**4. Contourlet transform method**

The contourlet transform (CT) is a multiscale and multidirectional framework of discrete image. It is the simple directional extension for wavelet that fixes its subband mixing problem and improves its directionality. Among the "beyond wavelet" techniques, contourlet allows for different and flexible number of directions at each scale, while achieving nearly critical sampling. The desirable properties of CT for image representation includes multiresolution,


**Table 4.** Performance comparison of the LP transform method and the wavelet transform based despeckling method

Difference image

The Laplacian pyramid transform is performed on the log transformed image. The Laplacian pyramidal decompositions up to six levels are obtained using biorthogonal filters with sufficient accuracy numbers such as the "9/7" and "5/3". Further, thresholding schemes such as hard thresholding, soft thresholding and semi soft thresholding is performed to reduce speckle. The threshold value is calculated using Bayes' shrinkage rule. The experimentation is carried out on 70 ultrasound images of liver and kidney. The performance evaluation of the proposed method is done in terms of variance, MSE, SNR, PSNR, CC values that are computed from despeckled image. The Laplacian pyramid transform with 1 level decomposition and

The Table 4. shows the performance comparison of the proposed LP transform based des‐ peckling method with the wavelet transform based despeckling method [36]. It is noticed that, in comparison with the despeckling medical ultrasound images based on WT, the despeckling based on LP method yields poor results. Because multiplicative noise is a particular type of signal dependent noise, in which the amplitude of the noise term is proportional to the value of the noise free signal having nonzero mean. Therefore, a band pass representation like LP is

In order to capture smooth contours in the images, the contourlet transforms, which allow directional decompositions, are employed for despeckling medical ultrasound images in the

The contourlet transform (CT) is a multiscale and multidirectional framework of discrete image. It is the simple directional extension for wavelet that fixes its subband mixing problem and improves its directionality. Among the "beyond wavelet" techniques, contourlet allows for different and flexible number of directions at each scale, while achieving nearly critical sampling. The desirable properties of CT for image representation includes multiresolution,

**Figure 7.** Block diagram of speckle noise suppression using Laplacian pyramid transform

hard thresholding is observed to be better than other thresholding methods.

not suitable for multiplicative noise model, So the method needs to be improved.

Thresholdin g

Inverse LP transform

Output despeckled image Y

Exp transfor m

Coarse approximation image

Laplacian pyramid transform

Input image X

next section.

**4. Contourlet transform method**

Log transfor m

218 Advancements and Breakthroughs in Ultrasound Imaging

N-Level LP

> allowing images to be approximated in a coarse to fine fashion; localization of the basis vectors in both space and frequency; low redundancy, so as not to increase the amount of data to be stored; directionality, allowing representation with basis elements oriented in a variety of directions; and anisotropy, the ability to capture smooth contours in images, using basis elements that are a variety of elongated shapes with different aspect ratios [37].

> The contourlet transform has been developed to overcome the limitations of the wavelets, and hence, the new algorithms based on the contourlet transform are more efficient than wavelet methods. In [38], the authors have presented a contourlet based speckle reduction method for denoising ultrasound images of breast. The double iterated filter bank structure and a small redundancy at most 4/3 using two thresholding methods shows a great promise for speckle reduction. In [39], the despeckling medical ultrasound images using contourlet transform using Bayes' shrinkage rule is investigated. The algorithm is also tested on ovarian ultrasound images to demonstrate improvements in the segmentation that yields good classification for follicle detection in an ovarian image [40].

> In [41], speckle reduction based on contourlet transform using scale adaptive threshold for medical ultrasound image has been examined, where in the subband contourlet coefficients of the ultrasound images after logarithmic transform are modelled as generalized Gaussian distribution. The scale adaptive threshold in Bayesian framework is applied. The method is tested on both synthetic and clinical ultrasound images interms of S/MSE and edge preserva‐ tion parameter. The proposed method exhibits better performance on speckle suppression than the wavelet based method, while it does well preserve the feature details of the image.

> The contourlet transform can be divided into two main steps: Laplacian pyramid decomposi‐ tion and directional filter banks. Contourlet transform is a multi scale and directional image representation that uses first a wavelet like structure for edge detection, and then a local directional transform for contour segment detection. A double filter bank structure of the contourlet obtains sparse expansions for typical images having smooth contours. In the double filter bank structure, the Laplacian pyramid is used to decompose an image into a number of radial subbands, and the directional filter banks decompose each LP detail subband into a number of directional subbands. The band pass images (dj [n]) from the LP are fed into a DFB so that directional information can be captured. The scheme can be iterated on the coarse image

(cj [n]). The combined result is a double iterated filter bank structure, named pyramidal directional filter bank (PDFB), which decomposes images into directional subbands at multiple scales. The general model for despeckling an image using contourlet transform is shown in the Figure 8.

**Figure 8.** The general model for speckle reduction using contourlet transform.

In the Figure 8, the CT based despeckling method consists of the log transformed original ultrasound image being subjected to contourlet transform, to obtain contourlet coefficients. The transformed image is denoised by applying thresholding techniques on individual band pass sub bands using a Bayes shrinkage rule, derived from the local statistics of the signal in the transform domain. Bayes' shrink was proposed by [29]. The goal of Bayes' shrinkage method is to minimize the Bayesian risk, and hence its name, Bayes' shrink. Further, thresh‐ olding schemes such as hard thresholding, soft thresholding or semi soft thresholding is performed to reduce speckle. The exponential operation is performed on the filtered output to obtain the despeckled image.

**Thresholding methods Levels PSNR Variance MSE CC SNR Computational**

**Figure 9.** a) Original ultrasound image. (b) Despeckled image using wavelet transform using subband Bayes soft thresholding (level 3). (c) Despeckled image using contourlet transform using soft thresholding. (d) Despeckled image using contourlet transform using hard thresholding. (e) Despeckled image using contourlet transform using semi soft

(a) (b) (c)

Hard thresholding L2-11 34.21 0.0203 0.00037 0.9990 27.74 1.64 Soft thresholding L2-31 32.49 0.0382 0.00056 0.9983 24.14 1.68 Semi-soft thresholding L2-11 33.31 0.0292 0.00046 0.9987 25.91 1.74

(d) (e)

**Table 5.** The results obtained for the optimal optimal decomposition of LP levels and directional decompositions in terms of image quality assessment parameters using contourlet method based on different thresholding techniques

The frequency bands obtained by using optimal level L2-11 of contourlet decomposition are as follows: the 2nd level has 1 approximation band of size 128 x128 and 4 detail components (2 of 128 x 256, 2 of 256 x 512). The reconstructed image is the despeckled image. The hard thresholding is better than other thresholding methods, because small coefficients are removed while others are left untouched in HT, while in ST or SST coefficients above the threshold are shrunk by absolute value of threshold. Further, it is found that the despeckling based on contourlet transform gives better results than the speckle reduction method based on wavelet transform in particular. The wavelet based Bayes' shrink thresholding method is based on

with Bayes' shrinkage rule.

thresholding.

**time(in Secs.)**

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The experimentation is carried out on 70 ultrasound images of liver and kidney. The six levels of Laplacian pyramidal decompositions are performed using biorthogonal filters with sufficient accuracy numbers such as the "9/7". The directional decompositions up to eight are performed in all the pyramidal levels, using two dimensional ladder filters. The contourlet transform uses the "9/7" filters in LP stage because, in the multiscale decomposition stage, it significantly reduces all inter scale, inter location and inter direction mutual information of contourlet coefficients. Similarly, in directional decomposition stage, the ladder structure PKVA filters [42] are more effective in localizing edge direction as these filters reduce the inter direction mutual information. Further, thresholding schemes such as hard thresholding, soft thresholding or semi soft thresholding is performed to reduce speckle. The threshold value is calculated using Bayes' shrinkage rule. The PSNR is calculated up to 6 LP decompositions. The PSNR value increases up to 2 decompositions using HT, ST and SST, and thereafter reduces. Hence, the optimal level of LP decomposition is 2. Further, it is observed from Table 5, that the 2-level Laplacian pyramidal decomposition and 4 directional bandpass subbands (2 at level 1, 2 at level 2) using hard thresholding yield better results than soft thresholding and semi soft thresholding techniques.

(cj

the Figure 8.

Input image X

Log transform

220 Advancements and Breakthroughs in Ultrasound Imaging

N-Level CT

to obtain the despeckled image.

thresholding techniques.

 [n]). The combined result is a double iterated filter bank structure, named pyramidal directional filter bank (PDFB), which decomposes images into directional subbands at multiple scales. The general model for despeckling an image using contourlet transform is shown in

> Contourlet transform

Coarse approximation

> Bandpass subbands

In the Figure 8, the CT based despeckling method consists of the log transformed original ultrasound image being subjected to contourlet transform, to obtain contourlet coefficients. The transformed image is denoised by applying thresholding techniques on individual band pass sub bands using a Bayes shrinkage rule, derived from the local statistics of the signal in the transform domain. Bayes' shrink was proposed by [29]. The goal of Bayes' shrinkage method is to minimize the Bayesian risk, and hence its name, Bayes' shrink. Further, thresh‐ olding schemes such as hard thresholding, soft thresholding or semi soft thresholding is performed to reduce speckle. The exponential operation is performed on the filtered output

The experimentation is carried out on 70 ultrasound images of liver and kidney. The six levels of Laplacian pyramidal decompositions are performed using biorthogonal filters with sufficient accuracy numbers such as the "9/7". The directional decompositions up to eight are performed in all the pyramidal levels, using two dimensional ladder filters. The contourlet transform uses the "9/7" filters in LP stage because, in the multiscale decomposition stage, it significantly reduces all inter scale, inter location and inter direction mutual information of contourlet coefficients. Similarly, in directional decomposition stage, the ladder structure PKVA filters [42] are more effective in localizing edge direction as these filters reduce the inter direction mutual information. Further, thresholding schemes such as hard thresholding, soft thresholding or semi soft thresholding is performed to reduce speckle. The threshold value is calculated using Bayes' shrinkage rule. The PSNR is calculated up to 6 LP decompositions. The PSNR value increases up to 2 decompositions using HT, ST and SST, and thereafter reduces. Hence, the optimal level of LP decomposition is 2. Further, it is observed from Table 5, that the 2-level Laplacian pyramidal decomposition and 4 directional bandpass subbands (2 at level 1, 2 at level 2) using hard thresholding yield better results than soft thresholding and semi soft

**Figure 8.** The general model for speckle reduction using contourlet transform.

Output despeckle d image Y

Thresholding

Inverse CT

Exp transform

Directional decompositio n

**Figure 9.** a) Original ultrasound image. (b) Despeckled image using wavelet transform using subband Bayes soft thresholding (level 3). (c) Despeckled image using contourlet transform using soft thresholding. (d) Despeckled image using contourlet transform using hard thresholding. (e) Despeckled image using contourlet transform using semi soft thresholding.


**Table 5.** The results obtained for the optimal optimal decomposition of LP levels and directional decompositions in terms of image quality assessment parameters using contourlet method based on different thresholding techniques with Bayes' shrinkage rule.

The frequency bands obtained by using optimal level L2-11 of contourlet decomposition are as follows: the 2nd level has 1 approximation band of size 128 x128 and 4 detail components (2 of 128 x 256, 2 of 256 x 512). The reconstructed image is the despeckled image. The hard thresholding is better than other thresholding methods, because small coefficients are removed while others are left untouched in HT, while in ST or SST coefficients above the threshold are shrunk by absolute value of threshold. Further, it is found that the despeckling based on contourlet transform gives better results than the speckle reduction method based on wavelet transform in particular. The wavelet based Bayes' shrink thresholding method is based on separable 2D wavelet transform that has limited directions (Horizontal, Vertical and Direc‐ tional). Speckle noise in medical ultrasound images will generate significant coefficients in wavelet domain just like true detail features, such as edges. However, the speckle noise is less likely to generate significant coefficients in the contourlet method, and thus, it directly leads to better performance in suppressing noise than the wavelet based Bayes' shrink thresholding scheme.

where B is the series of bit shifts in the ith row and jth column directions. If one decomposes an image of size (N, N) using the contourlet transform, the maximum number of decomposition levels in the LP stage will be B, and therefore, the maximum number of shifts are (B, B) in the row and column directions. After a B number of bit shifts, which depends on the level of decomposition, the transform output degrades. Hence, the cycle spinning has to be stopped after a certain number of bit shifts. The Figure 10. shows the block diagram for speckle reduction method based on contourlet transform with cycle spinning. The cycle spinning is applied to the log transformed image. It performs two dimensional circular shift in ith row and

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th column directions. The circular shifting is performed up to B number of bit shifts, where B depends on the level of decomposition. The transform output degrades as B increases. Hence, the cycle spinning has to be stopped after a certain number of bit shifts. Then contourlet transform is performed using double filter bank structure. The six levels of Laplacian pyra‐ midal decompositions are performed using biorthogonal filters with sufficient accuracy numbers such as the "9-7".The directional decompositions up to six is performed in the lowest pyramidal level, using two dimensional ladder filters designed in [42]. Further, a thresholding scheme either hard thresholding, soft thresholding or semi soft thresholding, is performed to reduce speckle. The threshold value is calculated using Bayes' shrinkage rule. The results obtained for the optimal bit shifts of cycle spinning using contourlet method based on hard thresholding, soft thresholding and semi soft thresholding using Bayes' rule are presented. The results obtained for different reconstruction methods are shown in Figures 11-15, which exhibit graphs of statistical features PSNR, SNR, variance, MSE and CC, respectively, for different levels of Laplacian pyramid decompositions and directional decompositions corresponding to optimal results of contourlet transform with cycle spinning based despeck‐ ling method [46]. The optimal reconstruction method is determined by the criteria, namely, lower variance and MSE, higher SNR and PSNR values, Correlation Coefficient is nearly equal to one. The contourlet transform with 2 level of pyramidal decomposition and two directional decompositions in the finest scale and hard thresholding technique with Bayes' shrinkage rule has yielded better results in comparison contourlet transform based methods [47]. In the Figures 11-15, the horizontal axis label CYC-HT-Bn indicates cycle spin (CYC), thresholding (HT,ST,SST), n number of bit shifts in cycle spinning. From Figures 11-15, it is observed that, 4 bit cycle spinning, having the 2-level of Laplacian pyramidal decomposition with 4 direc‐ tional bandpass sub bands (2 at level 1, 2 at level 2) subject to soft thresholding, yields optimal results for speckle reduction. The computational time (in Secs.) of the cycle spinning based CT method is shown in Figure 15. The CT based despeckling method takes less computational

The Figure 16. illustrates the resultant despeckled images of an ultrasound image obtained by the cycle spinning based CT method using hard, soft and semi soft thresholding with Bayes' shrinkage rule, and also that obtained by the CT method [47], for comparison by visual inspection The despeckling method based on cycle spinning using contourlet coefficient shrinkage (Figure16.(b)) performs better and appears to be an improvement over direct

time as compared to cycle spinning based CT method.

contourlet transform based despeckling method.

j

Another way to analyze the effects of despeckling techniques is to study the despeckled images. In the Figure 9., the resultant images of a sample medical ultrasound image are presented to compare the results of different despeckling techniques by visual inspection. In the Figure 9(b) and (c), the speckle is reduced considerably, but the structures are blurred and some visible artifacts are introduced. However, in Figure 9(e), the speckle is reduced well and structures are enhanced. But some details are lost and some are over enhanced. It is encour‐ aging to note that in the Figure 9(d), the speckle is effectively reduced and also structures are enhanced with almost no loss or noticeable artifact.

#### **4.1. Cycle spinning based contourlet transform**

The CT is not translation invariant. This means that the errors after denoising will be sensitive to the positions of discontinuities in the data. In order to avoid such effects, it is necessary to build translation invariant version of the transform. Translation invariance is achieved through several ways. For example, in [43], the time invariant schemes of wavelet based decomposi‐ tions have been proposed and have been often referred to as cycle spinning. Unfortunately, due to the downsamplers and upsamplers present in the directional filter banks of CT, the CT is not shift invariant, which is important in image denoising by thresholding and normally causes pseudo-Gibbs phenomenon. In [44], the cycle spinning algorithm is utilized in devel‐ oping a translation invariant contourlet based denoising technique. The experimental results clearly demonstrate the capability of the proposed scheme in image denoising, especially for detailed texture images. It is shown that most of the visual artifacts resulting from the contourlet transform denoising process are eliminated. In [45], a cycle spinning method is used to compensate for the lack of translation invariance property of sharp frequency localized contourlet. Experimental results demonstrate that cycle spinning is a simple and efficient way to average out the pseudo-Gibbs phenomena, which are around singularities and produced by the down sampling and up sampling of directional filter banks, and improve the denoising performance interms of visual quality and PSNR.

To compensate for the lack of translation invariance property of the contourlet transform, we apply the principle of cycle spinning to contourlets. Suppose X and Y are original and despeck‐ led images, F and F−1 are forward and inverse contourlet transform, Si,j is the 2D circular shift in i th row and jth column directions, λ is the threshold operator in contourlet transform domain. The cycle spinning based contourlet transform for image denoising could be described as

$$\mathbf{Y} = \frac{1}{\mathbf{B}^2} \sum\_{\mathbf{i}, \mathbf{j}=1}^{\mathbf{B}} \mathbf{C} \mathbf{S}\_{\cdot \mathbf{i}, \mathbf{j}} \Big\{ \mathbf{F}^{-1} (\lambda \langle \mathbf{F} (\mathbf{C} \mathbf{S}\_{\mathbf{i}, \mathbf{j}} (\mathbf{X})) )\Big\} \tag{31}$$

where B is the series of bit shifts in the ith row and jth column directions. If one decomposes an image of size (N, N) using the contourlet transform, the maximum number of decomposition levels in the LP stage will be B, and therefore, the maximum number of shifts are (B, B) in the row and column directions. After a B number of bit shifts, which depends on the level of decomposition, the transform output degrades. Hence, the cycle spinning has to be stopped after a certain number of bit shifts. The Figure 10. shows the block diagram for speckle reduction method based on contourlet transform with cycle spinning. The cycle spinning is applied to the log transformed image. It performs two dimensional circular shift in ith row and j th column directions. The circular shifting is performed up to B number of bit shifts, where B depends on the level of decomposition. The transform output degrades as B increases. Hence, the cycle spinning has to be stopped after a certain number of bit shifts. Then contourlet transform is performed using double filter bank structure. The six levels of Laplacian pyra‐ midal decompositions are performed using biorthogonal filters with sufficient accuracy numbers such as the "9-7".The directional decompositions up to six is performed in the lowest pyramidal level, using two dimensional ladder filters designed in [42]. Further, a thresholding scheme either hard thresholding, soft thresholding or semi soft thresholding, is performed to reduce speckle. The threshold value is calculated using Bayes' shrinkage rule. The results obtained for the optimal bit shifts of cycle spinning using contourlet method based on hard thresholding, soft thresholding and semi soft thresholding using Bayes' rule are presented. The results obtained for different reconstruction methods are shown in Figures 11-15, which exhibit graphs of statistical features PSNR, SNR, variance, MSE and CC, respectively, for different levels of Laplacian pyramid decompositions and directional decompositions corresponding to optimal results of contourlet transform with cycle spinning based despeck‐ ling method [46]. The optimal reconstruction method is determined by the criteria, namely, lower variance and MSE, higher SNR and PSNR values, Correlation Coefficient is nearly equal to one. The contourlet transform with 2 level of pyramidal decomposition and two directional decompositions in the finest scale and hard thresholding technique with Bayes' shrinkage rule has yielded better results in comparison contourlet transform based methods [47]. In the Figures 11-15, the horizontal axis label CYC-HT-Bn indicates cycle spin (CYC), thresholding (HT,ST,SST), n number of bit shifts in cycle spinning. From Figures 11-15, it is observed that, 4 bit cycle spinning, having the 2-level of Laplacian pyramidal decomposition with 4 direc‐ tional bandpass sub bands (2 at level 1, 2 at level 2) subject to soft thresholding, yields optimal results for speckle reduction. The computational time (in Secs.) of the cycle spinning based CT method is shown in Figure 15. The CT based despeckling method takes less computational time as compared to cycle spinning based CT method.

separable 2D wavelet transform that has limited directions (Horizontal, Vertical and Direc‐ tional). Speckle noise in medical ultrasound images will generate significant coefficients in wavelet domain just like true detail features, such as edges. However, the speckle noise is less likely to generate significant coefficients in the contourlet method, and thus, it directly leads to better performance in suppressing noise than the wavelet based Bayes' shrink thresholding

Another way to analyze the effects of despeckling techniques is to study the despeckled images. In the Figure 9., the resultant images of a sample medical ultrasound image are presented to compare the results of different despeckling techniques by visual inspection. In the Figure 9(b) and (c), the speckle is reduced considerably, but the structures are blurred and some visible artifacts are introduced. However, in Figure 9(e), the speckle is reduced well and structures are enhanced. But some details are lost and some are over enhanced. It is encour‐ aging to note that in the Figure 9(d), the speckle is effectively reduced and also structures are

The CT is not translation invariant. This means that the errors after denoising will be sensitive to the positions of discontinuities in the data. In order to avoid such effects, it is necessary to build translation invariant version of the transform. Translation invariance is achieved through several ways. For example, in [43], the time invariant schemes of wavelet based decomposi‐ tions have been proposed and have been often referred to as cycle spinning. Unfortunately, due to the downsamplers and upsamplers present in the directional filter banks of CT, the CT is not shift invariant, which is important in image denoising by thresholding and normally causes pseudo-Gibbs phenomenon. In [44], the cycle spinning algorithm is utilized in devel‐ oping a translation invariant contourlet based denoising technique. The experimental results clearly demonstrate the capability of the proposed scheme in image denoising, especially for detailed texture images. It is shown that most of the visual artifacts resulting from the contourlet transform denoising process are eliminated. In [45], a cycle spinning method is used to compensate for the lack of translation invariance property of sharp frequency localized contourlet. Experimental results demonstrate that cycle spinning is a simple and efficient way to average out the pseudo-Gibbs phenomena, which are around singularities and produced by the down sampling and up sampling of directional filter banks, and improve the denoising

To compensate for the lack of translation invariance property of the contourlet transform, we apply the principle of cycle spinning to contourlets. Suppose X and Y are original and despeck‐ led images, F and F−1 are forward and inverse contourlet transform, Si,j is the 2D circular shift in

th row and jth column directions, λ is the threshold operator in contourlet transform domain. The cycle spinning based contourlet transform for image denoising could be described as

CS-i,-j(F-1(λ(F(CSi,j(X))))) (31)

enhanced with almost no loss or noticeable artifact.

**4.1. Cycle spinning based contourlet transform**

222 Advancements and Breakthroughs in Ultrasound Imaging

performance interms of visual quality and PSNR.

Y= <sup>1</sup> <sup>B</sup><sup>2</sup> <sup>∑</sup> i,j=1 B

scheme.

i

The Figure 16. illustrates the resultant despeckled images of an ultrasound image obtained by the cycle spinning based CT method using hard, soft and semi soft thresholding with Bayes' shrinkage rule, and also that obtained by the CT method [47], for comparison by visual inspection The despeckling method based on cycle spinning using contourlet coefficient shrinkage (Figure16.(b)) performs better and appears to be an improvement over direct contourlet transform based despeckling method.

**Figure 10.** The block diagram of despeckling method based on contourlet transform with cycle spinning.

Among the transform domain filters developed in the previous sections, the contourlet transform with cycle spinning yields better visual quality enhancement of the despeckled images. However, there is still a need to remove Gaussian noise inherent in the medical ultrasound images, which is addressed in the next section.

**Figure 13.** The values of Correlation Coefficient for various thresholding methods obtained by cycle spinning based CT

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**Figure 14.** The values of variance for various thresholding methods obtained by cycle spinning based CT method and

**Figure 15.** The values of computational time for various thresholding methods obtained by cycle spinning based CT

method and that by direct CT based method.

that by CT based method.

method and that by CT based method.

**Figure 11.** The values of PSNR and SNR for various thresholding methods obtained by cycle spinning based CT method and that by direct CT based method.

**Figure 12.** The values of MSE for various thresholding methods obtained by cycle spinning based CT method and that by direct CT based method.

**Figure 13.** The values of Correlation Coefficient for various thresholding methods obtained by cycle spinning based CT method and that by direct CT based method.

Among the transform domain filters developed in the previous sections, the contourlet transform with cycle spinning yields better visual quality enhancement of the despeckled images. However, there is still a need to remove Gaussian noise inherent in the medical

**Figure 10.** The block diagram of despeckling method based on contourlet transform with cycle spinning.

spinning CT Thresholding

Inverse CT

**Figure 11.** The values of PSNR and SNR for various thresholding methods obtained by cycle spinning based CT method

**Figure 12.** The values of MSE for various thresholding methods obtained by cycle spinning based CT method and that

ultrasound images, which is addressed in the next section.

Inverse cycle spinning

and that by direct CT based method.

Output despeckled image Y image

Input

224 Advancements and Breakthroughs in Ultrasound Imaging

image X Cycle

by direct CT based method.

**Figure 14.** The values of variance for various thresholding methods obtained by cycle spinning based CT method and that by CT based method.

**Figure 15.** The values of computational time for various thresholding methods obtained by cycle spinning based CT method and that by CT based method.

**5.1. Pre- or post –processing**

Eq. (33)

image.

given by the Eq.(35).

Usually, medical ultrasound images are affected by the mixed noise, which is the combination of speckle noise and Gaussian noise. There are two factors that influence the usefulness of a smoothing filter. The first reduces the range of resolutions over which variations in the output appear by the filter variation ∆w, in the frequency domain be small and second factor is increase in spatial localization by a small spatial variance ∆x. These localization requirements in the spatial and frequency domain are conflicting and related by the uncertainty principle given in

1

It has been shown that Gaussian functions are the only ones that provide the optimal trade off

So Gaussian filters are widely used in image filtering. In [51], the removal of mixed noise using order statistic filter and wavelet domain Wiener filter is proposed. Authors have evaluated two methods. The first method comprises, order statistic prefilter and empirical Wiener filter, which is used to reduce the Gaussian noise. The disadvantage of this method is the higher time consumption. The second method is, order statistic filter for each decomposition level, where decomposition is carried out by the wavelet transform, followed by thresholding. The drawback of this method is that its efficiency is less than that of the first method (about 1dB) in removing the mixed noise. In [52], denoising of mixed noise in ultrasound images is presented. Combined Bayesian maximum a posterior (MAP) estimator and ST-PCNN (Soft threshold pulse coupled neural networks) method has been used for mixed noise reduction. The method removes the speckle noise considerably than the Gaussian noise that degrades the ultrasound images. The drawback of the method is either Gaussian noise or speckle noise is removed. Hence we present a method to remove residual Gaussian noise from despeckled

Two alternative algorithms are developed for reducing mixed noise in medical ultrasound images [53]. In the first alternative, the denoising method reduces the Gaussian noise by applying Gaussian filter in pre processing stage, then despeckling is performed using either wavelet transform, Laplacian pyramid transform or contourlet transform. The noise model for the first alternative (i.e. Gaussian noise removal in pre processing followed by despeckling) is

where Xij represents the noisy pixel in the image X, fij represents the noise free pixel, nij and gij represent the multiplicative speckle noise and additive Gaussian noise, respectively. The

Xij =fij <sup>4</sup><sup>π</sup> (33)

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<sup>4</sup>*<sup>π</sup>* (34)

nij + gij (35)

∆w ∆x ≥

<sup>∆</sup>*<sup>w</sup>* <sup>∆</sup> *<sup>x</sup>* <sup>=</sup> <sup>1</sup>

between these conflicting requirements constrained by the Eq. (34) [50]..

**Figure 16.** a) Original ultrasound image (b) Despeckled image using cycle spinning based CT using ST method. (c)Des‐ peckled image using cycle spinning based CT using HT method. (d)Despeckled image using cycle spinning based CT using SST method. (e) Despeckled image using direct CT method.

#### **5. Gaussian model for speckle noise**

Gaussian probability density is often used in image filtering. The Gaussian has the unique ability not to create new edges as its scale (standard deviation) is increased. This property enables the extraction of edges that represent different levels of detail in an image. As the scale is increased, the number of extracted weak and false edges reduces. However, at the same time edges shift from their true positions. The amount of shift an edge makes not only depends upon the scale of the Gaussian filter, but also on the intensity distributions of the underlying image [48]. Mathematically, a Gaussian filter modifies the input signal by convolution with a Gaussian function. Smoothing is commonly undertaken using line‐ ar filters such as the Gaussian function (the kernel is based on the normal distribution curve), which tends to produce good results in reducing the influence of noise with re‐ spect to the image. The 2D Gaussian distributions, with standard deviation σ for image X, is given by Eq. (32) [49].

$$\text{G(X, m, } \sigma) = \frac{1}{\sqrt{2\pi}} \mathbf{e}^{-\frac{1}{\mathfrak{T}} \left(\frac{\chi\_{-\mathfrak{m}}}{\sigma}\right)^2} \tag{32}$$

#### **5.1. Pre- or post –processing**

**5. Gaussian model for speckle noise**

226 Advancements and Breakthroughs in Ultrasound Imaging

using SST method. (e) Despeckled image using direct CT method.

is given by Eq. (32) [49].

Gaussian probability density is often used in image filtering. The Gaussian has the unique ability not to create new edges as its scale (standard deviation) is increased. This property enables the extraction of edges that represent different levels of detail in an image. As the scale is increased, the number of extracted weak and false edges reduces. However, at the same time edges shift from their true positions. The amount of shift an edge makes not only depends upon the scale of the Gaussian filter, but also on the intensity distributions of the underlying image [48]. Mathematically, a Gaussian filter modifies the input signal by convolution with a Gaussian function. Smoothing is commonly undertaken using line‐ ar filters such as the Gaussian function (the kernel is based on the normal distribution curve), which tends to produce good results in reducing the influence of noise with re‐ spect to the image. The 2D Gaussian distributions, with standard deviation σ for image X,

(a) (b) (c)

(d) (e)

**Figure 16.** a) Original ultrasound image (b) Despeckled image using cycle spinning based CT using ST method. (c)Des‐ peckled image using cycle spinning based CT using HT method. (d)Despeckled image using cycle spinning based CT

G(X, m, σ)= <sup>1</sup>

<sup>σ</sup> <sup>2</sup><sup>π</sup> <sup>e</sup> - 1 <sup>2</sup> ( <sup>X</sup>-m <sup>σ</sup> )<sup>2</sup>

(32)

Usually, medical ultrasound images are affected by the mixed noise, which is the combination of speckle noise and Gaussian noise. There are two factors that influence the usefulness of a smoothing filter. The first reduces the range of resolutions over which variations in the output appear by the filter variation ∆w, in the frequency domain be small and second factor is increase in spatial localization by a small spatial variance ∆x. These localization requirements in the spatial and frequency domain are conflicting and related by the uncertainty principle given in Eq. (33)

$$
\Delta \mathbf{w} \,\, \Delta \mathbf{x} \ge \frac{1}{4\pi} \tag{33}
$$

It has been shown that Gaussian functions are the only ones that provide the optimal trade off between these conflicting requirements constrained by the Eq. (34) [50]..

$$
\Delta w \, \Delta \chi = \frac{1}{4\pi} \tag{34}
$$

So Gaussian filters are widely used in image filtering. In [51], the removal of mixed noise using order statistic filter and wavelet domain Wiener filter is proposed. Authors have evaluated two methods. The first method comprises, order statistic prefilter and empirical Wiener filter, which is used to reduce the Gaussian noise. The disadvantage of this method is the higher time consumption. The second method is, order statistic filter for each decomposition level, where decomposition is carried out by the wavelet transform, followed by thresholding. The drawback of this method is that its efficiency is less than that of the first method (about 1dB) in removing the mixed noise. In [52], denoising of mixed noise in ultrasound images is presented. Combined Bayesian maximum a posterior (MAP) estimator and ST-PCNN (Soft threshold pulse coupled neural networks) method has been used for mixed noise reduction. The method removes the speckle noise considerably than the Gaussian noise that degrades the ultrasound images. The drawback of the method is either Gaussian noise or speckle noise is removed. Hence we present a method to remove residual Gaussian noise from despeckled image.

Two alternative algorithms are developed for reducing mixed noise in medical ultrasound images [53]. In the first alternative, the denoising method reduces the Gaussian noise by applying Gaussian filter in pre processing stage, then despeckling is performed using either wavelet transform, Laplacian pyramid transform or contourlet transform. The noise model for the first alternative (i.e. Gaussian noise removal in pre processing followed by despeckling) is given by the Eq.(35).

$$\mathbf{X}\_{\rm ij} = \mathbf{f}\_{\rm ij} \mathbf{n}\_{\rm ij} + \mathbf{g}\_{\rm ij} \tag{35}$$

where Xij represents the noisy pixel in the image X, fij represents the noise free pixel, nij and gij represent the multiplicative speckle noise and additive Gaussian noise, respectively. The indices i, j represent the spatial position over the image. We use transform domain filtering techniques [36,54,47] for despeckling along with Gaussian filter in pre processing stage for removal of Gaussian noise.

In the second alternative, the despeckling of medical ultrasound images is performed ei‐ ther using wavelet transform, Laplacian pyramid transform or contourlet transform and, then, it is followed by postprocessing stage in which Gaussian filter removes Gaussian noise from the despeckled image. However, the second alternative assumes the noise model given by the Eq.(36)

$$\mathbf{X}\_{\rm ij} = \left(\mathbf{f}\_{\rm ij} + \mathbf{g}\_{\rm ij}\right) \mathbf{n}\_{\rm ij} \tag{36}$$

well preserving edges and structures in the image. The parameters in the model are estimated

**Table 6.** Comparison of performance of denoising methods based on Gaussian filtering with despeckling methods.

(a) (b)

**Denoising methods SNR PSNR CC MSE Variance Computational**

25.22 30.23 0.9992 0.00050 0.0128 0.91

20.07 31.66 0.9965 0.00106 0.0426 0.38

29.99 35.15 0.9998 0.00031 0.0071 1.62

26.32 32.54 0.9995 0.00050 0.0120 0.89

19.76 30.12 0.9959 0.00133 0.0440 0.76

28.89 34.35 0.9996 0.00036 0.0088 2.25

19.06 29.94 0.9954 0.00121 0.0368 2.80

18.23 28.44 0.9972 0.00141 0.0416 2.24

27.74 34.21 0.9990 0.00037 0.0203 1.64

In [56], the authors have developed the adaptive weights smoothing algorithm, which is an iterative procedure in which the size of a neighbourhood is adaptive to the surface smoothness. In [57], the estimation of jump surfaces by local piecewise linear kernel smoothing is examined.

through an efficient iterative scheme.

0.0128 0.0876 0.0128

0.075 0.124 0.075

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0.124 0.204 0.124

0.075 0.124 0.075

**time (in Secs.)**

0.0876 0.5986 0.0876

0.0128 0.0876 0.0128

**Figure 17.** The 3×3 kernel with (a) σ = 0.5, (b) σ = 1.

First alternative with wavelet

First alternative with LP

First alternative with CT

Second alternative with WT

Second alternative with LP

Second alternative with CT

Despeckling method (WT

Despeckling method (LP transform) [54]

Despeckling method (CT

transform

transform

transform

transform

transform

transform

transform ) [36]

transform) [47]

The second alternative is investigated for image quality enhancement, due to noise removal, in an ultrasound image.

The experimentation is carried out using various kernel sizes and different values of σ. Larger values of σ produce a wider peak influencing the greater blurring. Kernel size is increased with increasing σ to maintain the Gaussian nature of the filter. Gaussian kernel coefficients depend on the value of σ. The Figure 17. shows different convolution kernels that approximate a Gaussian with σ. Gaussians are locally sensitive and can be made more spatially localized by decreasing parameter σ. It is observed that the kernel size 3×3 with σ = 0.5 yields better results than other kernels. It is found that larger kernels of size 5×5 or 7×7 produce better denoising effect but make the image more blurred. Thus, the empirically determined kernel size 3×3 and σ=0.5 are used in two alternative methods (Gaussian filter in Pre or Post processing).The two alternative methods are evaluated in terms of filter assessment parameters, namely, PSNR, SNR, MSE, variance and CC. The comparisons of the performance of the both alternatives with the despeckling methods discussed in [36,54, 47] are given in the Table 6. From the Table 6, it is observed that the Gaussian filter in pre processing stage is found to be more effective than that in despeckling based on Laplacian pyramid transform and contourlet transform. How‐ ever, the Gaussian filter in postprocessing stage is found to be more effective in despeckling based on wavelet transform. Thus, the Gaussian filter improves the performance of despeck‐ ling methods, because Gaussian noise is characterized by adding to each image pixel a value from a zero mean Gaussian distribution. The zero mean property of the distribution allows such noise to be removed by locally averaging pixel values [55]. Further, it is observed that, the Gaussian filter in pre processing stage followed by contourlet transform based despeckling method yields better visual enhancement than the other denoising methods, which is illus‐ trated in the Figure 18. The denoising and visual enhancement techniques developed in this study lead to improvement in the accuracy and reliability of automatic methods for medical ultrasound imaging systems.

#### **5.2. Linear regression model**

We present a linear regression based approach for clinical ultrasound image despeckling in the spatial domain. We propose a linear regression model for Gaussian noise representation of speckle noise for medical ultrasound images. This approach introduces an adaptive filter,

#### Speckle Noise Reduction in Medical Ultrasound Images http://dx.doi.org/10.5772/56519 229


**Figure 17.** The 3×3 kernel with (a) σ = 0.5, (b) σ = 1.

indices i, j represent the spatial position over the image. We use transform domain filtering techniques [36,54,47] for despeckling along with Gaussian filter in pre processing stage for

In the second alternative, the despeckling of medical ultrasound images is performed ei‐ ther using wavelet transform, Laplacian pyramid transform or contourlet transform and, then, it is followed by postprocessing stage in which Gaussian filter removes Gaussian noise from the despeckled image. However, the second alternative assumes the noise

The second alternative is investigated for image quality enhancement, due to noise removal,

The experimentation is carried out using various kernel sizes and different values of σ. Larger values of σ produce a wider peak influencing the greater blurring. Kernel size is increased with increasing σ to maintain the Gaussian nature of the filter. Gaussian kernel coefficients depend on the value of σ. The Figure 17. shows different convolution kernels that approximate a Gaussian with σ. Gaussians are locally sensitive and can be made more spatially localized by decreasing parameter σ. It is observed that the kernel size 3×3 with σ = 0.5 yields better results than other kernels. It is found that larger kernels of size 5×5 or 7×7 produce better denoising effect but make the image more blurred. Thus, the empirically determined kernel size 3×3 and σ=0.5 are used in two alternative methods (Gaussian filter in Pre or Post processing).The two alternative methods are evaluated in terms of filter assessment parameters, namely, PSNR, SNR, MSE, variance and CC. The comparisons of the performance of the both alternatives with the despeckling methods discussed in [36,54, 47] are given in the Table 6. From the Table 6, it is observed that the Gaussian filter in pre processing stage is found to be more effective than that in despeckling based on Laplacian pyramid transform and contourlet transform. How‐ ever, the Gaussian filter in postprocessing stage is found to be more effective in despeckling based on wavelet transform. Thus, the Gaussian filter improves the performance of despeck‐ ling methods, because Gaussian noise is characterized by adding to each image pixel a value from a zero mean Gaussian distribution. The zero mean property of the distribution allows such noise to be removed by locally averaging pixel values [55]. Further, it is observed that, the Gaussian filter in pre processing stage followed by contourlet transform based despeckling method yields better visual enhancement than the other denoising methods, which is illus‐ trated in the Figure 18. The denoising and visual enhancement techniques developed in this study lead to improvement in the accuracy and reliability of automatic methods for medical

We present a linear regression based approach for clinical ultrasound image despeckling in the spatial domain. We propose a linear regression model for Gaussian noise representation of speckle noise for medical ultrasound images. This approach introduces an adaptive filter,

) nij (36)

Xij

=(fij + gij

removal of Gaussian noise.

228 Advancements and Breakthroughs in Ultrasound Imaging

model given by the Eq.(36)

in an ultrasound image.

ultrasound imaging systems.

**5.2. Linear regression model**


**Table 6.** Comparison of performance of denoising methods based on Gaussian filtering with despeckling methods.

well preserving edges and structures in the image. The parameters in the model are estimated through an efficient iterative scheme.

In [56], the authors have developed the adaptive weights smoothing algorithm, which is an iterative procedure in which the size of a neighbourhood is adaptive to the surface smoothness. In [57], the estimation of jump surfaces by local piecewise linear kernel smoothing is examined.

where a, b, c and d are the constants, which are the linear regression model parameters for Gaussian representation G(m, σ) of speckle noise in the medical ultrasound images. The Eqs. (37) and (38) represent the linear regression model for Gaussian representation G(m, σ) of speckle noise in a medical ultrasound image with its PSNR value determined already [62]. The

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**Algorithm 1.** Linear regression model for Gaussian representation G (m, σ) of speckle noise.

Output: Linear regression model parameters for Gaussian representation G (m, σ) of speckle noise.

Step 2: Input despeckled ultrasound image Y obtained by using contourlet transform with cycle spinning.

Step 7: Subtract the simulated Gaussian noise G from despeckled image Y to obtain the resultant image (Y

Step 11: The Gaussian noise with mean m and standard deviation σ determined in the Step 10 is the modelled

Step 12: Repeat the Steps 1-11 for all the ultrasound images X and their corresponding despeckled images Y in

Step 13: Obtain the lines of best fit for the both PSNR vs. m and PSNR vs. σ obtained for all the ultrasound images

the Eqs.(37) and (38), which are the lines of best fit that form the linear regression model for representing the speckle noise in ultrasound image as Gaussian noise. The lines of best fit are found

) and the corresponding values of m and σ. Store the values of maximum PSNR and corresponding

(n1) in the Step 12 using the method of least square errors. i. e. determine the constants a, b, c and d of

^ ), i. e.

steps involved in this procedure are given in the Algorithm 1.

Input : Medical ultrasound image.

Step 1: Input medical ultrasound image X.

Step 4: Find the percentage of black pixels in Z.

obtained in the Step 5.

(i.e. Z = X – Y).

Y ^ =Y-G.

(Y ^

(Y ^

Stop

values of m and σ.

the data set.

Gaussian noise for the image X.

using goodness of fit static.

Step 3: Find the error image Z as difference between X and Y

Step 5: Find the mean (m) and standard deviation (σ) of Z.

) obtained in the Step 7. Set Y = Y

Step 6: Simulate the Gaussian noise G with mean m and standard deviation σ

Step 8: Find the PSNR, mean and standard deviation of the resultant image

Step 9: Repeat the Steps 3-8 until the percentage of black pixels in Z reach 99.9%.

Step 14: Output linear regression model parameters a, b, c and d of Eqs.(37) and (38).

^ .

Step 10: Find the maximum of the PSNR values obtained for the iterated despeckled images

Start

**Figure 18.** a) Original ultrasound image,(b) Denoised image using Gaussian filter, (c) Denoised image using 1st alterna‐ tive with WT method (d) Denoised image using 1st alternative with LP method. (e) Denoised image using 1st alternative with CT method. (f) Denoised image using 2nd alternative with WT method. (g) Denoised image using 2nd alternative with LP method. (h) Denoised image using 2nd alternative with CT method.

In [58, 59], an anisotropic diffusion algorithm has been proposed for Gaussian noise removal. In [60], a bilateral filter to remove Gaussian noise is developed. In [61], a window based linear regression filter for echo cardiographic image denoising is proposed. The main draw backs of the above algorithms are that they need more computational time and complex circuit to implement them.

We consider a medical ultrasound image X and the corresponding despeckled image Y obtained by using the contourlet transform with cycle spinning [46]. The subtracted image Z=X-Y is the error image containing speckle noise. We find the mean m and standard deviation σ of Z and then simulate Gaussian noise G with these values of m and σ. The removal of this Gaussian noise G from despeckled image Y yields the new despeckled image Y ^ , i.e. Y ^ =Y-G, which is further subtracted from the original image X to obtain the new error image Z containing the residual speckle noise. This procedure is repeated until the percentage of black pixels in error image Z reaches 99.9. We determine the maximum value of PSNR and the corresponding values of mean m and standard deviation σ using the iterated despeckled images Y ^ . This procedure is applied for all the medical ultrasound images Xi , i=1,..., 63, in the dataset, yielding the two sets of data points (PSNRi,mi ) and (PSNRi ,σi ), i=1,...,n1 exhibits linear correlation. Using the meth‐ od of least square errors, we obtain the lines of best fit for these data, namely:

$$\mathbf{a} \text{ m=a } \text{"PSNR+b"} \tag{37}$$

$$\circ \circ \circ \circ \text{PSNR} + \text{d} \tag{38}$$

where a, b, c and d are the constants, which are the linear regression model parameters for Gaussian representation G(m, σ) of speckle noise in the medical ultrasound images. The Eqs. (37) and (38) represent the linear regression model for Gaussian representation G(m, σ) of speckle noise in a medical ultrasound image with its PSNR value determined already [62]. The steps involved in this procedure are given in the Algorithm 1.

**Algorithm 1.** Linear regression model for Gaussian representation G (m, σ) of speckle noise.


Input : Medical ultrasound image.

In [58, 59], an anisotropic diffusion algorithm has been proposed for Gaussian noise removal. In [60], a bilateral filter to remove Gaussian noise is developed. In [61], a window based linear regression filter for echo cardiographic image denoising is proposed. The main draw backs of the above algorithms are that they need more computational time and complex circuit to

(e) (f) (g) (h)

**Figure 18.** a) Original ultrasound image,(b) Denoised image using Gaussian filter, (c) Denoised image using 1st alterna‐ tive with WT method (d) Denoised image using 1st alternative with LP method. (e) Denoised image using 1st alternative with CT method. (f) Denoised image using 2nd alternative with WT method. (g) Denoised image using 2nd alternative

(a) (b) (c) (d)

We consider a medical ultrasound image X and the corresponding despeckled image Y obtained by using the contourlet transform with cycle spinning [46]. The subtracted image Z=X-Y is the error image containing speckle noise. We find the mean m and standard deviation σ of Z and then simulate Gaussian noise G with these values of m and σ. The removal of this Gaussian

further subtracted from the original image X to obtain the new error image Z containing the residual speckle noise. This procedure is repeated until the percentage of black pixels in error image Z reaches 99.9. We determine the maximum value of PSNR and the corresponding values

^ , i.e. Y ^

, i=1,..., 63, in the dataset, yielding the two sets

), i=1,...,n1 exhibits linear correlation. Using the meth‐

m=a \* PSNR + b (37)

σ =c \* PSNR + d (38)

^

=Y-G, which is

. This procedure is

noise G from despeckled image Y yields the new despeckled image Y

of mean m and standard deviation σ using the iterated despeckled images Y

,σi

od of least square errors, we obtain the lines of best fit for these data, namely:

applied for all the medical ultrasound images Xi

) and (PSNRi

with LP method. (h) Denoised image using 2nd alternative with CT method.

230 Advancements and Breakthroughs in Ultrasound Imaging

implement them.

of data points (PSNRi,mi

The data points (PSNRi,mi ) and (PSNRi ,σi ), i=1,...,63 obtained for 63 typical ultrasound images of the image data set are stored and then used for regression of PSNR on m and also PSNR on σ. In the ultrasound image data set used for building regression model for Gaussian repre‐ sentation of speckle noise, we select a reference image Xref for which the PSNR value is minimum. Given an arbitrary input medical ultrasound image X, we compute the PSNR of X with respect to the reference image Xref. Using linear regression model (Eqs. (37) and (38)), we estimate the values of mean m and standard deviation σ of the Gaussian noise, which is removed from the input original image. The resultant image is the despeckled image (Y). The 'goodness of fit' statistic for the lines of regression is given by two quantities, namely, the sum of squares due to error of the fit (SSE) and root mean squared error (RMSE).The SSE is given by Eq. (39) :

$$\text{SSE} \sim \sum\_{i=1}^{n} \left( \mathbf{y}\_{i} \cdot \hat{\mathbf{Y}}\_{i} \right)^{2} \tag{39}$$

**Algorithm 2.** Despeckling based on linear regression model for Gaussian noise estimation.

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Step 2: Compute PSNR of image X with respect to the reference image Xref prescribed by

Step 3: Obtain Gaussian noise estimation G(m, σ) corresponding to the PSNR computed in

Step 4: Apply Gaussian filter with estimated mean (m) and standard deviation (σ), obtained n the Step 3, to the input image X, which yields the despeckled image Y.

Step 4: Apply Gaussian filter with estimated mean (m) and standard deviation (σ), obtained in the Step 3, to the

The linear regression model parameters a, b, c and d for Gaussian representation of speckle noise are computed for the dataset of 63 ultrasound images and the linear regression model equations are (Eqs.(37) and (38)),where a=-6.129e-007, b=2.742e-005, c=-0.0002192, d=0.01004, with the measures of 'best fit' are SSE=4.682e-009, RMSE=8.833e-006 for mean vs. PSNR and SSE=0.0006471, RMSE=0.003284 for standard deviation vs. PSNR. The Figures(20 and 21) show the lines of best fit for mean vs. PSNR and standard deviation vs. PSNR, respectively, which

the Step 2 using the regression model (Eqs.(37) and (38))

input image X, which yields the despeckled image Y.

are used for Gaussian noise estimation and removal.

Input: Medical ultrasound image. Output: Despeckled image.

Step 1: Input medical ultrasound image X.

the regression model.

Step 5: Output the despeckled image Y.

**Figure 20.** Linear regression of mean on PSNR

Start

Stop.

where Yi =actual mean value of Gaussian noise, Y ^ i= estimated mean value of Gaussian noise, and

n1= total number of ultrasound images in the dataset.

The RMSE is given by Eq. (40) :

$$\text{RMSE} = \sqrt{\frac{\text{SSE}}{n\_1}} \tag{40}$$

If the SSE and RMSE values are closer to zero, they indicate better fit. The general model for Gaussian noise estimation and removal in despeckling ultrasound image is shown in the Figure 19.

**Figure 19.** The general model for Gaussian representation of speckle noise.

The steps involved in the denoising process, shown in Figure 19 are given in the Algorithm 2.

**Algorithm 2.** Despeckling based on linear regression model for Gaussian noise estimation.

Input: Medical ultrasound image.

Output: Despeckled image.

Start

The data points (PSNRi,mi

232 Advancements and Breakthroughs in Ultrasound Imaging

by Eq. (39) :

where Yi

Figure 19.

and

) and (PSNRi

=actual mean value of Gaussian noise, Y

n1= total number of ultrasound images in the dataset.

Output despeckled image Y

**Figure 19.** The general model for Gaussian representation of speckle noise.

The RMSE is given by Eq. (40) :

Input image X ,σi

of the image data set are stored and then used for regression of PSNR on m and also PSNR on σ. In the ultrasound image data set used for building regression model for Gaussian repre‐ sentation of speckle noise, we select a reference image Xref for which the PSNR value is minimum. Given an arbitrary input medical ultrasound image X, we compute the PSNR of X with respect to the reference image Xref. Using linear regression model (Eqs. (37) and (38)), we estimate the values of mean m and standard deviation σ of the Gaussian noise, which is removed from the input original image. The resultant image is the despeckled image (Y). The 'goodness of fit' statistic for the lines of regression is given by two quantities, namely, the sum of squares due to error of the fit (SSE) and root mean squared error (RMSE).The SSE is given

> <sup>µ</sup> ( ) 2 i i =1 n SSE= Y -Y i

> > ^

1

If the SSE and RMSE values are closer to zero, they indicate better fit. The general model for Gaussian noise estimation and removal in despeckling ultrasound image is shown in the

The steps involved in the denoising process, shown in Figure 19 are given in the Algorithm 2.

Compute PSNR

), i=1,...,63 obtained for 63 typical ultrasound images

<sup>å</sup> (39)

SSE RMSE= <sup>n</sup> (40)

Estimate m, σ using linear regression model for noise

Apply Gaussian filter with estimated m, σ

i= estimated mean value of Gaussian noise,


Stop.

The linear regression model parameters a, b, c and d for Gaussian representation of speckle noise are computed for the dataset of 63 ultrasound images and the linear regression model equations are (Eqs.(37) and (38)),where a=-6.129e-007, b=2.742e-005, c=-0.0002192, d=0.01004, with the measures of 'best fit' are SSE=4.682e-009, RMSE=8.833e-006 for mean vs. PSNR and SSE=0.0006471, RMSE=0.003284 for standard deviation vs. PSNR. The Figures(20 and 21) show the lines of best fit for mean vs. PSNR and standard deviation vs. PSNR, respectively, which are used for Gaussian noise estimation and removal.

**Figure 20.** Linear regression of mean on PSNR

(a) (b) (c)

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**Figure 22.** a) Original ultrasound image (b) Despeckled image using the contourlet transform with cycle spinning (c) Denoised image using the proposed linear regression model. The box Indicates the region of image in (b) and (c)

(a) (b) (c)

(d) (e) (f)

**Figure 23.** a) A sub image of original ultrasound image (b) sub image of despeckling using Wiener filter (c) sub image of despeckling using WT method (d) sub image of despeckling using CT method (e) sub image of despeckling using

cycle spinning based CT method (f) sub image of despeckling using proposed linear regression model.

showing prominent visual enhancement due to the despeckling.

**Figure 21.** Linear regression of standard deviation on PSNR

The comparison of the results of the proposed method with the contourlet transform method (with cycle spinning) is given in the Table 7. It is observed that the image quality enhancement obtained by the despeckling method based on linear regression model is better than that obtained by the contourlet transform method in terms of PSNR and computational time required for denoising.


**Table 7.** Comparison of performance of despeckling based on contourlet transform and proposed method based on linear regression model.

The Figure 22. shows a sample medical ultrasound image, its despeckled image using contourlet transform with cycle spinning and the denoised image using the linear regression model respectively. The visual quality of image enhancement can also be observed from the sample image and its denoised image. The anatomical structures are more clearly visible in the Figure 22.(c) than that in Figure 22.(b). The box indicates the region of image in (b) and (c) showing prominent visual enhancement due despeckling methods.

The Figure 23 shows the visual enhancement due to various despeckling methods for com‐ parison. (a) shows the sub image of original image, The Figure 23 (b)-(f) indicates the sub image showing visual enhancement due to different despeckling methods namely, Wiener filter with (3X3),wavelet transform method, contourlet transform method, cycle spinning based contour‐

**Figure 22.** a) Original ultrasound image (b) Despeckled image using the contourlet transform with cycle spinning (c) Denoised image using the proposed linear regression model. The box Indicates the region of image in (b) and (c) showing prominent visual enhancement due to the despeckling.

**Figure 21.** Linear regression of standard deviation on PSNR

234 Advancements and Breakthroughs in Ultrasound Imaging

**Denoising methods**

required for denoising.

linear regression model.

The comparison of the results of the proposed method with the contourlet transform method (with cycle spinning) is given in the Table 7. It is observed that the image quality enhancement obtained by the despeckling method based on linear regression model is better than that obtained by the contourlet transform method in terms of PSNR and computational time

Contourlet transform using cycle spinning 35.91 24.09 Proposed method based on linear regression model 36.98 0.34

**Table 7.** Comparison of performance of despeckling based on contourlet transform and proposed method based on

The Figure 22. shows a sample medical ultrasound image, its despeckled image using contourlet transform with cycle spinning and the denoised image using the linear regression model respectively. The visual quality of image enhancement can also be observed from the sample image and its denoised image. The anatomical structures are more clearly visible in the Figure 22.(c) than that in Figure 22.(b). The box indicates the region of image in (b) and (c)

The Figure 23 shows the visual enhancement due to various despeckling methods for com‐ parison. (a) shows the sub image of original image, The Figure 23 (b)-(f) indicates the sub image showing visual enhancement due to different despeckling methods namely, Wiener filter with (3X3),wavelet transform method, contourlet transform method, cycle spinning based contour‐

showing prominent visual enhancement due despeckling methods.

**PSNR (in dB)** **Computational Time (in Secs.)**

**Figure 23.** a) A sub image of original ultrasound image (b) sub image of despeckling using Wiener filter (c) sub image of despeckling using WT method (d) sub image of despeckling using CT method (e) sub image of despeckling using cycle spinning based CT method (f) sub image of despeckling using proposed linear regression model.

let transform method and proposed linear regression model. The prominent visual enhance‐ ment is observed using the proposed linear regression model.

war Scanning Centre, Bijapur, Karnataka, India, for providing the ultrasound images of

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237

and Sharan Badiger3

1 Dept. of P.G. Studies and Research in Computer Science, Gulbarga University, Gulbarga,

3 Dept. of Medicine, Sri.B.M.Patil Medical College, BLDE University, Bijapur, Karnataka,

[1] Suetens Paul, Fundamentals of Medical Imaging, (1st Edition), Cambridge universi‐

[2] Hedrick W.R. and Hykes D.L., Image and signal processing in diagnostic ultrasound

[3] Godman J.W., Some Fundamental Properties of Speckle, Jl. Opt. Soc. Am. 1976; 66

[4] Burckhardt C.B., Speckle in Ultrasound B Mode Scans, IEEE Trans. Sonics Ultrason‐

[5] Yongjian Y. and Acton S.T. Speckle reducing anisotropic diffusion, IEEE Trans. Im‐

[6] Prager R.W., Gee A.H., Treece G.M., and Berman L., Speckle detection in ultrasound images using first order statistics, GUED/ F-INFENG/TR 415, University of Cam‐

[7] Zong X., Laine A. F., and Geiser E. A. Speckle reduction and contrast enhancement of echocardiograms via multiscale nonlinear processing, IEEE Trans.on Medical. Imag‐

imaging, Journal of Diagnostic Medical Sonography 1989; 5(5): 231–239.

2 Dept. of Computer Science and Engineering, B.L.D.E.A's Dr. P.G.H. College of

kidney, liver and also for helpful discussions.

P.S. Hiremath1\*, Prema T. Akkasaligar2

Engineering, Bijapur, Karnataka, India

ty, U.K.; 2002.p145-182.

(11):1145-1149.

ics 1978;25:1-6.

ing., 1998;17:532-540.

\*Address all correspondence to: hiremathps53@yahoo.com

age Processing. Nov. 2002; 11(11):1260–1270.

bridge, Dept. of Engineering, July, 2002: 1–17.

**Author details**

Karnataka, India

India

**References**

The proposed method estimates the Gaussian noise content in the input medical ultrasound image for denoising the image efficiently. Hence, it is easily amenable for building embedded system software for ultrasound imaging equipments in order to display the high quality images, which helps the medical expert in the diagnosis with greater accuracy.
