Spatial Frequency Representation

Chapter 2

Abstract

Upendra Kumar

distribution, urban planning

niques have been developed [1].

existing evaluation indicators.

2. Pan-sharpening techniques

1. Introduction

11

Pan-sharpening Using

Spatial-frequency Method

spectral fidelity pan-sharpened images over spatial-scale methods.

Keywords: pan sharpening, spatial scale, spatial frequency analysis, discrete wavelet transform, non–subsampled contourlet transform, pseudo-Wigner

Earth resource satellites provide data covering different parts of the electromagnetic spectrum at different spatial, spectral, and temporal resolutions. To utilize these different types of image data effectively, a number of pan-sharpening tech-

Further, in order to benchmark different image fusion techniques, image quality

metrics have been used. There are two types of metrics used to evaluate image quality, namely, subjective (qualitative) and objective (quantitative). The objective of this chapter is to discuss the methodology of some of the prevalent existing techniques, as well as the mathematical representation of some of the standard

Pan sharpening is also known as image fusion, image integration, and multisensor data fusion. Over the years, a large number of pan-sharpening techniques have been developed and have placed into different categorizes. In this study, multiscale transform (MST)-based techniques have been discussed.

Over the years, researchers have formulated various techniques for pan sharpening that attempt to minimize the spectral distortion, i.e., retain the maximum spectral fidelity of the MS images. On the other hand, if the use of the PAN-sharpened image is just to produce maps for better visual interpretation, then the spectral distortion is not of much concern, as the goal is to produce images with high contrast. To solve the color distortion problem, methods based on spatial frequency domain have been introduced and have demonstrated superior performance in terms of producing high

## Chapter 2

## Pan-sharpening Using Spatial-frequency Method

Upendra Kumar

## Abstract

Over the years, researchers have formulated various techniques for pan sharpening that attempt to minimize the spectral distortion, i.e., retain the maximum spectral fidelity of the MS images. On the other hand, if the use of the PAN-sharpened image is just to produce maps for better visual interpretation, then the spectral distortion is not of much concern, as the goal is to produce images with high contrast. To solve the color distortion problem, methods based on spatial frequency domain have been introduced and have demonstrated superior performance in terms of producing high spectral fidelity pan-sharpened images over spatial-scale methods.

Keywords: pan sharpening, spatial scale, spatial frequency analysis, discrete wavelet transform, non–subsampled contourlet transform, pseudo-Wigner distribution, urban planning

## 1. Introduction

Earth resource satellites provide data covering different parts of the electromagnetic spectrum at different spatial, spectral, and temporal resolutions. To utilize these different types of image data effectively, a number of pan-sharpening techniques have been developed [1].

Further, in order to benchmark different image fusion techniques, image quality metrics have been used. There are two types of metrics used to evaluate image quality, namely, subjective (qualitative) and objective (quantitative). The objective of this chapter is to discuss the methodology of some of the prevalent existing techniques, as well as the mathematical representation of some of the standard existing evaluation indicators.

### 2. Pan-sharpening techniques

Pan sharpening is also known as image fusion, image integration, and multisensor data fusion. Over the years, a large number of pan-sharpening techniques have been developed and have placed into different categorizes. In this study, multiscale transform (MST)-based techniques have been discussed.

#### 2.1 Multiscale transform-based pan-sharpening techniques

In recent years, multiscale transform (MST)-based pan-sharpening techniques have received a lot of attention, since they preserve the spectral fidelity in the pansharpened images. Further, it is more suitable for information representation, interpretation, and analysis [2, 3].

Many variations of the multiscale transform-based techniques exist, such as discrete wavelet transform (DWT), stationary wavelet transform (SWT), curvelet transform (CVT), contourlet transform (CT), and Non–subsampled contourlet transform (NSCT) [4]. The next subsections give a descriptive overview and methodology of MST-based pan-sharpening techniques which are selected for this study.

#### 2.1.1 Discrete wavelet transform (DWT)

Before discussing about discrete wavelet transform, first of all, it would be appropriate to discuss in general regarding Fourier transform (FT).

Fourier transform (FT) was first invented by French mathematician and physicist Jean Baptiste Joseph Fourier in 1822. Fourier stated that any periodical function can be represented as a sum of sine and cosine of different frequencies, each multiplied by a different coefficient [5, 6]. Fourier transform converts a signal from the time-amplitude domain to the frequency-amplitude domain. Images are considered as 2-D discrete functions. To use Fourier transform to analyze images, discrete Fourier transform (DFT) is used. FT is a reversible transform, which means the original signal can be recovered through the inverse discrete Fourier transform (IDFT) [7, 8].

However, FT has a drawback, i.e., it does not provide the information about the time at which the particular frequency exists in the signal. Fourier transform only captures the different frequencies in a signal and cannot detect when those frequencies occurred. To overcome this drawback, wavelet transform (WT) was introduced. Wavelet transform (WT) can be more useful than Fourier transform, since it is based on functions that are localized in both space and frequency/scale [9]. Wavelet transform brings a multiresolution framework. With this setting, the signal can be decomposed into components that collect the information at a specified scale, i.e., different frequencies are analyzed with different resolutions [2–6]. The WT has numerous applications in remote sensing such as image registration, spatial and spectral fusion, feature extraction, speckle reduction, texture classification, and crop phenology detection [7].

Wavelet transform can be broadly classified into two main groups, i.e., continuous wavelet transform (CWT) and discrete wavelet transform (DWT). Since CWT is continuous, as a result, there are an infinite number of scale and translation parameters which leads to an infinite number of possible wavelet functions. To overcome the shortcoming of CWT, DWT was introduced.

In the DWT algorithm, an image can be analyzed by passing it through an analysis filter bank followed by decimation operation. The analysis filter bank consists of low pass and high pass filter at each decomposition stage. When a signal passes through these filters, it splits in to two signals. The low pass filter, which corresponds to an averaging operation, extracts the coarse (average) information of the signal. The high pass filter, which corresponds to a differencing operation, extracts the detail information of the signal such as edges, points, and lines. The output of the filtering operation is then decimated by two, i.e., a 2-D transform is accomplished by performing two separate one-dimensional transform [9–12]. First

Pan-sharpening Using Spatial-frequency Method DOI: http://dx.doi.org/10.5772/intechopen.80637

of all, the image is filtered along the row and decimated by two, and it is then followed by filtering the subimage along the column and decimated by two.

This operation splits the image into four bands namely one approximation band, which contains coarse information and three detail bands, horizontal, vertical, and diagonal, respectively, which contain information about the salient features of the image such as edges, points, and lines [5, 8]. A J-level decomposition can be performed resulting in ð3j þ 1Þ different frequency bands. At each level of decomposition, the image is split into high and low frequency components; the lowfrequency components can be further decomposed until the desired resolution is reached [13–15]. The pan-sharpening procedure for the pan sharpening of panchromatic (PAN) and multispectral (MS) images using DWT has been explained in Section 3.1 (Figure 1).

#### 2.1.2 Stationary wavelet transform (SWT)

It is observed that discrete wavelet transform (DWT) is not a shift-invariant transform. Therefore, in order to get rid of this problem, stationary wavelet transform (SWT)-based fusion technique, an extension of DWT scheme, also known as "à trous" algorithm, has been introduced [10, 11]. In the "à trous" algorithm, the downsampling step is suppressed and instead the filter is upsampled by inserting zeros between the filter coefficients (Figure 2).

In the SWT algorithm, it uses a two-dimensional filter derived from the scaling function. This produces two images, of which one is an approximation image while the other is a detailed image called the wavelet plane. A wavelet plane represents the horizontal, vertical, and diagonal detail between 2<sup>j</sup> and 2<sup>j</sup>�<sup>1</sup> resolution and is

Figure 1. Decomposition of an image using DWT.

Figure 2. Structure of "à trous" filters.

computed as the difference between two consecutive approximations Il˜<sup>1</sup> and Il levels. All the approximation images obtained, by applying this decomposition, have the same number of columns and rows as the original image, since filters at each level are upsampled by inserting zeros between the filter coefficients and make the size of the image same [16–19].

This is a consequence of the fact that the "à trous" algorithm is a nonorthogonal, redundant oversampled transform [19–21]. The "à trous" decomposition process is shown in Figure 2.

The procedure for the pan sharpening of PAN and MS images using SWT can be summarized as follows (Figure 3):


The SWT eliminates the shift sensitivity problem at the cost of an overcomplete signal representation. However, it does not resolve the problem of feature orientation. In addition, the discrete wavelet transform (DWT), and stationary wavelet transform (SWT), cannot capture curves and edges of images well. Wavelets perform well only at representing point singularities, i.e., appropriate to represent linear edges, since they ignore the geometric properties of structures and do not exploit the regularity of edges.

For curved edges, the accuracy of edge localization in the wavelet transform is low. So, there is a need for an alternative approach, which has the potential or capability to detect, represent, and process high-dimensional data. In order to solve this problem, multiscale geometric analysis has been further investigated. As a result, Candès and Donoho [22] have proposed the concept of curvelet transform (CVT).

Further, in order to solve the problem of curvelet transform, which is first developed in continuous domain and then does discretization of images or signals of interest, Yang et al. [23] and Do and Vetterli [24] presented a flexible multiresolution, local, and directional image expansion using contour segments, named contourlet transform. However, due to the downsampling and upsampling, the CT lacks shift invariance and thus results in ringing artifacts [16]. To overcome the weakness of wavelets, curvelets, and contourlets, Cunha et al. [25] proposed

Figure 3. Methodology adopted for SWT-based pan-sharpening.

non–subsampled contourlet transform (NSCT), based on non–subsampled pyramid decomposition (NSPD) and non–subsampled filter bank (NSFB).

### 2.1.3 Non–subsampled contourlet transform (NSCT) technique

In order to reduce the frequency aliasing of contourlets and enhance directional selectivity and shift invariance, Holschneider and Tchamitchian [17] proposed non–subsampled contourlet transform. This is based on the non– subsampled pyramid filter banks (NSPFBs) and the non–subsampled directional filter banks (NSDFBs) structure. The former provides multiscale decomposition using two-channel non–subsampled 2-D filter banks, while the later provides directional decomposition, i.e., it is used to split band pass subbands in each scale into different directions [25, 26].

As a result, NSCT is shift invariant and leads to have better frequency selectivity and regularity than CT [25–28]. The scheme of NSCT structure is shown in Figure 4 (a). The NSCT structure classifies two-dimensional frequency domain into wedgeshaped directional subband as shown in Figure 4(b).

In order to provide more practical and flexible solution to the existing problem as stated above, there is a need for an improved or a new fusion technique, which is superior among all the existing pan-sharpening techniques. A new pan-sharpening technique should ideally possess properties of shift invariance, directionality, low computational complexity, and low computational time, applicable to real-time image processing tool, and is also efficient in capturing intrinsic geometrical structures of the natural image along the smooth contours. Moreover, it should perform efficiently under all categories of datasets, such as very high, high, and medium resolution satellite datasets. A spatial frequency-based technique should ideally possess properties, such as shift invariance, directionality, low computational complexity, and low computational time, applicable to real-time image processing tool, and is also efficient in capturing intrinsic geometrical structures of the natural image along the smooth contours [27, 28]. Thus, in order to resolve the existing problems, pan-sharpening method based on joint spatial frequency domain such as pseudo-Wigner distribution has been introduced.

## 3. Spatial-frequency based pan-sharpening technique

Analysis of non-stationary 2-D signals (image) is a challenging job, as their spectral properties change with time. Such signals cannot be analyzed well by pure spatial domain and frequency domain representations. The joint spatial frequency domain-based image analysis methods, such as Wigner Ville distribution (WVD)

Figure 4.

Two level NSCT decomposition. (a) NSFB structure that implements the NSCT and (b) the corresponding frequency partition.

and pseudo-Wigner distribution (PWD), have been proven to be a powerful tool for analyzing, understanding, and detection of spatial frequency characteristics of non-stationary images in a more comprehensive manner.

The use of Wigner Ville distribution for image processing was first suggested by [18]. It was shown that WVD is a very efficient and powerful tool for capturing the essential non-stationary image structures [29] and appears as a new promising method for the characterization of local spectral properties of images. The Wigner Ville distribution has many interesting properties related to translation, modulation, scaling, convolution, and localization in spatial frequency space, real-valued function and contains phase information, which motivates its use in the field of image analysis applications. Since WVD suffers with the serious problem of interference that makes the interpretation impossible, thus to resolve the limitation of WVD, pseudo-Wigner distribution (PWD) was introduced.

## 3.1 Pseudo-Wigner Distribution (PWD) technique

Spatial frequency information of a non-stationary image can be effectively extracted with one of the well-known spatial frequency technique known as pseudo-Wigner distribution (PWD). PWD is ideally suited for representing a nonstationary image in the spatial frequency domain and is carried out by adapting the fast Fourier transform (FFT) algorithm. The significant properties of PWD motivate its use in the field of image processing, especially for the fusion of satellite images [30, 31]. These properties are as follows:


With reference to Table 1, pseudo-Wigner distribution (PWD) overcomes the shortcomings of the traditional Fourier-based methods, discrete wavelet transform (DWT), stationary wavelet transform (SWT), curvelet transform (CT), contourlet

Figure 5. Concept of shift variant and shift invariant.

Pan-sharpening Using Spatial-frequency Method DOI: http://dx.doi.org/10.5772/intechopen.80637


Table 1.

Shortcomings of existing pan-sharpening methods.

transform (CT), and non–subsampled contourlet transform (NSCT). Consequently, it is not based on a multiscale decomposition procedure as wavelets and contourlets are. Further, one of the most challenging applications that comes across by the remote sensing experts is to fuse MS and PAN images collected from different or same satellite sensor with each other to achieve a pan-sharpened image, without introducing artifacts or inconsistencies; otherwise it may damage the quality of the fused image.

Thus, the goal of pan-sharpening is to produce pan-sharpened images with the highest spectral fidelity possible, as the importance of such images in various applications, ranging from land use/land cover classification to road extraction. Therefore, preserving the spectral information of the original MS images in the pansharpened images is of great importance [31–33]. Therefore, an attempt to utilize the concept of pseudo-Wigner distribution (PWD) for the pan-sharpening of highresolution PAN image with a low-resolution MS image has been introduced.

#### 3.1.1 Mathematical background of pseudo-Wigner distribution

Let us consider an arbitrary 1-D discrete function v nð Þ. The PWD of a given array v nð Þ of N pixels is given by Eq. (1).

$$W(n,m) = 2\sum\_{k=-\frac{N}{2}}^{\frac{N}{2}-1} v(n+k)v^\*(n-k) \times \exp\left(-2i\left(\frac{2\pi k}{N}\right)m\right) \tag{1}$$

where n and m represent the spatial and frequency discrete variables,respectively, and k is a shifting parameter. Eq. (1) can be interpreted asthe discrete Fouriertransform (DFT) of the product v nð þ <sup>k</sup>Þv∗ð<sup>n</sup> � <sup>k</sup>Þ. Here, <sup>v</sup><sup>∗</sup> indicatesthe complex conjugate of 1-D sequence, v.Wðn; mÞis a matrixwhere every row representsthe pixel-wise PWD of the pixel at position n. Further, v n½ � is a 1-D sequence of data from the image, containing the gray values ofNpixels, aligned in the desired direction.By scanning the imagewith a 1-D window of N pixels, i.e.,shifting the window to all possible positions overthe full image, the full pixel-wise PWD of the image is produced. The window can be tilted in any direction to obtain a directional distribution [34, 35]. Further, the reasonsfor selecting short 1-D window for PWD analysis are asfollows:


The general pan-sharpening procedure adopted for the pan sharpening of PAN and MS images using DWT, NSCT, and PWD [35] techniques can be summarized as follows (Figure 6):

i. Coregister both the source images and resample the multispectral image to make its pixel size equal to that of the PAN, in order to avoid the problem of misregistration.

#### Figure 6.

General methodology adopted for DWT-, NSCT-, and PWD-based pan-sharpening.


As a result, a new multispectral image with higher spatial resolution is obtained. This process is repeated for each individual MS and PAN band pair. Finally, all the new fused bands are concatenated to form a new fused multispectral image.

Pan-sharpening Using Spatial-frequency Method DOI: http://dx.doi.org/10.5772/intechopen.80637


#### Table 2.

Assessment of image quality by qualitative method.

It may be noted that each MST technique (DWT, NSCT, and PWD) has its unique mathematical properties, which leads to different image decomposition procedure of an image.

#### 3.2 Comparative assessment of various pan-sharpening techniques

Pan-sharpening techniques, belonging to color, statistical, and multiscale transform-based techniques, have been evaluated in terms of certain parameters, such as spectral distortion, shift invariance, directionality, and computational complexity. Comparative assessment of various pan-sharpening techniques has been shown in Table 2.

#### 4. Fusion rules

There are various fusion rules to combine the fused coefficients. Let <sup>W</sup><sup>P</sup> <sup>ð</sup>x; <sup>y</sup><sup>Þ</sup> <sup>A</sup> and WMS <sup>B</sup> ðx; yÞ denote the coefficients for higher spatial resolution PAN image and for the lower spatial resolution MS image, and <sup>W</sup><sup>F</sup>ðx; <sup>y</sup><sup>Þ</sup> denotes the coefficient of the fused image. Using these notations, following fusion rules can be summarized as follows:

#### i. Average fusion rule

The average fusion rule takes the average of the coefficients of the <sup>W</sup><sup>P</sup> <sup>ð</sup>x; <sup>y</sup>Þ, <sup>A</sup> PAN, and <sup>W</sup>MSðx; <sup>y</sup>Þ, MS images, which is given by Eq. (2). <sup>B</sup>

$$\mathcal{W}^F(\mathbf{x}, \boldsymbol{\mathcal{y}}) = \left(\mathcal{W}\_A^{\text{PAN}}(\mathbf{x}, \boldsymbol{\mathcal{y}}) + \mathcal{W}\_B^{\text{MS}}(\mathbf{x}, \boldsymbol{\mathcal{y}})\right) / \mathcal{D} \tag{2}$$

#### ii. Maximum fusion rule

The maximum fusion rule compares the coefficients from the <sup>W</sup>PANðx; <sup>y</sup>Þ, PAN, <sup>A</sup> and WMS <sup>B</sup> ðx; yÞ, MS images, and picks the larger magnitudes as the fused coefficients, which is given by Eq. (3).

$$\boldsymbol{W}^{F}(\boldsymbol{x},\boldsymbol{y}) = \begin{cases} \boldsymbol{W}\_{A}^{\mathrm{PANS}}(\boldsymbol{x},\boldsymbol{y}), \text{ if } \left| \boldsymbol{W}\_{A}^{\mathrm{P}}(\boldsymbol{x},\boldsymbol{y}) \right| \succ \left| \boldsymbol{W}\_{B}^{\mathrm{MS}}(\boldsymbol{x},\boldsymbol{y}) \right| \\\ \boldsymbol{W}\_{B}^{\mathrm{MS}}(\boldsymbol{x},\boldsymbol{y}), \text{ if } \left| \boldsymbol{W}\_{A}^{\mathrm{P}}(\boldsymbol{x},\boldsymbol{y}) \right| \leq \left| \boldsymbol{W}\_{B}^{\mathrm{MS}}(\boldsymbol{x},\boldsymbol{y}) \right| \end{cases} \tag{3}$$

Here, both the fusion rules are chosen as the basic fusion rule throughout this study, which are explained by Eqs. (2) and (3).

#### 5. Assessment of accuracy for pan-sharpening techniques

Pan-sharpening algorithms are designed to produce good-quality pansharpened images. A fused image would be considered perfect quality if the spatial detail missing in the MS image is transferred from the panchromatic image without distorting the spectral content of the multispectral image [26]. Unfortunately, this is not possible. There is a trade-off between enhancement of spatial detail and spectral distortion. A fully spatially enhanced fused image would be the panchromatic (PAN) image itself, while an image free of spectral distortion would be the original multispectral (MS) image [36].

The diversity of datasets has contributed to the development of different types of techniques and procedures for the implementation of image fusion. In order to benchmark different pan-sharpening techniques, image quality metrics have been used, i.e., quality metrics are required to evaluate the quality of the fused images [37, 38]. There are two types of metrics used to evaluate image quality:


#### 5.1 Qualitative evaluation

Qualitative analysis deals with the visual comparison of the original PAN and MS images with that of the fused image, in terms of spectral and spatial distortion. The evaluation results vary depending on the intensity, sharpness, existence of noisy areas, missing spatial detail, and distortions in the geometry of the objects and display conditions of the image. A number of viewers will be shown the images and asked to judge the image quality. These may also vary from observer to observer, i.e., interpretation of image quality may be influenced or varied by personal preference [39, 40]. Therefore, an exact decision cannot be given. Further, these methods are time-consuming, inconvenient, and expensive.

On the basis of expert/observer personal preference, quality of fused image has been ranked in terms of "Grade," "Absolute Measure," and "Relative Measure" [41], as shown in Table 2.

#### 5.2 Quantitative evaluation metrics

It is evident that, in most cases, there is slight difference among fusion results, i.e., quantitative evaluation methods sometimes produce results that cannot be sustained by visual inspection. However, there is no universally accepted metric to objectively evaluate the image fusion results. The generated pan-sharpened images are compared from diverse perspectives of image visualization, coherence, structural similarity, and spectral information content.

The well-known full-reference objective metrics are correlation coefficient (CC), root mean square error, peak signal-to-noise ratio [41]. The reason behind selecting these evaluation indicators is that they measure the statistical, structural similarity, and spectral distortion introduced by the pan-sharpening process. The quantitative metrics that are used in this study, as well as the mathematical representation of these measures, have been discussed below.

#### 5.2.1 Root mean square error

Root mean square error (RMSE) is a frequently used measure of the differences between the fused and the original images. RMSE is a good measure of accuracy [41]. Smaller RMSE value represents a greater accuracy measure and is explained by Eq. (4).

$$RMSE = \sqrt{\sum\_{i=1}^{m} \sum\_{j=1}^{n} \frac{\left(F(i,j) - R\_o(i,j)\right)^2}{c}} \tag{4}$$

where m � n indicates size of the image and F ið Þ ; j and Roð Þ i; j indicate the fused image and the original image, respectively.

#### 5.2.2 Peak signal-to-noise ratio

Peak signal-to-noise ratio (PSNR) indices reveal that the radiometric distortion of the fused image is compared to the original image. PSNR can reflect the quality of reconstruction. The larger value of PSNR indicates less amount of image distortion [41] and is given by Eq. (5).

$$PSNR = 10\ \log\left(\frac{L}{RMSE}\right)^2\tag{5}$$

where L is related to the radiometric resolution of the sensor; for example, L is 255 for an 8-bit sensor and 2047 for a 16-bit sensor.

#### 5.2.3 Correlation coefficient

The correlation coefficient (CC) of two images is often used to indicate their degree of correlation. If the correlation coefficient of two images approaches one, it indicates that the fused image and original image match perfectly [40, 41]. High value of the correlation shows that the spectral characteristic of the multispectral image has been preserved well. The correlation coefficient is represented by Eq. (6)

$$corr(\mathbf{x}, \boldsymbol{y}) = \frac{\sum\_{i=1}^{m} \sum\_{j=1}^{n} (\mathbf{x}(i, j) - \overline{\mathbf{x}}) \left( \boldsymbol{y}(i, j) - \overline{\mathbf{y}} \right)}{\sqrt{\sum\_{i=1}^{m} \sum\_{j=1}^{n} \left( \mathbf{x}(i, j) - \overline{\mathbf{x}} \right)^{2} \sum\_{i=1}^{m} \sum\_{j=1}^{n} \left( \boldsymbol{y}(i, j) - \overline{\mathbf{y}} \right)^{2}}} \tag{6}$$

where x ið Þ ; j and y ið Þ ; j are the elements of the images x and y, respectively, and x and y stand for their mean values.

#### 5.2.4 Spatial correlation coefficient

In order to assess the spatial quality of the fused image quantitatively, procedure proposed by [42] has been adopted. This approach is used to measure the amount of edge information from the PAN image, which is transferred into the fused images. The high spatial resolution information missing in the MS image is present in the high frequencies of the PAN image. The pan-sharpening process inserts the higher frequencies from the PAN image into the MS image. Therefore, the CC between the high pass filtered PAN and the fused images would indicate how much spatial information from the PAN image has been incorporated into the MS image. A higher correlation between the two high pass filtered images implies that the spatial


Table 3.

The ideal and error value of different quantitative indicators.

information has been retained faithfully. This CC is called the spatial correlation coefficient (SCC). In order to extract the spatial detail of the images to be compared, following Laplacian filter has been used and is represented by Eq. (7).

$$\mathbf{M}\mathbf{s}\mathbf{k} = \begin{bmatrix} -\mathbf{1} & -\mathbf{1} & -\mathbf{1} \\ -\mathbf{1} & \mathbf{8} & -\mathbf{1} \\ -\mathbf{1} & -\mathbf{1} & -\mathbf{1} \end{bmatrix} \tag{7}$$

The pan-sharpened image which will best preserve the spectral and structural information of the original low resolution MS image is the one that has satisfied the following conditions (Table 3).

#### 6. Summary

This chapter provides the methodology of the proposed approaches for the pansharpening of satellite images, along with the discussion of some prevalent existing multisensor pan-sharpening techniques and well-known evaluation indicators.

### Author details

Upendra Kumar Ambalika Institute of Management and Technology, Lucknow, India

\*Address all correspondence to: upendra2122@gmail.com

© 2019 The Author(s). Licensee IntechOpen. This chapteris distributed underthe terms oftheCreative 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.

Pan-sharpening Using Spatial-frequency Method DOI: http://dx.doi.org/10.5772/intechopen.80637

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[27] Ding S, Zhao X, Xu H, Zhu Q , Xue Y. NSCT-PCNN image fusion based on image gradient motivation. IET Computer Vision. 2017;12(4):377-383

[28] Shabanzade F, Ghassemian H. Combination of wavelet and contourlet transforms for PET and MRI image fusion. In: Artificial Intelligence and Signal Processing Conference (AISP). IEEE; 2017. pp. 178-183

[29] Claasen TACM, Mecklenbrauker WFG. The Wigner distribution—A tool for time–frequency analysis. Philips Journal of Research. 1980;35(3):217-250

[30] Gabarda S, Cristobal G. On the use of a joint spatial-frequency representation for the fusion of multifocus images. Pattern Recognition Letters. 2005;26(16):2572-2578

[31] Rajput UK, Ghosh SK, Kumar A. Multi-sensor fusion of satellite images for urban information extraction using pseudo-Wigner distribution. Journal of Applied Remote Sensing. 2014;8(1): 083-668

[32] Gabarda S, Cristóbal G. Blind image quality assessment through anisotropy. Journal Optical of Society America A. 2007;24(12):B42-B51

[33] Redondo R, Fischer S, Sroubek F, Cristobal G. 2D Wigner distribution based multi-size windows technique for image fusion. Journal of Visual Communication and Image Representation. 2008;19(1):12-19

Pan-sharpening Using Spatial-frequency Method DOI: http://dx.doi.org/10.5772/intechopen.80637

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[36] Vijayaraj V, Younan N, O'Hara C. Quantitative analysis of pan-sharpened images. Optical Engineering. 2006; 45(4):46-202

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land sat TM and SPOT panchromatic data. International Journal of Remote Sensing. 1998;19(4):743-757

Chapter 3

Abstract

prediction

1. Introduction

often, e.g., each week.

27

Distortions

Benoit Vozel and Kacem Chehdi

verified for test and real life images.

Lossy Compression of Remote

Sensing Images with Controllable

Vladimir Lukin, Alexander Zemliachenko, Sergey Krivenko,

In this chapter, approaches to provide a desired quality of remote sensing images compressed in a lossy manner are considered. It is shown that, under certain conditions, this can be done automatically and quickly using prediction of coder performance parameters. The main parameters (metrics) are mean square error (MSE) or peak signal-to-noise ratio (PSNR) of introduced losses (distortions) although prediction of other important metrics is also possible. Having such a prediction, it becomes possible to set a quantization step of a coder in a proper manner to provide distortions of a desired level or less without compression/decompression iterations for single-channel image. It is shown that this approach can be also exploited in three-dimensional (3D) compression of multichannel images to produce a larger

compression ratio (CR) for the same or less introduced distortions as for component-wise compression of multichannel data. The proposed methods are

Keywords: lossy compression, remote sensing, image processing, performance

A huge amount of data is provided nowadays by existing remote sensing (RS) sensors, both spaceborne and airborne [1, 2]. Data volume is especially large if images are hyperspectral (i.e., having hundreds sub-band images) and/or high resolution ones. Note that both tendencies (to create and exploit multichannel systems as well as to produce high resolution data) are typical for recent years. Volume of acquired data additionally increases due to more frequent observations of sensed terrains [2]—it has become a usual practice to monitor a territory quite

The obtained RS data have to be transferred, stored and/or disseminated. For each of this operation, data compression can be desirable [1, 3, 4]. Meanwhile, there are several obstacles that can prevent efficient execution of these operations. Concerning data transferring: bandwidth of a communication channel used to transfer data can be limited, time for transferring can be restricted, time and power for compression can be limited as well [1, 3]. The same can relate to data dissemination although the limitations are usually less strict compared to downlink data

## Chapter 3

## Lossy Compression of Remote Sensing Images with Controllable Distortions

Vladimir Lukin, Alexander Zemliachenko, Sergey Krivenko, Benoit Vozel and Kacem Chehdi

## Abstract

In this chapter, approaches to provide a desired quality of remote sensing images compressed in a lossy manner are considered. It is shown that, under certain conditions, this can be done automatically and quickly using prediction of coder performance parameters. The main parameters (metrics) are mean square error (MSE) or peak signal-to-noise ratio (PSNR) of introduced losses (distortions) although prediction of other important metrics is also possible. Having such a prediction, it becomes possible to set a quantization step of a coder in a proper manner to provide distortions of a desired level or less without compression/decompression iterations for single-channel image. It is shown that this approach can be also exploited in three-dimensional (3D) compression of multichannel images to produce a larger compression ratio (CR) for the same or less introduced distortions as for component-wise compression of multichannel data. The proposed methods are verified for test and real life images.

Keywords: lossy compression, remote sensing, image processing, performance prediction

### 1. Introduction

A huge amount of data is provided nowadays by existing remote sensing (RS) sensors, both spaceborne and airborne [1, 2]. Data volume is especially large if images are hyperspectral (i.e., having hundreds sub-band images) and/or high resolution ones. Note that both tendencies (to create and exploit multichannel systems as well as to produce high resolution data) are typical for recent years. Volume of acquired data additionally increases due to more frequent observations of sensed terrains [2]—it has become a usual practice to monitor a territory quite often, e.g., each week.

The obtained RS data have to be transferred, stored and/or disseminated. For each of this operation, data compression can be desirable [1, 3, 4]. Meanwhile, there are several obstacles that can prevent efficient execution of these operations. Concerning data transferring: bandwidth of a communication channel used to transfer data can be limited, time for transferring can be restricted, time and power for compression can be limited as well [1, 3]. The same can relate to data dissemination although the limitations are usually less strict compared to downlink data

transferring. Memory for RS data storage can be a problem too despite of rapid development of new facilities in recent years [2].

Therefore, it is often desired to compress RS images [4, 5]. As known, there are lossless and lossy image compression techniques [1]. Limits attainable by lossless compression are practically reached [1]. Compression ratio (CR) for the existing methods rarely reaches 5 even for compressing hyperspectral data when inter-band correlation is exploited in full extent [4]. However, larger CR values are required often. Then, lossy compression of acquired RS data has to be applied.

The main peculiarity of lossy compression is that it introduces losses (distortions, degradations) into RS images. Then, it can be useful only under condition that introduced losses do not sufficiently negatively influence the goals the acquired RS data are intended for (terrain classification and/or parameter estimation, specific object detection, etc.). One assumption is that introduced losses have to be of the same level or smaller than degradations due to noise in original data [6]. Therefore, noise characteristics have to be taken into consideration and, thus, they should be known in advance or pre-estimated [7–11]. This also means that it is necessary to be able to control introduced distortions and/or to provide a desired level of losses. Moreover, often this should be done automatically, e.g., in on-board compression [3, 12].

A slightly other assumption is possible if compressed images are subject to visual inspection and analysis. Then, introduced distortions should be such that they do not degrade image visual quality [13]. Then, one has to take into account both specific properties of component images, e.g., variations of their dynamic range [7, 14, 15] and peculiarities of human vision system (HVS).

Finally, one more assumption is that introduced distortions should be such that they do not have (noticeable) negative impact on classification accuracy or performance of other operations of RS data processing at final stages. Note that classification accuracy reduction is connected with metrics characterizing introduced distortions [16].

Thus, introduced distortions should be controlled for all aforementioned strategies. Here by "controlled" we mean several aspects. First, distortions have to be measured or estimated or predicted to ensure that they are not larger than allowed threshold according to a certain metric (criterion) [17, 18]. Second, introduced distortions can be accurately measured only if compression and decompression are already done. Then, if distortion level has to be changed, coder parameters have to be changed and metric calculation has to be done after next iteration of compression/decompression [18]. This is often impractical, especially on-board. Then, it is more reasonable to talk about distortion estimation or prediction without compression and decompression but with approximate providing of a desired quality of compressed data.

Certainly, CR can be important as well. Then, an appropriate compromise has to be provided between CR and introduced losses. Note that CR also depends upon a used coder and a way data redundancy is exploited. In this sense, it is worth incorporating inter-channel correlation inherent for multichannel RS data that can be done in different ways [19–21]. It is possible to apply different transforms [11, 22–24] or to carry out different groupings of component images [11, 25, 26].

Lossy compression of images with taking into account noise type [27] and characteristics has been paid considerable attention [28–30]. Possible existence of optimal operation point (OOP) and its prediction have been claimed and studied [13, 18]. Problems of CR prediction and its providing for coders based on discrete cosine transform (DCT) have been considered [18, 31]. Meanwhile, problems of prediction of compressed image quality and providing a desired quality have not been thoroughly analyzed yet.

Lossy Compression of Remote Sensing Images with Controllable Distortions DOI: http://dx.doi.org/10.5772/intechopen.82361

In this direction, a certain work has been done. In particular, an approach to quality prediction for wavelet based compression of remote sensing images has been put forward [32]. Prediction of mean square error (MSE) of introduced losses for JPEG has been done [33]. However, control and prediction of metric values for more advanced coders as AGU [34] and ADCT [35] that outperform JPEG considerably [36] were not developed till last 2 years. Since providing of a desired metric value using iterative (multiple) compression/decompression requires sufficient time and resources [36], it was decided to design a new approach without iterations [37]. Later this approach has been further advanced [38–40], mainly for singlecomponent (grayscale) images in 8-bit representation and with taking into account possible presence of noise.

In this chapter, we consider application of the designed approach to RS images including multichannel data and keeping in mind the following: (1) dynamic range of component images in multichannel data varies in wide limits and 16-bit representation is often used for them; (2) in many component images of multichannel (e.g., hyperspectral) data, input peak signal-to-noise ratio (PSNR) is high and noise influence is negligible; (3) there is essential correlation of signal component in neighbor sub-band images of multichannel images. We show that by taking into account these properties, it is possible to carry out efficient compression of multichannel RS data with controllable quality.

### 2. Peculiarities of RS image lossy compression

To understand the problem of lossy compression, some preliminaries are needed.

First, lossy compression introduces distortions due to which a decompressed image differs from the corresponding original one (subject to compression). These distortions are introduced at the stage of quantization of coefficients of a used orthogonal transform: wavelet, DCT or some other [34, 35, 41]. If DCT serves as the basis of lossy compression, quantization step (QS) or scaling factor (SF) serve as parameter that controls compression (PCC). A larger QS or SF leads, in general, to greater introduced distortions and a larger CR [34, 35] but MSE of introduced losses and attained CR values considerably depend upon complexity of a compressed image and noise presence.

Figure 1 presents three images: noise-free image Frisco of low complexity, the same image corrupted with additive white Gaussian noise with zero mean and variance 100, and noise-free image Airfield of quite high complexity (it contains a lot of edges and fine details).

Figure 1. Noise-free and noisy (σ = 10) test images Frisco and the test image airfield.

Figure 2 shows dependences of mean square error MSEout between original and compressed images on QS for the case the advanced DCT (ADCT) coder [42] is applied. It is seen well that smaller distortions are introduced if an image is noisefree and has a simpler structure. The values of MSEoutðQSÞ for the same QS can differ by several times and, thus, i.e., QS itself does not determine MSEoutðQSÞ.

Dependences CRðQSÞ for the same images are presented in Figure 3. It is seen that the simple structure noise-free image Frisco is compressed in the best way whilst the complex structure image Airfield is compressed with the smallest CR. The reason is that the percentage of DCT coefficients that are assigned zero values after quantization increases if image complexity is lower, noise intensity is less, and QS is larger [31, 43]. Thus, the rate/distortion curve is individual for each particular image and QS has to be adapted to image and noise properties to provide a desired compromise or to satisfy imposed requirements.

We have already mentioned that compression of noisy images has several peculiarities. Suppose that an acquired (noisy) image in a k-th component is image is represented as [8, 10]

$$\mathbf{I}\_{\rm kj}^{\rm noisy} = \mathbf{I}\_{\rm kj}^{\rm true} + \mathbf{n}\_{\rm kj} \Big( \mathbf{I}\_{\rm kj}^{\rm true} \Big), \mathbf{i} = \mathbf{1}, \dots, \mathbf{I}, \mathbf{j} = \mathbf{1}, \dots, \mathbf{J}, \mathbf{k} = \mathbf{1}, \dots, \mathbf{K} \tag{1}$$

where I noisy is the ijth sample of the kth component image, nkij is the ijth value of kij the in the kth component image supposed dependent on I true - the true value for the kij kijth voxel, I and J define the image size, K is the number of components. One can determine input MSE for each component image as

Figure 2. Dependences MSE vs QS for noise-free and noisy images Frisco and noise-free image airfield.

Figure 3. Dependences CR vs QS for noise-free and noisy images Frisco and noise-free image airfield.

Lossy Compression of Remote Sensing Images with Controllable Distortions DOI: http://dx.doi.org/10.5772/intechopen.82361

$$\text{MSE}\_{\mathbf{k}}^{\text{imp}} = \sum\_{i=1}^{1} \sum\_{\mathbf{j}=1}^{I} \left( \text{I}\_{\text{kij}}^{\text{noisy}} - \text{I}\_{\text{kij}}^{\text{true}} \right)^{2} / (\text{I}), \quad \mathbf{k} = \mathbf{1}, \dots, \mathbf{K} \tag{2}$$

and, respectively, input PSNR

$$\text{PSNR}\_{\text{k}}^{\text{imp}} = \mathbf{10} \log\_{10} \left( \text{D}\_{\text{k}}^{2} / \text{MSE}\_{\text{k}}^{\text{imp}} \right), \text{k} = \text{1}, \dots, \text{K}, \tag{3}$$

where Dk is image dynamic range assumed individual for each component image (Dk ¼ I max I min � where <sup>I</sup> max and I min k k <sup>k</sup> <sup>k</sup> are maximal and minimal values in the kth image, respectively).

Earlier analysis [7, 44] has shown that MSEinp, <sup>k</sup> <sup>¼</sup> <sup>1</sup>, …,<sup>K</sup> and PSNRinp, <sup>k</sup> <sup>¼</sup> <sup>1</sup>, <sup>k</sup> <sup>k</sup> … ,K in very wide limits for such typical examples of multichannel RS data as images provided by hyperspectral sensors AVIRIS [45] and Hyperion [46]. For more than 80% of component images, input PSNR exceeds 40 dB. This means that, most probably [42], OOPs for these component images do not exist, i.e. ˜ °<sup>2</sup> I <sup>c</sup> <sup>I</sup> MSE<sup>c</sup> true <sup>k</sup> <sup>¼</sup> <sup>∑</sup><sup>I</sup> <sup>i</sup><sup>¼</sup> <sup>j</sup>¼<sup>1</sup> kij � kij <sup>=</sup>ð Þ steadily increases if QS becomes larger <sup>1</sup>∑<sup>J</sup> IJ (fI c kij, i ¼ 1, … ,I, j ¼ 1, … , J, k ¼ 1, … ,Kg denotes compressed image in a k-th channel; OOP exists for <sup>a</sup> k-th component image if MSE<sup>c</sup> <sup>ð</sup>QSÞ) has one minimum). <sup>k</sup>

If so, i.e. if quality of the compressed noisy image steadily decreases with QS growth, there should be some reasonable strategy to carry out compression for such an image or a group of images with similar properties. Here it is worth recalling the following. Analysis done in the paper [16] has shown that lossy compression has practically no negative impact on image classification accuracy if the metric PSNR-HVS-M [47] is not less than 42–44 dB.

<sup>k</sup>=MSEHVSM The metric PSNR-HVS-M (PSNR � HVS � <sup>M</sup><sup>c</sup> <sup>k</sup> <sup>¼</sup> <sup>10</sup> log <sup>10</sup>˛ D<sup>2</sup> ˝ , <sup>k</sup> <sup>k</sup> <sup>¼</sup> <sup>1</sup>, …,K, MSEHVSM is MSE with taking into consideration specific features of <sup>k</sup> human vision system (HVS)) takes into account two important peculiarities of human vision system: less sensitivity to degradations in high spatial frequencies and masking effect of textures. One can be surprised that visual quality metric has been used in analysis. This can be explained by the fact that the required values of PSNR-HVS-M > 42 dB mean that quality of a compressed image is such that introduced distortions are invisible. According to PSNR, this happens if PSNR<sup>c</sup> k exceeds 35–37 dB [48].

Thus, we need to provide a desired (controlled) quality of compressed images. This should be done quickly (desirably, without iterative compression/decompression), rather accurately, and with producing a large CR. We expect that CR increase can be gained due to grouping of component images.

#### 3. An approach to providing controlled losses

Let us start from considering lossy compression of a single-channel noise-free image in 8-bit representation. After compression, one obtains fI <sup>c</sup> <sup>1</sup>, … ,I, <sup>j</sup> <sup>¼</sup> <sup>1</sup>, kij, <sup>i</sup> <sup>¼</sup> … , J, k ¼ 1, … ,Kg where quality of this image becomes worse for a larger CR or smaller bpp that takes place for larger QS or SF if a DCT-based coder is applied. Let us see how this happens for JPEG with uniform quantization of DCT coefficients. Suppose that an image to be compressed is divided into N=IJ/4 non-overlapping blocks of the size 8 � 8 pixels. Then, in each block, we have DCT coefficients fD nð ; k; lÞ; n ¼ 1; …; N; k ¼ 1; …; 7; l ¼ 1; …; 7g . After quantization, we have

� � Dqðn; k; lÞ; n ¼ 1; …; N; k ¼ 1; …; 7; l ¼ 1; …; 7 . Then, MSE of losses can be determined as

$$\text{MSE} = \frac{\mathbf{1}}{\mathbf{N}} \quad \sum\_{\mathbf{n}=1}^{\text{N}} \text{MSE}\_{\mathbf{n}} = \sum\_{\mathbf{n}=1}^{\text{N}} \sum\_{\mathbf{k}=0}^{\text{7}} \sum\_{\mathbf{l}=0}^{\text{7}} \left( \Delta \mathbf{D}\_{\mathbf{q}}(\mathbf{n}, \mathbf{k}, \mathbf{l}) \right)^{2} \tag{4}$$

where

$$\mathbf{D}\_{\mathbf{q}}(\mathbf{n}, \mathbf{k}, \mathbf{l}) = \ [\mathbf{D}(\mathbf{n}, \mathbf{k}, \mathbf{l})/\mathbf{Q}\mathbf{S}], \mathbf{k} = \mathbf{0}, \dots, 7, \mathbf{l} = \mathbf{0}, \dots, 7,$$

$$\Delta \mathbf{D}\_{\mathbf{q}}(\mathbf{n}, \mathbf{k}, \mathbf{l}) = \mathbf{Q}\mathbf{S} \times \mathbf{D}\_{\mathbf{q}}(\mathbf{n}, \mathbf{k}, \mathbf{l}) - \mathbf{D}(\mathbf{n}, \mathbf{k}, \mathbf{l}), \mathbf{k} = \mathbf{0}, \dots, 7, \mathbf{l} = \mathbf{0}, \dots, 7.$$

and [] denotes rounding-off to the nearest integer, n denotes the block index.

A usual assumption concerning distribution of quantization errors is that it is uniform or close to uniform. Then, MSE is about QS2 =12: This is true for quite small QS (see data in Figure 2) but, for larger QS, MSE becomes smaller than QS2 =12: The main reason is that distributions of alternating current (AC) DCT coefficients differ a lot depending upon an image. Figure 4 presents these distributions using the same scale for the three considered images (Figure 1). Obviously, these distributions differ from Gaussian and from Laplacian (assumed in the paper [33]) as well. For the simple structure image, the distribution is quite narrow and it has heavy tails. If noise is present, the distribution "widens" and becomes closer to Gaussian.

It is seen from analysis of distribution in Figure 4a that if QS is about 10, most of AC DCT coefficients become zeros after quantization. Thus, we have decided to analyze quantization errors more in detail. Histograms of these errors for four cases are given in Figure 5. The histogram in Figure 5a shows that error distribution is close to uniform for the noise-free image Airfield that has wide distribution of AC DCT coefficients (Figure 4c). The distribution is also practically uniform for noisy image Frisco (noise standard deviation equals to 5, Figure 5d). Then, MSE of introduced losses is really close to QS2 =12 (see data in Figure 2). In other cases (Figure 5b and c), the distributions sufficiently differ from uniform. This happens for noise-free image Frisco. Thus, introduced losses MSE is less than QS<sup>2</sup> =12:

Hence, MSE ≈ QS<sup>2</sup> =12 can be treated as the upper limit of introduced losses. Note that this is valid not only for JPEG but for the coders AGU and ADCT [38–40]. This means that having a desired (threshold) MSEdes, it is possible to easily calculate <sup>p</sup>ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi QS as <sup>12</sup> MSEdes. <sup>A</sup> question is when the approximation MSE<sup>≈</sup> QS<sup>2</sup> =12 is valid? Note that if MSE is smaller than QS<sup>2</sup> =12, one can benefit from using a larger QS and providing a larger CR. Clearly, that if a desired PSNRdes has to be provided, it has to be recalculated to MSEdes taking into account dynamic range for a given image as MSE <sup>ð</sup> PSNRdes=10<sup>Þ</sup> des <sup>¼</sup> D2 =10 .

Our idea [38–40] is that MSE can be predicted in one of two ways. The first way is determined as

$$\text{MSE}\_{\text{pred}} = \frac{1}{\text{R}} \quad \sum\_{\mathbf{r}=1}^{\text{R}} \text{MSE}\_{\mathbf{r}} = \frac{1}{64 \text{R}} \sum\_{\mathbf{r}=1}^{\text{R}} \sum\_{\mathbf{k}=0}^{7} \sum\_{\mathbf{l}=0}^{7} \left(\Delta \text{D}\_{\mathbf{q}}(\mathbf{n}, \mathbf{k}, \mathbf{l})\right)^{2} \tag{5}$$

$$\Delta \mathbf{D}\_{\mathbf{q}}(\mathbf{r}, \mathbf{k}, \mathbf{l}) = \mathbf{Q} \mathbf{S} \times \mathbf{D}\_{\mathbf{q}}(\mathbf{r}, \mathbf{k}, \mathbf{l}) - \mathbf{D}(\mathbf{r}, \mathbf{k}, \mathbf{l}), \mathbf{k} = \mathbf{0}, \dots, 7, \mathbf{l} = \mathbf{0}, \dots, 7, \mathbf{r} = \mathbf{1}, \dots, \mathbf{R} \tag{6}$$

where R is the number of analyzed blocks (R ≪ N), C is a correcting factor used for a given coder. In other words, we employ statistics of DCT coefficients calculated in a limited number R of analyzed blocks of size 8x8 pixels. According to our studies [38, 40], it is enough to have R about 500 where analyzed blocks are

Lossy Compression of Remote Sensing Images with Controllable Distortions DOI: http://dx.doi.org/10.5772/intechopen.82361

#### Figure 4.

Distributions of AC DCT coefficients for the noise-free image Frisco (a), noise-free image airfield (b) and noisy image airfield (c), all in the same limits from �200 to 200.

randomly distributed over area of an image to be compressed to have prediction accurate enough. Taking into account that number of 8 � 8 pixel blocks in compressed images usually exceeds several thousands, prediction occurs to be much faster than even compression by JPEG. Certainly, prediction is much faster than compression by AGU (uses 32 � 32 blocks, efficient coding and deblocking after decompression) and, especially, ADCT (exploits partition scheme optimization).

Expressions (5 and 6) allow predicting MSE for a given QS. But they do not allow direct setting of QS. One has to apply an iterative procedure that starts <sup>p</sup>ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi from QS <sup>¼</sup> <sup>12</sup> MSEdes. If the predicted MSEpred (5) occurs to be considerably (e.g., by 15–20% or more) smaller than MSEdes, then a larger QS has to be tried with calculating (6) for all analyzed blocks and (5) again. Since the already calculated DCT coefficients are available, the procedure is quite fast.

The second way is the following. Suppose that the predicted MSE can be presented as

$$\text{MSE}\_{\text{pred}} = \left( \text{QS}^2 / \text{12} \right) \mathbf{f}\_0(\mathbf{X}) \tag{7}$$

where f0ð Þ X is a function of one or two parameters X that can be easily and quickly calculated for DCT coefficients determined in analyzed blocks. Then one has to find such parameter(s) and the function. To solve this task, we have exploited our earlier experience in predicting filtering efficiency [49] and compression ratio [18] by simple analysis of DCT statistics in 8x8 pixel blocks and regression analysis [50, 51].

The prediction strategy is the following. We suppose that there is an input parameter (or a few parameters) that can characterize a compressed image. It is also assumed that output (predicted) parameter (MSE, PSNR, CR, or another metric) is

#### Lossy Compression of Remote Sensing Images with Controllable Distortions DOI: http://dx.doi.org/10.5772/intechopen.82361

strictly connected with this (these) input parameter(s). This connection (prediction approximation) is available to the moment to carry out prediction, i.e., in our case, the function f0ð Þ X has been obtained in advance (in off-line mode). Then, one has to calculate input parameter(s) for a given QS and insert it (them) into f0ð Þ X .

It has been shown in [52] that a good parameter integrally characterizing an image (its complexity) is probability P0 that AC DCT coefficients after quantization become equal to zero (this parameter can be also treated as probability that AC DCT coefficient absolute values are smaller than QS/2). It is obvious that P0 can be very easily calculated. Keeping these properties of P0 in mind, we have obtained scatter plots of 12MSE=Q S2 to estimate f0ð Þ P0 . <sup>A</sup> wide set of test noise-free images has been used that included standard optical images, test RS images and test medical image (this was done to understand does the image nature (origin) influence performance of lossy compression; in fact, very similar results have been obtained for test images of different origin; the main factor is image complexity). Each point of the scatter plot corresponds to one test image compressed with some QS where vertical coordinate is P0 determined for this case).

Figure 6 presents scatter plots obtained for AGU and ADCT coders with examples of fitted curves. The main and very important observation is that the scatter plots behave in a compact manner, i.e. points that have approximately the same arguments have close values of 12MSE=Q S<sup>2</sup> . Another observation is that the scatter plots for two considered coders behave in a very similar manner, i.e. there is a tendency to monotonous decreasing of 12MSE=Q S<sup>2</sup> if P0 increases. Finally, the scatter plots confirm that, in many practical situations, MSE≈ QS<sup>2</sup> =12. At least, this is true for P0 , 0:6.

It is worth recalling here that P0 , 0:6 corresponds to rather small QS. To prove this, Figure 7 presents the scatter plot from [48] and the fitted curve. As it is seen, for P0 , 0:6, CR does not exceed 5. If P0 ≥0:6, there is the tendency of reduction of f0ð Þ P0 . The scatter plot points are placed not so compactly here. Thus, prediction using only f0ð Þ P0 becomes less accurate. Nevertheless, the following prediction procedure can be proposed:

Figure 6. Scatter plots for AGU (a) and ADCT (b) coders.

Figure 7. The scatter plot of CR on P0 and the fitted curve for the coder AGU.


As it is seen, all the operations are very easy and fast since they are performed for a limited number of AC DCT coefficients. Moreover, using the same parameter, it is possible to predict both MSE and CR. Then, it is easy to find a proper compromise depending upon priority of requirements and imposed restrictions.

One question is what curves to fit and what are criteria of fitting quality to be used. There are different approaches but we employed goodness-of-the-fit R2 and RMSE [50] as two main criteria (the former one has to be maximized and the latter one minimized for a given scatter plot). Without going to details, we can state the following. For each scatter plot, usually there are several functions able to provide approximately the same R<sup>2</sup> and RMSE. Sums of two exponentials (see an example in Figure 7), polynomials of low order, Fourier series, power functions are good candidates to be tested. Using the corresponding tools of Matlab or Excel, it is possible to quickly find optimal or, at least, appropriately good solution.

#### 4. Peculiarities of compression

#### 4.1 Visual quality metrics

� � We have already mentioned that it is often desirable to predict visual quality metrics. To check whether or not this is possible, the scatter plot was got for MSEHVS�<sup>M</sup><sup>=</sup> Q S2 =12 vs. P0 (Figure 8). As it is seen, this ratio is about 0.05 for small P0 (this happens for small QS and/or complex structure images), i.e. PSNR-HVS-M is by about 13 dB larger than PSNR. This means that introduced losses are masked by image content well and, most probably, they cannot be noticed visually. Lossy Compression of Remote Sensing Images with Controllable Distortions DOI: http://dx.doi.org/10.5772/intechopen.82361

Figure 8. ˜ ° The scatter plot MSEHVS�<sup>M</sup><sup>=</sup> Q S2 =12 vs. P0 and the fitted curve, AGU coder.

The difference in PSNR-HVS-M and PSNR decreases to 5–7 dB for P0 . 0:5, i.e. typical conditions of lossy compression. The scatter plot and the fitted curve show that MSEHVS�<sup>M</sup> can be predicted well for a given QS. In other words, visual metrics can be predicted too using the proposed approach. Again, the sum of two exponentials (just this case is presented in Figure 8) can serve well as approximation curve with quite small number of varied parameters.

#### 4.2 Experimental data for component-wise compression

Let us present the results of applying the proposed approach to real-life hyperspectral data. Images of Hyperion sensor dataset EO1H1800252002116110KZ have been compressed. Hyperion sensor produces data of bad quality (very noisy) in sub-bands with indices k = 1,…,12 and k = 58,…,76. The images in these sub-bands are often discarded in analysis, so we have not compressed them.

Then, two approaches to compression have been compared. Both presume component-wise compression. The first one has been proposed earlier [11]. Images are compressed after applying variance stabilizing transform that takes into account signal-dependent noise properties and converts this noise to additive with variance approximately equal to unity. Then, the recommended QS = 3.5 (this notation is used in figures below). Inverse transform is applied component-wise after decompression. For the proposed method, the component-wise images have been transformed to the interval from 0 to 255. Then, for each of them, AGU coder has been applied with QS = 17 that approximately corresponds to PSNRdes ¼ 34:5 dB (MSEdes ≈24≈ 17 � 17=12Þ. The notation QS = 17 is used for the corresponding data.

The obtained PSNR values calculated between compressed and original component images are presented in Figure 9. As it is seen, PSNR for the method [11] in most sub-bands occurs to be considerably larger than PSNRdes set by us. Only in some sub-bands (indices 165–185) where input PSNR is quite small the determined PSNR values are about 40 dB (i.e., the introduced losses are invisible in decompressed images). For the proposed approach, PSNR for the introduced losses is considerably smaller but, for all sub-band images, PSNR anyway exceeds 35 dB. As it follows from analysis of data in Figure 10, CR for all sub-bands exceeds 5 (a more detailed study shows that P0 . 0:6 in all cases). Thus, MSE is smaller than Q S<sup>2</sup> =12 (see data in Figure 6) and the provided PSNR is larger than expected.

The main observation for data in Figure 10 is that CR for the proposed method is by several times larger than for the prototype method for almost all sub-bands except the bands with small input PSNR. Thus, we have gained essential benefit in CR sense while introduced distortions remained invisible.

#### Figure 9.

PSNR for component-wise compression by the method ([11], QS = 3.5), the proposed component-wise approach (QS = 17), and the proposed 3D compression method (QS = 17, bl = 4).

#### Figure 10.

CR for component-wise compression by the method ([11], QS = 3.5), the proposed approach (QS = 17), and the proposed 3D compression method (QS = 17, bl = 4).

We do not present examples of original and compressed component images because visually they are identical. Note that setting a larger PSNRdes leads to larger PSNR of introduced losses and smaller CR for each component image, respectively. By setting a larger PSNRdes one can ensure that classification accuracy does not make worse.

#### 4.3 3D compression

Consider now possibilities of 3D compression in groups. There are many different options [11]. We have analyzed one of the simplest ones where component images have been transformed to the 8-bit representation limits, then combined in 4-band groups, and then compressed by 3D version of AGU coder. After decompression the images have to be "stretched" to original limits.

Lossy Compression of Remote Sensing Images with Controllable Distortions DOI: http://dx.doi.org/10.5772/intechopen.82361

As previously in Section 4.2, we have employed QS = 17. For convenience of comparison, the obtained data are also presented in Figures 9 and 10, for 3D compression they are denoted as QS = 17, bl = 4. CR values for the 3D case are shown the same for all components of the same group. As it is seen, CR values for 3D compression are about two times larger than for the proposed component-wise compression. This is an obvious advantage of 3D compression. Meanwhile, there are also very interesting observations stemming from analysis of data for PSNR (Figure 9). As it is seen, there are many sub-bands for which PSNR for 3D compression is considerably larger (and the introduced losses are sufficiently smaller) than for component-wise compression. PSNR values are almost the same if subbands with small input PSNR are compressed. This is one more positive feature of 3D compression that should be studied more in detail in the future.

### 5. Conclusions

We have considered the task of lossy compression of RS images with controllable quality characterized by traditional metrics. It is shown that MSE and PSNR can be predicted for DCT-based coders and, due to this, it is possible to provide a desired MSE or PSNR without compression/decompression iterations quite quickly and accurately. Being applied to compress RS images without visible distortions, this approach allows providing CR considerably larger than for approach based on taking noise properties into account.

Moreover, it is demonstrated that prediction of some visual quality metrics is also possible. It is also shown that 3D compression of images collected into groups provides considerably better results. However, additional studies are needed to predict distortion parameters in this case. Examples for real-life data as hyperspectral image are presented.

This research has been partly supported by the Project M/29–2018 of Ukrainian-French program "Dnipro" and STCU Project No. 6386.

## Author details

Vladimir Lukin<sup>1</sup> \*, Alexander Zemliachenko<sup>1</sup> , Sergey Krivenko<sup>1</sup> , Benoit Vozel<sup>2</sup> and Kacem Chehdi<sup>2</sup>

1 National Aerospace University, Kharkiv, Ukraine

2 University of Rennes 1, Lannion, France

\*Address all correspondence to: lukin@ai.kharkov.com

© 2018 The Author(s). Licensee IntechOpen. This chapteris distributed underthe terms oftheCreative 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.

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**45**

Section 3

Ionospheric Monitoring

Systems

## Section 3
