed, where the *l*th solution *sl*, ∀*l* = 1, 2, . . . , *Fs*, in the *i*th dimension, ∀*i* = 1, . . . , *U*, is kept on the memory file at the *n*th iteration, as well as the respective cost function values *f*(*sl*). A schematic grid to understand the file format and the associate ACO

**Figure 1.** File structure for the ACO algorithm's solutions. Each line of the *Fs* × *U* matrix represents one solution for the problem of dimension *U*; each column represents one dimension, which, in turn, is sampled by each Gaussian kernel. Cost function vector **J** = [*J*(*s*1),..., *J*(*sl*),..., *J*(*sFs*)], dimension *Fs* × 1, represents the solution for the *n*-th iteration. Finally, each

The found solutions are used to generate PDFs dynamically, through a method based on the stored solutions. The vectors µ*<sup>i</sup>* and σ*<sup>i</sup>* at *i*th dimension and ω common for all dimensions are calculated through the solutions of the file at each algorithm iteration; so, the Gaussian

The solutions file must store *Fs* solutions. Note that this number is equal to the number of

kernel will have one Gaussian PDF sampling each *i*-th variable of each solution. Herein, the

). Thus, one can conclude that the *i*-th Gaussian

kernel can be built, and guide the ants throughout the dimensions of the problem.

<sup>−</sup>(*<sup>x</sup>* <sup>−</sup> *<sup>µ</sup><sup>i</sup> l* )2

Ant Colony Optimization for Resource Allocation and Anomaly Detection in Communication Networks

2*σi*<sup>2</sup> *l*

<sup>1</sup>, *<sup>σ</sup><sup>i</sup>*

<sup>2</sup>,..., *<sup>σ</sup><sup>i</sup>*

, *i* = 1, . . . , *U* (22)

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117

*Fs*] is the vector of standard

$$h\_{l}(\mathbf{p}) = \sum\_{j \neq i}^{\mathcal{U}} p\_{j} |g\_{ij}|^{2} + \sigma^{2} \qquad \text{and} \qquad f\_{l}(\mathbf{p}) = \theta F\_{l} \cdot p\_{l} |g\_{il}|^{2} + h\_{l}(\mathbf{p}), \qquad \forall i = 1, \ldots, \mathcal{U}. \tag{20}$$

Note that the Eq. (19) is the productory of linear fractional exponentiated functions, and the function ∏*<sup>U</sup> <sup>i</sup>*=1(*zi*)*vi* is an increasing function in a nonnegative real domain [12]. Based on these properties, problem (19) can be properly rewritten as:

$$\begin{aligned} f\_2(\mathbf{p}) &= \max \sum\_{i=1}^{U} v\_i [\log\_2 f\_i(\mathbf{p}) - \log\_2 h\_i(\mathbf{p})] \ &= \max \sum\_{i=1}^{U} v\_i [\bar{f}\_i(\mathbf{p}) - \bar{h}\_i(\mathbf{p})] \\ \text{s.t.} \quad &0 < p\_i \le p\_{i,\text{max}} \\ &\bar{f}\_i(\mathbf{p}) - \bar{h}\_i(\mathbf{p}) \ge r\_{i,\text{min}} \quad \forall i = 1, \dots, U \end{aligned} \tag{21}$$

This way, the cost function turns into a sum of logarithms, which results in a monotonic nondecreasing function. With no loss of generality, in this work *vi* <sup>=</sup> *<sup>U</sup>*<sup>−</sup>1, <sup>∀</sup>*i*, has been adopted.

## **2.5. The ACOR Metaheuristic**

The ACO**R** is a continuous-valued metaheuristic based on the ants behavior when looking for food. Note that it was first proposed for combinatorial optimization problems. In its discrete version, each ant walks through the points of the input set and deposits pheromone on its edges. The next point selection is done probabilistically, considering the amount of pheromone on each edge, jointly with the heuristic information available in the current algorithm iteration.

Given a set of points next to an ant, the probability of each of this points to be chosen forms a probability mass function (PMF). The main idea of the continuous version ACO**R** is the adaptation of this PMF to a Probability Density Function (PDF), allowing each ant to sample a continuous PDF instead of dealing with discrete sampling points. This is due to the fact that the continuous domain has infinite points to be chosen.

The PDF used in this work is Gaussian given its soft capacity in generating random numbers, and due to the fact that it has only one maximum point located at the mean of the process. Nevertheless, this last feature is not useful when the search space has more than one feasible region. To overcome this problem, the ACO**R** uses a Gaussian kernel pdf (a weighted sum of Gaussians) to sample each dimension of the problem. Each Gaussian kernel is defined as follows [19]:

$$\mathcal{G}^{i}(\mathbf{x}) = \sum\_{l=1}^{\text{Fs}} \omega\_{l} \eta\_{l}^{i}(\mathbf{x}) = \sum\_{l=1}^{\text{Fs}} \omega\_{l} \frac{1}{\sigma\_{l}^{i} \sqrt{2\pi}} \exp\left[-\frac{(\mathbf{x} - \mu\_{l}^{i})^{2}}{2\sigma\_{l}^{i}}\right], \qquad i = 1, \dots, \mathsf{U} \tag{22}$$

where *i* is the Gaussian kernel indexer, with *U* being the number of dimensions of the problem; ω = [*ω*1, *ω*2,..., *ωFs*] is the weight vector associated to each Gaussian in the kernel; µ*<sup>i</sup>* = [*µ<sup>i</sup>* <sup>1</sup>, *<sup>µ</sup><sup>i</sup>* <sup>2</sup>,..., *<sup>µ</sup><sup>i</sup> Fs*] is the vector of means and <sup>σ</sup>*<sup>i</sup>* = [*σ<sup>i</sup>* <sup>1</sup>, *<sup>σ</sup><sup>i</sup>* <sup>2</sup>,..., *<sup>σ</sup><sup>i</sup> Fs*] is the vector of standard deviations. Hence, the cardinality of both vectors is equal to the number of Gaussians in the set, |ω| = |µ*<sup>i</sup>* | = |σ*<sup>i</sup>* | = *Fs*.

8 Search Algorithms

*hi*(**p**) =

function ∏*<sup>U</sup>*

adopted.

algorithm iteration.

follows [19]:

*U* ∑ *j*�=*i*

*pj*|*gij*|

*J*2(**p**) = max

− *f i*

**2.5. The ACOR Metaheuristic**

transmit with satisfied QoS requirements, assumed normalized, such that ∑*<sup>U</sup>*

or service class. Hence, functions *fi*(**p**) and *hi*(**p**) can be readily defined as:

<sup>2</sup> + *σ*<sup>2</sup> and *fi*(**p**) = *θFi* · *pi*|*gii*|

these properties, problem (19) can be properly rewritten as:

that the continuous domain has infinite points to be chosen.

*U* ∑ *i*=1

(**p**) <sup>−</sup> <sup>−</sup>

is noteworthy that the second restriction in Eq. (19) is easily obtained from Eqs. (11) and (12), where the minimum information rate given can be transformed in the minimum SNIR through the Shannon capacity equation, considering a maximum tolerable BER for each user

Note that the Eq. (19) is the productory of linear fractional exponentiated functions, and the

*vi*[log2 *fi*(**p**) <sup>−</sup> log2 *hi*(**p**)] = max

This way, the cost function turns into a sum of logarithms, which results in a monotonic nondecreasing function. With no loss of generality, in this work *vi* <sup>=</sup> *<sup>U</sup>*<sup>−</sup>1, <sup>∀</sup>*i*, has been

The ACO**R** is a continuous-valued metaheuristic based on the ants behavior when looking for food. Note that it was first proposed for combinatorial optimization problems. In its discrete version, each ant walks through the points of the input set and deposits pheromone on its edges. The next point selection is done probabilistically, considering the amount of pheromone on each edge, jointly with the heuristic information available in the current

Given a set of points next to an ant, the probability of each of this points to be chosen forms a probability mass function (PMF). The main idea of the continuous version ACO**R** is the adaptation of this PMF to a Probability Density Function (PDF), allowing each ant to sample a continuous PDF instead of dealing with discrete sampling points. This is due to the fact

The PDF used in this work is Gaussian given its soft capacity in generating random numbers, and due to the fact that it has only one maximum point located at the mean of the process. Nevertheless, this last feature is not useful when the search space has more than one feasible region. To overcome this problem, the ACO**R** uses a Gaussian kernel pdf (a weighted sum of Gaussians) to sample each dimension of the problem. Each Gaussian kernel is defined as

*hi*(**p**) ≥ *ri*,min, ∀*i* = 1, . . . , *U*

*<sup>i</sup>*=1(*zi*)*vi* is an increasing function in a nonnegative real domain [12]. Based on

s.t. 0 < *pi* ≤ *pi*,max, (21)

*U* ∑ *i*=1 *vi*[ − *f i* *<sup>i</sup>*=<sup>1</sup> *vi* = 1. It

<sup>2</sup> + *hi*(**p**), ∀*i* = 1, . . . , *U*. (20)

(**p**) <sup>−</sup> <sup>−</sup>

*hi*(**p**)]

For discrete combinatorial optimization problems, the pheromone informations are kept in a table. This is not possible when we need to deal with continuous problems, since there are an infinite number of points to keep, and as a consequence, an infinite ways to evolve. Thus, a solution file is deployed, where the *l*th solution *sl*, ∀*l* = 1, 2, . . . , *Fs*, in the *i*th dimension, ∀*i* = 1, . . . , *U*, is kept on the memory file at the *n*th iteration, as well as the respective cost function values *f*(*sl*). A schematic grid to understand the file format and the associate ACO input parameters is sketched in Fig. 1.

**Figure 1.** File structure for the ACO algorithm's solutions. Each line of the *Fs* × *U* matrix represents one solution for the problem of dimension *U*; each column represents one dimension, which, in turn, is sampled by each Gaussian kernel. Cost function vector **J** = [*J*(*s*1),..., *J*(*sl*),..., *J*(*sFs*)], dimension *Fs* × 1, represents the solution for the *n*-th iteration. Finally, each layer (in depth) shows the solution file in each iteration from *n* = 1 to *N*.

The found solutions are used to generate PDFs dynamically, through a method based on the stored solutions. The vectors µ*<sup>i</sup>* and σ*<sup>i</sup>* at *i*th dimension and ω common for all dimensions are calculated through the solutions of the file at each algorithm iteration; so, the Gaussian kernel can be built, and guide the ants throughout the dimensions of the problem.

The solutions file must store *Fs* solutions. Note that this number is equal to the number of Gaussian PDFs (*g<sup>i</sup> l* (*x*)) in the *i*-th kernel (*G<sup>i</sup>* ). Thus, one can conclude that the *i*-th Gaussian kernel will have one Gaussian PDF sampling each *i*-th variable of each solution. Herein, the greater the value of *Fs*, the greater will be the number of Gaussian PDFs on the algorithm. Therefore, the parameter *Fs* leads to the complexity of the set.

(more diversity), at the cost of lower convergence speed. Thus, the ACO**R**'s *q* parameter corresponds to the best-so-far solution and iteration-best solution concepts. So, Eq. (23) gives rise to an important equilibrium between *q* and *Fs* parameters, making their individual calibration sensitive to each one, in order to achieve a good tradeoff among robustness and

Ant Colony Optimization for Resource Allocation and Anomaly Detection in Communication Networks

Furthermore, the suitable choice for the population size *m* plays an important role in order to improve the robustness-speed tradeoff in conjunction with the best attainable *q* · *Fs* calibration. Note that *m* might be overloaded in order to increase the algorithm capacities, at the undesirable cost of greater computational complexity. Finally, the algorithm robustness R can be thought as the ratio between the number of convergence success S to the total

<sup>T</sup> · <sup>100</sup> [%] @*<sup>N</sup>* iterations

and the speed as the average number of iterations needed to the algorithm achieves

Next, important steps of the ACO algorithm are briefly discussed, such as general ACO algorithm structure, the σ*<sup>i</sup>* vector computation, as well as the sampling process; the last one will be presented directly on the algorithm structure. The algorithm organization common to various implemented versions of continuous ACO is described in Algorithm 1, in which

• *AntBasedSolutionConstruction()*: Through the decision variables of each solution *s<sup>i</sup>*

1, . . . , *U*, each ant builds the solution by *U* steps. Since ACO**R** uses a Gaussian mixture in each problem dimension (Eq. (22)), and that the number of Gaussians on the mixture is equal to the size *Fs* of the solutions file, we conclude that at each step *i* we will have a

this work, the vector ω will not be used, and the explanation for that is given in the next

In practice, the sample process is made on three stages: First, the elements of ω vector must be calculated, where should be noted that the solutions ranking parameter *l* will never change, independently of the change of the solutions order on the file. On the second stage, each ant must choose a solution of the file aiming to sample it, and the probability of this choose must be relative to the normalization of each solution weight

*l* , *i* =

, σ*<sup>i</sup>* and ω must be updated. In

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119

convergence velocity for each specific optimization problem.

<sup>R</sup> <sup>=</sup> <sup>S</sup>

the functions performed inside are briefly described in following.

number of process realizations T :

**Algorithm 1** Overview of ACO

*PheromoneUpdate() DaemonActions()*

different sample of *G<sup>i</sup>*

for the sum of all weights:

**end while**

paragraph.

**while** *The end conditions aren't met* **do** *AntBasedSolutionConstruction()*

.

In order to sample the Gaussian kernel, the vectors µ*<sup>i</sup>*

convergence in T trials for a given problem.

A detailed description for the evolution of the ACO solution structure is given in the following. From Fig. 1, note that for an *U*-dimensional problem, the file solution stores *Fs* different solutions, the values of its *U* variables and the results of each solution applied to the cost function. So, *s<sup>i</sup> <sup>l</sup>* is the value of the *i*-th variable of the *l*-th solution of the file, and *J*(*sl*) is the result of the *l*-th solution applied to the cost function. For each dimension *i* = 1, . . . , *U* of the problem (in this case, each column of the table), there is a different Gaussian Kernel PDF (*G<sup>i</sup>* ) defined. So, for each *G<sup>i</sup>* , the values of the *i*-th variable of all solutions becomes the elements of the mean vector, µ*<sup>i</sup>* = [*µ<sup>i</sup>* <sup>1</sup>,..., *<sup>µ</sup><sup>i</sup> Fs*]=[*s<sup>i</sup>* 1,...,*s<sup>i</sup> Fs*], i.e., the *l*-th value of dimension *i* is the mean of the *l*-th gaussian of *G<sup>i</sup>* .

Furthermore, the number of ants (or particles) *m* is another important input parameter of the ACO algorithm to be adjusted. The ants are responsible for the sampling of *G<sup>i</sup>* , and thus, for the algorithm evolution as well. In that way, on each iteration, each ant chooses one solution of the file probabilistically, through a method based on the weight vector ω. Since the ant has chosen one solution of the file, the next step consists of sampling through the Gaussian kernel. After that, a new solution is generated and attached to the end of the file. As the last ant finishes its sampling, the solution file is sorted based on the value entries in the cost function matrix **J** = [*J*(*s*1),..., *J*(*sl*),..., *J*(*sFs*)]. Hence, for both problems treated in Eq. (17) and (21), the matrix **J** must be sorted decreasingly, i.e., *J*(*s*1) ≥ *J*(*s*2) ≥ ... ≥ *J*(*sFs*).

When the sorting process is completed, a number of worst solutions ordered at the end of the file is discarded, in which is done equal to the number of solutions added on the sampling process. Note that since each ant samples only one solution on each iteration, the number of solutions to be discarded is equal to the number of ants.

At this point, a Gaussian kernel (*G<sup>i</sup>* ) is defined for each dimension of the problem, which the *l*-th variable becomes an element of the µ*<sup>i</sup>* vector. Thus, considering the *G<sup>i</sup>* defined on the *i*-th dimension, The weight *wl* of each solution is calculated as follows:

$$\omega\_{l} = \frac{1}{q \text{F} s \sqrt{2\pi}} \exp\left[-\frac{(l-1)^{2}}{2q^{2} \text{F} s^{2}}\right], \qquad l = 1, \ldots, \text{F}s \tag{23}$$

The weight of the *l*-th solution can be seen as the probability of a solution to be chosen and sampled by an ant. Hence, the *l*-th solution's rank in the file, also is the input parameter in Eq. (23), which is a Gaussian PDF with mean 1 and standard deviation *q* · *Fs*, where *q* is an input parameter of the ACO algorithm. The *q* parameter can be interpreted as a diversification parameter, where low values of *q* enhances the convergence speed of the algorithm; on the other hand, high values entries for *q* enhances the process robustness. This is due to the fact that, on the normal function for the *l*-th solution's weight calculation in Eq. (23), the higher the standard deviation values are, more chances to select solutions that are not so near to the mean of the process, which, in turn, is the first solution of the file. So, when the standard deviation *q* · *Fs* assumes small values, only the best and a few solutions of the file will be sampled, enabling the algorithm to converge faster. On the other hand, when high-valued standard deviations are admitted, the probability of the file solutions to be chosen becomes more uniform, which makes the algorithm search in a larger space (more diversity), at the cost of lower convergence speed. Thus, the ACO**R**'s *q* parameter corresponds to the best-so-far solution and iteration-best solution concepts. So, Eq. (23) gives rise to an important equilibrium between *q* and *Fs* parameters, making their individual calibration sensitive to each one, in order to achieve a good tradeoff among robustness and convergence velocity for each specific optimization problem.

10 Search Algorithms

PDF (*G<sup>i</sup>*

the cost function. So, *s<sup>i</sup>*

) defined. So, for each *G<sup>i</sup>*

elements of the mean vector, µ*<sup>i</sup>* = [*µ<sup>i</sup>*

At this point, a Gaussian kernel (*G<sup>i</sup>*

*i* is the mean of the *l*-th gaussian of *G<sup>i</sup>*

greater the value of *Fs*, the greater will be the number of Gaussian PDFs on the algorithm.

A detailed description for the evolution of the ACO solution structure is given in the following. From Fig. 1, note that for an *U*-dimensional problem, the file solution stores *Fs* different solutions, the values of its *U* variables and the results of each solution applied to

is the result of the *l*-th solution applied to the cost function. For each dimension *i* = 1, . . . , *U* of the problem (in this case, each column of the table), there is a different Gaussian Kernel

*Fs*]=[*s<sup>i</sup>*

Furthermore, the number of ants (or particles) *m* is another important input parameter of the

the algorithm evolution as well. In that way, on each iteration, each ant chooses one solution of the file probabilistically, through a method based on the weight vector ω. Since the ant has chosen one solution of the file, the next step consists of sampling through the Gaussian kernel. After that, a new solution is generated and attached to the end of the file. As the last ant finishes its sampling, the solution file is sorted based on the value entries in the cost function matrix **J** = [*J*(*s*1),..., *J*(*sl*),..., *J*(*sFs*)]. Hence, for both problems treated in Eq. (17)

When the sorting process is completed, a number of worst solutions ordered at the end of the file is discarded, in which is done equal to the number of solutions added on the sampling process. Note that since each ant samples only one solution on each iteration, the number of

*l*-th variable becomes an element of the µ*<sup>i</sup>* vector. Thus, considering the *G<sup>i</sup>* defined on the

<sup>−</sup>(*<sup>l</sup>* <sup>−</sup> <sup>1</sup>)<sup>2</sup> 2*q*2*Fs*<sup>2</sup>

The weight of the *l*-th solution can be seen as the probability of a solution to be chosen and sampled by an ant. Hence, the *l*-th solution's rank in the file, also is the input parameter in Eq. (23), which is a Gaussian PDF with mean 1 and standard deviation *q* · *Fs*, where *q* is an input parameter of the ACO algorithm. The *q* parameter can be interpreted as a diversification parameter, where low values of *q* enhances the convergence speed of the algorithm; on the other hand, high values entries for *q* enhances the process robustness. This is due to the fact that, on the normal function for the *l*-th solution's weight calculation in Eq. (23), the higher the standard deviation values are, more chances to select solutions that are not so near to the mean of the process, which, in turn, is the first solution of the file. So, when the standard deviation *q* · *Fs* assumes small values, only the best and a few solutions of the file will be sampled, enabling the algorithm to converge faster. On the other hand, when high-valued standard deviations are admitted, the probability of the file solutions to be chosen becomes more uniform, which makes the algorithm search in a larger space

<sup>1</sup>,..., *<sup>µ</sup><sup>i</sup>*

.

ACO algorithm to be adjusted. The ants are responsible for the sampling of *G<sup>i</sup>*

and (21), the matrix **J** must be sorted decreasingly, i.e., *J*(*s*1) ≥ *J*(*s*2) ≥ ... ≥ *J*(*sFs*).

solutions to be discarded is equal to the number of ants.

*<sup>ω</sup><sup>l</sup>* <sup>=</sup> <sup>1</sup>

*qFs*√2*<sup>π</sup>*

*i*-th dimension, The weight *wl* of each solution is calculated as follows:

exp 

*<sup>l</sup>* is the value of the *i*-th variable of the *l*-th solution of the file, and *J*(*sl*)

1,...,*s<sup>i</sup>*

, the values of the *i*-th variable of all solutions becomes the

) is defined for each dimension of the problem, which the

, *l* = 1, . . . , *Fs* (23)

*Fs*], i.e., the *l*-th value of dimension

, and thus, for

Therefore, the parameter *Fs* leads to the complexity of the set.

Furthermore, the suitable choice for the population size *m* plays an important role in order to improve the robustness-speed tradeoff in conjunction with the best attainable *q* · *Fs* calibration. Note that *m* might be overloaded in order to increase the algorithm capacities, at the undesirable cost of greater computational complexity. Finally, the algorithm robustness R can be thought as the ratio between the number of convergence success S to the total number of process realizations T :

> <sup>R</sup> <sup>=</sup> <sup>S</sup> <sup>T</sup> · <sup>100</sup> [%] @*<sup>N</sup>* iterations

and the speed as the average number of iterations needed to the algorithm achieves convergence in T trials for a given problem.

Next, important steps of the ACO algorithm are briefly discussed, such as general ACO algorithm structure, the σ*<sup>i</sup>* vector computation, as well as the sampling process; the last one will be presented directly on the algorithm structure. The algorithm organization common to various implemented versions of continuous ACO is described in Algorithm 1, in which the functions performed inside are briefly described in following.


• *AntBasedSolutionConstruction()*: Through the decision variables of each solution *s<sup>i</sup> l* , *i* = 1, . . . , *U*, each ant builds the solution by *U* steps. Since ACO**R** uses a Gaussian mixture in each problem dimension (Eq. (22)), and that the number of Gaussians on the mixture is equal to the size *Fs* of the solutions file, we conclude that at each step *i* we will have a different sample of *G<sup>i</sup>* .

In order to sample the Gaussian kernel, the vectors µ*<sup>i</sup>* , σ*<sup>i</sup>* and ω must be updated. In this work, the vector ω will not be used, and the explanation for that is given in the next paragraph.

In practice, the sample process is made on three stages: First, the elements of ω vector must be calculated, where should be noted that the solutions ranking parameter *l* will never change, independently of the change of the solutions order on the file. On the second stage, each ant must choose a solution of the file aiming to sample it, and the probability of this choose must be relative to the normalization of each solution weight for the sum of all weights:

$$p\_l = \omega\_l \cdot \left(\sum\_{r=1}^{k} \omega\_r\right)^{-1} \tag{24}$$

• *DaemonActions()*: This is the optional component of ACO**R** that can be used to implement centralized actions of the algorithm that aren't accessible for the ants. In this stage, the found solution must be updated and returned as the final solution. Besides, it is possible to implement local search methods here, but this aspect is not exploited in this current

Ant Colony Optimization for Resource Allocation and Anomaly Detection in Communication Networks

This subsection is divided in two parts. The first one deals with ACO typical performance and its input parameters optimization; the second part numerical simulations results for both power and rate allocation problems are presented and the NMSE is compared with the

The simulations were carried out in the MatLab 7.0 platform and the scenario parameters are presented on Table 1. We assumed a rectangular cell with one base station in the center and users uniformly spread across all the cell extension. We considered that all mobile terminals

rate, and (∆*t*)*<sup>c</sup>* is the coherence time of the channel2. This is part of the SNIR estimation process, which means that the channel is constant in each optimization window, assumed herein equal to 667*µs*. Thus, the ACO algorithm must converge to the solution within each

Simulation experiments were carried out in order to determine the suitable values for the ACO input parameters for each problem, such as file size (*Fs*), pheromone evaporation coefficient (*ξ*), population (*m*) and the diversity parameter (*q*). The best parameters combination was chosen considering the solutions quality measured by the normalized mean

where **p** is the solution found through the ACO algorithm, **p**<sup>∗</sup> the analytical (optimal) solution and **E** the mathematical expectation operator. In order to find the best parameters

A typical convergence behavior for the RA-ACO under equal-rate power control problem is shown in Fig. 2. At a first glance, power allocation for *U* = 5 users (lightly loading system) is performed by a) RA-ACO algorithm, b) RA-PSO algorithm from [1]. One can

<sup>2</sup> Corresponds to the time interval in which the channel characteristics do not suffer expressive variations.


NMSE = **E**

slot is the time slot duration, *R*slot is the transmitted power vector update

*<sup>T</sup>*slot <sup>&</sup>lt; (∆*t*)*<sup>c</sup>* (26)

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121

(27)

experience slow fading channels, i.e. the following relation is always satisfied:

work, since we look firstly for low computational complexity.

**2.6. Numerical Results**

RA-PSO algorithm.

where *<sup>T</sup>*slot <sup>=</sup> *<sup>R</sup>*−<sup>1</sup>

667*µs* interval.

*2.6.1. RA-ACO Input Parameters Optimization*

for both problems non-exhaustive tests were conducted.

squared error (NMSE), defined as:

that is, the probability of each solution being chosen can be thought as a random number generator of normal distribution, with mean 1 and standard deviation *q* × *Fs*, since the choose probability of each rank will never change. Adopting this strategy, the *ω* vector as well as the first stage of the sampling process will no longer be needed.

Thus, since the ant chosen its solution, it must be sampled stepwise using a normal random number generator. The chosen solution must be sampled dimensionally (*gi l* , *i* = 1, . . . , *U*), causing each Gaussian mixture's parameters to be seen only in one dimension a time, smoothing the calculation of the pattern deviation and allowing linear transformations on the problem without result changes.

Therefore, let *sl* (Fig. 1) to be the solution chosen by an ant during the ACO's evolution process. It is known that the ant will sample *sl* dimensionally, as well as that the sampling is done through a Gaussian function parametrized by the *µ<sup>i</sup> <sup>l</sup>* and *<sup>σ</sup><sup>i</sup> <sup>l</sup>* values. Thus, the *<sup>σ</sup><sup>i</sup> l* is calculated for dimension *i* as follows:

$$
\sigma\_l^i = \xi \sum\_{\varepsilon=1}^k \frac{|s\_\varepsilon^i - s\_l^i|}{k-1} \tag{25}
$$

Herein, the *σ<sup>i</sup> <sup>l</sup>* value for *<sup>s</sup><sup>i</sup> <sup>l</sup>* is the mean distance from *<sup>s</sup><sup>i</sup> <sup>l</sup>* to the other values of dimension *i* in the other solutions of the file. The process is repeated until the last dimension of the file is reached. This way, the higher the variability of the different solutions, higher will be the standard deviation value *σ<sup>i</sup> l* . Note that the *ξ* ∈ [0, 1] parameter aims to reduce the standard deviation, working as a learning factor. Since the *σ<sup>i</sup> <sup>l</sup>* value is calculated, the ant will sample the Gaussian PDF *g<sup>i</sup> l* (*µi l* , *σi l*).

The parameter *ξ* is the same for all dimensions of all solutions, and corresponds to the pheromone evaporation rate, or to the inverse of the learning rate. This way, when *ξ* is low valued the algorithm speed is enhanced, and when it is high-valued, its robustness is enhanced. It is noteworthy that the algorithm converges when *σ* → 0 throughout all dimensions of the file.

• *PheromoneUpdate()*: The ACO**R** algorithm updates its pheromone informations as follows: At the beginning of the algorithm, the file is initialized with *Fs* solutions uniformly random distributed. From this, the pheromone updating is done adding the new solutions generated by the ants, as well as removing the same number of worst solutions.

Finally, the size of the solutions file is a parameter of the algorithm, and must not be smaller than the number of dimensions of the problem if it is enabled to handle variable correlation, and to support linear transformations on the problem being optimized. Nevertheless, these techniques are not used in this work. Furthermore, the size of the file leads to the algorithm diversity, since a big file will cover a greater region of the search space than the small one, enabling the algorithm to overcome local optima, but on the other hand, a small file will make the algorithm to be faster than the big one.

• *DaemonActions()*: This is the optional component of ACO**R** that can be used to implement centralized actions of the algorithm that aren't accessible for the ants. In this stage, the found solution must be updated and returned as the final solution. Besides, it is possible to implement local search methods here, but this aspect is not exploited in this current work, since we look firstly for low computational complexity.

## **2.6. Numerical Results**

12 Search Algorithms

(*gi l*

Herein, the *σ<sup>i</sup>*

*pl* = *ω<sup>l</sup>* ·

well as the first stage of the sampling process will no longer be needed.

*σi <sup>l</sup>* = *ξ*

*l*

standard deviation, working as a learning factor. Since the *σ<sup>i</sup>*

*l* (*µi l* , *σi l*).

transformations on the problem without result changes.

is calculated for dimension *i* as follows:

*<sup>l</sup>* value for *<sup>s</sup><sup>i</sup>*

be the standard deviation value *σ<sup>i</sup>*

will sample the Gaussian PDF *g<sup>i</sup>*

dimensions of the file.

is done through a Gaussian function parametrized by the *µ<sup>i</sup>*

 *k* ∑ *r*=1 *ωr*

that is, the probability of each solution being chosen can be thought as a random number generator of normal distribution, with mean 1 and standard deviation *q* × *Fs*, since the choose probability of each rank will never change. Adopting this strategy, the *ω* vector as

Thus, since the ant chosen its solution, it must be sampled stepwise using a normal random number generator. The chosen solution must be sampled dimensionally

Therefore, let *sl* (Fig. 1) to be the solution chosen by an ant during the ACO's evolution process. It is known that the ant will sample *sl* dimensionally, as well as that the sampling

> *k* ∑ *e*=1

*<sup>l</sup>* is the mean distance from *<sup>s</sup><sup>i</sup>*


*i* in the other solutions of the file. The process is repeated until the last dimension of the file is reached. This way, the higher the variability of the different solutions, higher will

The parameter *ξ* is the same for all dimensions of all solutions, and corresponds to the pheromone evaporation rate, or to the inverse of the learning rate. This way, when *ξ* is low valued the algorithm speed is enhanced, and when it is high-valued, its robustness is enhanced. It is noteworthy that the algorithm converges when *σ* → 0 throughout all

• *PheromoneUpdate()*: The ACO**R** algorithm updates its pheromone informations as follows: At the beginning of the algorithm, the file is initialized with *Fs* solutions uniformly random distributed. From this, the pheromone updating is done adding the new solutions

Finally, the size of the solutions file is a parameter of the algorithm, and must not be smaller than the number of dimensions of the problem if it is enabled to handle variable correlation, and to support linear transformations on the problem being optimized. Nevertheless, these techniques are not used in this work. Furthermore, the size of the file leads to the algorithm diversity, since a big file will cover a greater region of the search space than the small one, enabling the algorithm to overcome local optima, but on

generated by the ants, as well as removing the same number of worst solutions.

the other hand, a small file will make the algorithm to be faster than the big one.

*<sup>l</sup>* and *<sup>σ</sup><sup>i</sup>*

. Note that the *ξ* ∈ [0, 1] parameter aims to reduce the

*<sup>k</sup>* <sup>−</sup> <sup>1</sup> (25)

*<sup>l</sup>* to the other values of dimension

*<sup>l</sup>* value is calculated, the ant

*<sup>l</sup>* values. Thus, the *<sup>σ</sup><sup>i</sup>*

, *i* = 1, . . . , *U*), causing each Gaussian mixture's parameters to be seen only in one dimension a time, smoothing the calculation of the pattern deviation and allowing linear

<sup>−</sup><sup>1</sup>

(24)

*l*

This subsection is divided in two parts. The first one deals with ACO typical performance and its input parameters optimization; the second part numerical simulations results for both power and rate allocation problems are presented and the NMSE is compared with the RA-PSO algorithm.

The simulations were carried out in the MatLab 7.0 platform and the scenario parameters are presented on Table 1. We assumed a rectangular cell with one base station in the center and users uniformly spread across all the cell extension. We considered that all mobile terminals experience slow fading channels, i.e. the following relation is always satisfied:

$$T\_{\rm slot} < (\Delta t)\_{\rm c} \tag{26}$$

where *<sup>T</sup>*slot <sup>=</sup> *<sup>R</sup>*−<sup>1</sup> slot is the time slot duration, *R*slot is the transmitted power vector update rate, and (∆*t*)*<sup>c</sup>* is the coherence time of the channel2. This is part of the SNIR estimation process, which means that the channel is constant in each optimization window, assumed herein equal to 667*µs*. Thus, the ACO algorithm must converge to the solution within each 667*µs* interval.

## *2.6.1. RA-ACO Input Parameters Optimization*

Simulation experiments were carried out in order to determine the suitable values for the ACO input parameters for each problem, such as file size (*Fs*), pheromone evaporation coefficient (*ξ*), population (*m*) and the diversity parameter (*q*). The best parameters combination was chosen considering the solutions quality measured by the normalized mean squared error (NMSE), defined as:

$$\text{NMSE} = \mathbb{E}\left[\frac{||\mathbf{p} - \mathbf{p}^\*||^2}{||\mathbf{p}^\*||}\right] \tag{27}$$

where **p** is the solution found through the ACO algorithm, **p**<sup>∗</sup> the analytical (optimal) solution and **E** the mathematical expectation operator. In order to find the best parameters for both problems non-exhaustive tests were conducted.

A typical convergence behavior for the RA-ACO under equal-rate power control problem is shown in Fig. 2. At a first glance, power allocation for *U* = 5 users (lightly loading system) is performed by a) RA-ACO algorithm, b) RA-PSO algorithm from [1]. One can

<sup>2</sup> Corresponds to the time interval in which the channel characteristics do not suffer expressive variations.


**Power Allocation (PA) Problem.** Under the first resource allocation problem posed by Eq. (9) or (17), Fig. 3 depicts the associated NMSE under different ACO input parameter values combination taking into account different loading system, i.e., *U* = 5, 10 and 20, respectively. Note that the population size *m* and file size *Fs* parameters (*m*, *Fs* ∈ **N**), both with entry values common for all the different {*q*, *ξ*} input parameters configurations, where chosen based on the problem dimensionality. Numerical experiments have shown that different entries around the ones chosen do not affect substantially the NMSE results as the different entries for *q* and *ξ* parameters do. It is worth noting that the PA problem in (9) presents a non-convex characteristic; hence, the value entries for the population size *m* and file size *Fs* parameters assume relative high values regarding the dimensions of the problem, meaning that both parameters are of the order of problem dimension, {*m*, *Fs*}≈O[*U*]. It means that RA-ACO can solve the non-convex PA problem in DS/CDMA systems but with input

Ant Colony Optimization for Resource Allocation and Anomaly Detection in Communication Networks

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123

Herein, a parameter calibration strategy was adopted in order to find the best tradeoff for the {*q*; *Fs*} set, given in Eq. (23). Since the parameters *Fs* and *m* are directly related to the computational complexity of the algorithm, finding a suitable parameter set with *Fs* entries

On the other hand, the population size *m* parameter has a small or even no influence on the any other ACO input parameter (as *q* and *Fs* interfere each other). Although the *m* entries values directly increases the algorithm computational complexity. Therefore, the parameters *m* and *Fs* were fixed at low values and then the best *q* and *ξ* combination for it was sought. Hence, based on the NMSE *versus* convergence speed results obtained in Fig. 3, the optimized RA-ACO input *q* and *ξ* parameters for the power control problem in DS/CDMA networks

> *U* (users) 5 10 20 *q* 0.61 0.40 0.40 *ξ* 1.00 0.82 0.75 *m* 7 15 35 *Fs* 8 4 25 Robustness, R 100 % 100 % 30 %

Also, the robustness achieved by the RA-ACO for the power allocation problem is added to the Table 2. Herein, the success of convergence is reached when the NMSE of the algorithm's solution goes less than 10<sup>−</sup>2. Due to the non-convexity of the PA problem in (17), when the number of users grows from 10 to 20, the needed robustnes grows exponentially, thus, the

**Weighted Throughput Maximization (WTM) Problem.** For the weighted throughput maximization (WTM) problem posed in Eq. (21), Figure 4 shows different cost function evolutions when parameters *q* and *ξ* are combined under three distinct system loading, *U* = 20, 100 and 250 users. The average cost function evolution values where taken over

From Fig. 4-a it is clear that for *U* = 20 users, the *q* = 0.10 and *ξ* = 1.00 choice results in an average cost function value higher than the other ones. Besides, even in a

T = 1000 trials. Also, the correspondent sum rate difference (<sup>∆</sup> <sup>∑</sup>rate) is zoomed in.

under different level of interference could be found, as summarized in Table 2.

**Table 2.** Optimized RA-ACO input parameters and respective robustness for the Problem of Eq. (17).

algorithm's performance have a critical decay of 70%.

parameter loads relatively high.

as low as possible is of great interest.

**Table 1.** Multirate DS/CDMA system, channel and ACO input parameters

see the smooth-monotonic convergence of the RA-ACO algorithm toward the optimal power solution, in this case given by (16), in contrast to the non-monotonic oscillated convergence behavior presented by the RA-PSO algorithm. Besides, for *U* = 5 users power allocation problem, the ACO was able to achieve convergence after ≈ 250 iterations in contrast to the ≈ 450 iterations necessary for the RA-PSO convergence.

**Figure 2.** Power allocation for *U* = 5 users. Equally information rate among users is adopted. a) RA-ACO; b) RA-PSO algorithm from [1].

**Power Allocation (PA) Problem.** Under the first resource allocation problem posed by Eq. (9) or (17), Fig. 3 depicts the associated NMSE under different ACO input parameter values combination taking into account different loading system, i.e., *U* = 5, 10 and 20, respectively.

14 Search Algorithms

**Parameters Adopted Values**

Noise Power *Pn* = −63 [dBm] Chip rate *rc* = 3.84 × 10<sup>6</sup> Min. Signal-noise ratio *SNR*min = 4 dB Max. power per user *P*max = 1 [W] Min. Power per user *P*min = 0 [W]
