Efficient and robust similarity measure for denoising ultrasound images in non-local framework

Abstract

Denoising of medical image modalities is one among the foremost basic issues in medical image process. Medical modalities such as ultrasound images suffer from multiplicative speckle noise. This noise consequently reduces the contrast of ultrasound images and adversely affects the other medical image processing tasks such as medical image registration, image super-resolution, and image segmentation. Therefore, one of the important objectives of any denoising algorithm is to attenuate the speckle noise effectively and also preserve the significant medical details in the denoised image. The main focus of this paper is the reduction of speckle noise for ultrasound images using various similarity measures in non-local framework. Through exhaustive experiments conducted on real ultrasound images, B-mode and simulated synthetic images demonstrate that, the Square chord and Chi-square distance-based similarity measures are the most effective similarity measure used in non-local framework for denoising of ultrasound images.

Keywords

Image denoising speckle noise non-local means edge preservation

1 Introduction

Ultrasound (US) imaging is one among the outstanding diagnostic techniques by which we can view inside (like liver, pancreas, abdomen and kidney) of a human body. Also, US imaging technique is collectively the foremost powerful and non-invasive technique because it is relatively cheap and there are no ionizing radiations in US images. However, US images are often corrupted by speckle noise [1 –5] and various denoising algorithms have been developed to attenuate speckle noise. But, there is a scope of improvement in the denoising strength of various speckle noise removal algorithms. When the backscattered acoustic pulses received, this phenomenon leads to both constructive and destructive interference and appears as a granular pattern called speckle noise. Its multiplicative behavior in different conditions is explored well by Dainty [1] and Goodman [2]. Therefore, to extract the crucial information such as fine edge details from the US image which is corrupted by speckle noise is one amongst the difficult tasks.

Numerous approaches have been described in literature [6–16] to reduce the impact of speckle noise in ultrasound images, however most of them were unsuccessful due to elimination of diffused and weak edges. These weak edges play crucial role in the diagnosis to the patients. Due to elimination of weak edges, the ultrasound images are not easy to interpret particularly for physicians and doctors and hence not able to provide appropriate diagnosis to the patients. In recent years, wavelet based denoising methods are extensively used in the area of medical image processing because of multi-resolution, sparseness, edge preservation and time localization properties of wavelet transform. The aim of wavelet-based denoising techniques is to develop optimum shrinkage function to reduce the speckle noise without affecting the significant information present in the critical areas of medical images like edge lesion [17 –26]. The main drawback associated with wavelet-based approaches is that they are not able to manage the appropriate balance between feature preservation and speckle suppression in denoised images. Later, neighborhood-based noise filtering techniques came into existence in image denoising. The Bilateral (BF) and non-local means (NLM) filters are the two standard image denoising approaches which fall in the category of neighborhood filtering. In Bilateral filtering [27] approach, a weighted average is calculated by creating the square neighborhood around each pixel of noisy image and weights are computed by considering the both spatial (geometric) and range (intensity difference) distance between the noisy pixels. The noisy pixels present in neighborhood of a center pixel I_n (x, y) provide important structural information and hence improved denoising performance is observed in case of neighborhood-based noise filtering approaches. Recently, NLM filtering approach set the new trend in the field of neighborhood-based noise filtering approaches, because it effectively use the self-similarities and patterns exist in image in the form of local patches instead of using single pixel value (e.g. BF). In non-local means algorithm, patches of same size are created at each pixel. Then a patch is searched in a search window that have similar content information (edge/ fine structures) is compared to reference patch created at each pixel i of an image. Then calculate the weighted average of all those patches found in a search window to compute the denoised pixel value at i of a reference patch. Since, the time complexity of searching a patch in a search window is very high, therefore, we create small patches around the central pixel of a reference patch to compute the denoised pixel value. Although the NLM approach is very effective and simple, it has three major disadvantages. First, the time complexity of the algorithm is too high. Second, if there are two rotated patches such that one patch is rotated version of other patch but they have similar content then the NLM algorithm is misclassified those similar patches. Third, the denoising performance of NLM algorithm is not good on edge or fine structure related areas [28]. In NLM algorithm, the matching of two different patches is computed by Euclidean distance (similarity) measure using the pixel values of two patches. The similarity (distance) computation between two patches by any similarity measure is time consuming. To overcome from these limitations, Coupe et al. [29] introduced the optimized Bayesian NLM (OBNLM) approach, which uses Pearson distance to calculate the similarity present between two patches in the non-local manner. But, the time complexity of OBNLM approaches is high and to overcome from such computational burden, Singh et al. [28] designed the hybrid approach for denoising of US images. But, the use of Pearson distance measure in non-local framework motivates us to explore different similarity measure for denoising of speckle corrupted US images. In this paper, we explore different similarity measures for speckle noise reduction and through exhaustive experimental analysis we observed that the square chord, Chi-square and Bray-Curtis-based distance measures provides better denoising performance than the classical Euclidean norm which is used for patch matching and weight computation process in traditional nonlocal framework. Also, experiments performed on various test images reveal that the proposed measures have capabilities to restore the significant edge details as compared to traditional NLM filtering approach.

In the following section, the introduction of speckle noise model is presented. Material and methods is presented in section 3 with short explanation of non-local means filter and different similarity measures. In Section 4, a large number of experiments are performed and comparative performance analysis of robust and efficient similarity measure used in non-local framework with traditional speckle reduction (SR) approaches is presented. Finally, Section 5 concludes the paper.

2 Speckle noise model

In literature, we saw that speckle noise present in US images is roughly modeled as multiplicative noise which is expressed as follows: $g (x) \approx f (x) μ (x)$ (1)

In Equation (1), x is the pixel location; f ( x ) is the original image; g ( x ) is the observed image; and μ ( x ) is a Gaussian noise which follows $N (0, σ^{2})$ distribution having zero mean and variance σ². Recently, it is observed from the studies of speckle behavior done by various authors [1 –5] that, the above speckle noise model is also approximated by Rayleigh [30], Gamma [31] or Fisher–Tippett distribution [32]. Therefore, it is highly desirable to know the mathematical model of a speckle noise; hence, we choose the following speckle noise model which also successfully applied in many studies [1 –5]. $g^{noisy} (x) \approx f^{original} (x) + f^{γ} (x) μ^{noise} (x)$ (2)

Where, x is the pixel location, f^original ( x ) is the original image which is noise-free, μ^noise ( x ) is Gaussian noise having zero mean and variance σ². The value of γ depends on the ultrasound image capturing devices. In our experimental study, for real ultrasound and B-mode images, the value of γ is 0.5 and for synthetic images, the value of γ is 1. The model defined by Equation (2) is also suggested by Loupas et al. [6]. According to Loupes, this model is most appropriate for real US images and it is better from Rayleigh model when the factor γ is set to 0.5. Along these lines, in every experiment, we have chosen this model to simulate speckle noise in US images.

3 Material and methods

The neighborhood filtering based method uses patch based features approach to remove the noise and NLM filter belongs to the category of patch-based approach which is a special case of Yaroslavsky filter [33].

3.1 Non-local means (NLM) filter

In NLM approach [28], it considers the region based comparison instead of pixel based and the denoised pixel value is estimated by the weighted average of patches in a search window. The NLM filter is a strong denoising algorithm, but classical NLM approach is not applicable to deal with speckle noise directly. Consequently, many authors provide the different combination of NLM with local statistics [34] and maximum likelihood estimation [30, 35] to effectively reduce speckle noise from US images. Therefore, our main aim is to provide efficient and robust similarity measure in NLM framework for denoising of ultrasound images without using the combinations of different statistical techniques. Consider the vector representation of noisy image ${g (x), x \in R \times R}$ and formulation of classical non-local means filter is described as follows. ${\hat{f}}_{NLM} (x) = \frac{\sum_{y \in SW} w_{NLM} (x, y) g^{noisy} (y)}{\sum_{y \in SW} w_{NLM} (x, y)}$ (3) $\begin{matrix} w_{NLM} (x, y) = exp (- d (x, y)) \\ and \\ d (x, y) = \frac{{∥ P_{x} - P_{y} ∥}_{2, G α}^{2}}{h^{2}} \end{matrix}$ (4)

Where, SW is the search window, w_NLM ( x , y ) represent the similarities exist between two patches P _x and P _y centered at pixel x , y respectively and h is the parameter which controls the amount of smoothing required to filter out the noise from an image. In the following section, firstly we provide the importance of distance measure in the field of image processing and then, the case study of different similarity measure used for patch matching process in non-local framework is revealed.

3.2 Proposed efficient and robust similarity measure in NLM framework

A continuous accumulation of efforts are accomplished by various authors to provide the appropriate distance measures for solving various image processing problems like pattern classification, clustering, and image denoising. Since, distance measures provide the information about the closeness of two vectors or blocks or objects. Therefore, it is highly desirable to select appropriate distance measure according to the choice of problem we are dealing because it affects the overall accuracy of an algorithm. In most of the image processing problems like image denoising, Euclidean distance is a good choice to measure the similarity exists between two patches or blocks. It means smaller the Euclidean distance, higher is the possibility that the two image patches or blocks have same information content. However, once the noise is multiplicative (speckle), as in case of US images, Euclidean distance is not an appropriate distance measure to attenuate speckle noise and preserve fine edge details efficiently. Therefore, instead of using Euclidean as a distance measure for patch matching process used in NLM framework, there are several other distance measures which are more reliable to deal with multiplicative noise effectively and provide better denoising performance than the Euclidean distance measure. We investigated seven different distance measures and used it for weighted computation process in the NLM framework. Therefore, the distance d ( x , y ) is replaced by Dist_i ( x , y ) where i ∈ {1, 2, 3, . . . , 7}. Where,

$\begin{matrix} {Dist}_{i} (x, y) = \\ {\begin{matrix} i = 1, for Euclidean distance (ED) \\ i = 2, for SquareChord distance (SCD) \\ i = 3, for Manhattan distance (MD) \\ i = 4, for Chi - Square distance (CSD) \\ i = 5, for Pearson distancee (PD) \\ i = 6, for Bray - Curtis distance (BCD) \\ i = 7, for Geman - McClure distance (GMD) \end{matrix} \end{matrix}$ (5)

Dist_i ( x , y ) is the distance between two patches centered at pixel x , y . To compute the distance, we have converted 5 × 5 patch into 25 × 1 one-dimensional vector. Then, the distance is computed between two one-dimensional vectors, where, N is the number of samples present in a vector. The mathematical description of all seven distance measures are briefly discussed below.

3.2.1 Euclidean distance (ED)

Euclidean distance is straight-line distance between two vectors in n-dimension. It is like a distance between two points in x-y plane. For calculating the similarity between two patches, instead of using n-dimension we have used n gray values of each patch. ${Dist}_{1} (x, y) = \sqrt{\sum_{j = 1}^{N} (x_{j} - y_{j})^{2}}$ (6)

3.2.2 Square chord distance (SCD)

The Square-Chord distance is calculated by sum of the squares of the differences between the square-root of gray values of each patch. ${Dist}_{2} (x, y) = \sum_{j = 1}^{N} (\sqrt{x_{j}} - \sqrt{y_{j}})^{2}$ (7)

3.2.3 Manhattan distance (MD)

The Manhattan distance between two points in x-y plane is calculated along the axes at right angles. It is the sum of absolute difference between the gray values of two patches. In other words, it is a L₁-norm of the difference. We can also say that in a chess board, the distance covered by the rook from one square to another square is Manhattan distance. So, for n gray values of each patch, the Manhattan distance between two patches will be ${Dist}_{3} (x, y) = \sum_{j = 1}^{N} | x_{j} - y_{j} |$ (8)

3.2.4 Chi-Square distance (CSD)

Chi-square distance is a weighted Euclidean distance where weights are the inverse of sum of gray values from each patch. It can also be used as a measure of dissimilarity between two histograms of two different images. The Chi-square distance has the property of distributional equivalence, meaning that it ensures that the distances between rows and columns are invariant when two columns (or two rows) with identical profiles are aggregated. ${Dist}_{4} (x, y) = \sum_{j = 1}^{N} \frac{(x_{j} - y_{j})^{2}}{(x_{j} + y_{j})}$ (9)

3.2.5 Pearson distance (PD)

The Pearson distance is a correlation distance based on Pearson’s product-momentum correlation coefficient of the two sample vectors. ${Dist}_{5} (x, y) = \sum_{j = 1}^{N} \frac{(x_{j} - y_{j})^{2}}{y_{j}}$ (10)

3.2.6 Bray–curtis distance (BCD)

Bray–Curtis distance is often used by ecologists to calculate the dissimilarity between two large data sets. ${Dist}_{6} (x, y) = \frac{\sum_{j = 1}^{N} | x_{j} - y_{j} |}{\sum_{j = 1}^{N} (x_{j} - y_{j})}$ (11)

3.2.7 Geman-McClure distance (GMD)

The mathematical description of Geman-McClure distance formulation is represented below.

${Dist}_{7} (x, y) = \frac{y_{j}}{{(1 + {∥ x_{j} - y_{j} ∥}_{2}^{2})}^{2}}$ (12)

4 Experimental analysis

We have investigated different similarity measures in NLM framework to judge the performance for the denoising of speckle noise corrupted US images. All algorithms have been tested in Visual C++8.0.We have performed four different simulations on different test cases of ultrasound images to observe the efficacy of proposed different similarity measures. Experiment I is performed on the synthetic images having various level of speckle noise. B-mode ultrasound images are used in the experiment II and experiment III is conducted on simulated ultrasound kidney image. Both images are created using the Field II [36, 37] simulation. In experiment IV, we have tested all the denoising algorithms to real ultrasound images such as Pancreas and Tarsal Ganglion Cyst images. These real images are downloaded from [38]. In synthetic images, we have added speckle noise by setting γ = 1 and in real US images by setting γ = 0.5 in Equation (2). Also, a comparative performance analysis of different distance measures used in NLM are visually and quantitatively performed using well-known quantitative measures like signal-to-noise ratio (SNR) [39], Peak signal-to-noise-ratio (PSNR) [39], mean-square error (MSE), mean structural similarity index (MSSIM) [40] and method noise [41] which are briefly explained as follows.

PSNR: To evaluate the performance of denoising algorithms, PSNR is computed between the original image signal g and the denoised image signal $\hat{g}$ .

PSNR = 10 {log}_{10} (\frac{255^{2}}{MSE})

(13)

$MSE = \frac{1}{MN} \sum_{i = 0}^{MN - 1} {(g (i) - \hat{g} (i))}^{2}$ (14)

The PSNR is measured in decibels (dB) and a higher value indicates to a lower noise and therefore a higher quality.

SNR: The performance of denoising algorithms is also assessed using SNR [33] values, which is given below.

SNR = 10 {log}_{10} \frac{\sum_{i = 0}^{MN - 1} {(g (i)^{2} - \hat{g} (i)^{2})}^{2}}{\sum_{i = 0}^{MN - 1} {(g (i) - \hat{g} (i))}^{2}}

(15)

MSSIM: To estimate the similarity between two images on the basis of their structure information, structure similarity is used [40]. The SSIM is calculated between two patches by using mean, correlation and variance. The SSIM between two patches x and y is given below.

$SSIM (x, y) = \frac{(2 μ_{f} μ_{\hat{g}} + C_{1}) (2 σ_{f \hat{g}} + C_{2})}{(μ_{f}^{2} μ_{\hat{g}}^{2} + C_{1}) (σ_{f}^{2} σ_{\hat{g}}^{2} + C_{2})}$ (16)

Where, μ_f and $μ_{\hat{g}}$ are the mean while $σ_{f}^{2}$ and $σ_{\hat{g}}^{2}$ are the variance of gray values from patches x and y, respectively. $σ_{f \hat{g}}$ is the covariance of f and $\hat{g}$ and C₁ = (K₁L) ²,C₂ = (K₂L) ² are two parameters which avoid weak denominator condition. The value of parameter L is vary from the domain of pixel values (255 for 8-bit gray scale images) and K₁ = 0.01 and K₂ = 0.03 are default values, which are computed from extensive experiments [40]. The value of MSSIM is calculated using the value of SSIM from Equation (16). $MSSIM = \frac{1}{MN} \sum_{x = 0}^{M - 1} \sum_{y = 0}^{N - 1} SSIM (x, y)$ (17)

The MSSIM value occurs between [0, 1] and high value denotes better edge preservation.

Method noise: The denoising performance of various despeckling methods degrade the original edge and structural information in medical images, hence the loss of useful fine edge content is observed in denoised medical images. Therefore, method noise is being used as a similarity measure to assess the loss of various structural and edge information occurred by applying the denoising algorithm [41]. The formulation of method noise $\hat{n}$ is described in Equation (18):

\hat{n} = g^{noisy} - {\hat{g}}^{denoised}

(18)

Where, g^noisy is the observed image and ${\hat{g}}^{denoised}$ is the image obtained after applying denoising methods.

CNR: To assess deterioration of contrast and to determine noise in the image, contrast-to-noise ratio (CNR) is used. It is described as the ratio of the mean difference of two Region of Interest (ROI) and the background noise [42, 43].

CNR = \frac{m_{R} - m_{Re f}}{σ}

(19)

Where, m_R is the mean intensity of the ROI, m_Ref is the mean intensity of the background ROI and σ is the standard deviation of the image.

IEF: The performance of the restoration process can also be quantified using Image Enhancement factor (IEF). It is the ratio of mean square error of noisy and original image to the restored and original image. The higher value of IEF is the indication of better image restoration.

IEF = \frac{\sum_{i = 0}^{MN - 1} (N (i) - o (i))^{2}}{\sum_{i = 0}^{MN - 1} (R (i) - o (i))^{2}}

(20)

Where, N, R and O are the noisy, restored and original images respectively.

4.1 Experiments on synthetic images

We have used various quantitative and qualitative measures to test the performance of various distance measures. We have created a synthetic image using Scilab 6.0.1 [44], which contains concentric circles, triangles, lines and rectangles with size of 256 × 256 as shown in Fig. 1. Moreover, we have added different level of simulated speckle noise to synthetic image using Scilab 6.0.1 [44].

Fig. 1

Results of experiment I: (a) Noise-free image, (b) Image having simulated speckle noise variance σ² = 2.5; denoised images obtained from, (c) SRAD filter, (d) Kaun filter, (e) Frost filter, (f) Lee filter, (g) Bilateral filter, (h) Guided filter, (i) NLM (ED), (j) NLM (SCD), (k) NLM (MD), (l) NLM (CSD), (m) NLM (PD), (n) NLM (BCD), (o) NLM (GMD).

It is clearly observed from PSNR, SNR values obtained by different similarity measures as shown in Tables 1, 2 and from visual investigation of denoised images in Fig. 1, method noise images in Fig. 2 that the results obtained from classical speckle noise reduction filters and classical NLM filter are not satisfactory since a fair amount of speckle noise is still visible after denoising. The denoised result obtained by the different speckle noise reduction filters and different distance measures used in non-local framework is shown inFig. 1(c)–(o).

Table 1

Presents a comparison of PSNR values of various SR filters and different similarity measure used in NLM framework at various level (σ² = 0.5, 1.0, 1.5, . . . , 5.0)of speckle noise

Methods	0.5	1	1.5	2	2.5	3	3.5	4	4.5	5	Avg. PSNR
Noisy	34.52	28.49	24.99	22.46	20.56	18.93	17.66	16.49	15.53	14.73	21.436
Lee	35.12	29.58	28.86	28.03	27.12	26.61	26.27	25.87	25.63	25.41	27.85
Frost	35.56	33.08	30.71	29.80	28.82	28.15	27.72	27.36	27.15	27.09	29.544
Kaun	42.80	37.63	34.24	32.57	30.93	29.66	28.78	28.25	27.80	27.39	32.005
SRAD	36.97	32.51	30.63	29.31	28.01	27.11	25.73	25.02	23.86	23.52	28.267
Guided	41.96	36.31	33.02	30.82	29.32	28.21	27.55	26.99	26.59	26.3	30.707
Bilateral	45.12	39.29	35.36	32.11	29.75	28.18	27.25	26.48	25.9	25.39	31.483
NLM (ED)	47.26	41.4	38.38	36.09	33.95	31.92	30.49	28.98	27.94	27.31	34.372
NLM (SCD)	47.39	41.34	38.34	36.68	35.21	33.54	32.33	30.69	29.42	28.52	35.346
NLM (MD)	45.84	40.04	36.59	34.09	32.15	30.39	29.31	28.14	27.38	26.92	33.085
NLM (CSD)	47.36	41.46	38.38	36.16	34.15	32.32	31.02	29.55	28.48	27.76	34.664
NLM (PD)	47.29	41.38	38.02	36.11	33.98	32.08	30.85	29.11	28.05	27.48	34.435
NLM (BCD)	45.86	39.98	36.66	34.37	32.63	31.01	29.96	28.73	27.88	27.29	33.437
NLM (GMD)	45.79	40.12	36.67	33.99	31.9	30.09	29.01	27.90	27.19	26.79	32.945

Table 2

Presents a comparison of SNR values of various SR filters and different similarity measure used in NLM framework at various level (σ² = 0.5, 1.0, 1.5, . . . , 5.0) of speckle noise

Methods	0.5	1	1.5	2	2.5	3	3.5	4	4.5	5	Avg. SNR
Noisy	28.82	22.81	19.34	16.85	14.99	13.43	12.21	11.14	10.27	9.56	15.942
Lee	29.64	23.86	23.13	22.31	21.40	20.88	20.54	20.16	19.92	19.70	22.154
Frost	29.84	27.36	24.99	24.07	23.09	22.42	21.99	21.64	21.42	21.36	23.818
Kaun	37.10	31.92	28.52	26.86	25.20	23.94	23.06	22.51	22.08	21.67	26.286
SRAD	31.26	26.80	24.92	23.60	22.29	21.41	20.04	19.33	18.22	17.89	22.576
Guided	36.25	30.59	27.29	25.11	23.59	22.49	21.82	21.26	20.88	20.59	24.987
Bilateral	39.41	33.58	29.65	26.40	24.03	22.47	21.52	20.77	20.2	19.7	25.773
NLM (ED)	41.56	35.69	32.67	30.38	28.23	26.2	24.77	23.26	22.22	21.59	28.657
NLM (SCD)	41.68	36.63	32.63	30.97	29.51	27.84	26.63	25.01	23.74	22.85	29.748
NLM (MD)	40.14	34.33	30.87	28.38	26.43	24.67	23.58	22.42	21.66	21.21	27.369
NLM (CSD)	41.66	35.76	32.67	30.46	28.45	26.62	25.31	23.86	22.8	22.09	28.968
NLM (PD)	41.59	35.34	31.56	30.42	28.68	26.91	24.86	23.96	22.34	22.04	28.77
NLM (BCD)	40.15	34.27	30.95	28.67	26.93	25.32	24.26	23.05	22.21	21.64	27.745
NLM (GMD)	40.09	34.41	30.95	28.28	26.17	24.38	23.27	22.18	21.48	21.07	27.228

In the results we have seen that, popular speckle noise reduction methods and NLM filter are not able to manage the balance between feature preservation and speckle suppression. We can see the over smoothing in the results obtained from SRAD and NLM filters and after the denoising, weak edges are disappeared. Second order statistics based GF provides better edge preservation and denoising performance than the Bilateral in the case of higher speckle noise, however the comprehensive performance of this filter is unsatisfactory. We have also analyzed that at every noise level, the Square chord and Chi-square based distance measure performed much better than the classical Euclidean and other well-known distance measures. It means that for synthetic images, square chord-based and Chi-square-based distance measures are more successful to reduce the speckle noise.

4.2 Experiments II on B-mode ultrasound simulation image

A B-mode image is a cross-sectional image indicating organ and tissues boundaries within the human body. It is not same as synthetic image [28] due to the background. The B-mode US image is created in Scilab 6.0.1 [44] and simulated noise is added having variance 0.01.

Different denoising methods are applied on the noisy B-mode image. Denoised images and the denoising performance are demonstrated by Fig. 3 and Table 3. The comparison of denoising performance provided by all classical speckle noise reduction filters and different similarity measures used in non-local framework exhibits that the denoising performance provided by Bray-Curtis-based distance measure is much better than all other filters and also able to preserve different types of important features including fine edges and small objects in the filtered image. Furthermore, Bray-Curtis-based distance measure used in weight computation process in non-local framework enhances the quality and contrast level of denoised image which is very helpful to produce better tissue characterization as well as diagnosis for malignant lesions in US images [28].

Fig. 2

Method noise results of synthetic image: (a) noisy residual, (b) SRAD filter, (c) Kaun filter, (d) Frost filter, (e) Lee filter, (f) Bilateral filter, (g) Guided filter, (h) NLM (ED), (i) NLM (SCD), (j) NLM (MD), (k) NLM (CSD), (l) NLM (PD), (m) NLM (BCD), (n) NLM (GMD).

Fig. 3

Results of experiment II: (a) Noise-free image, (b) Noisy image; results after applying various denoising algorithms, (c) SRAD filter, (d) Kaun filter, (e) Frost filter, (f) Lee filter, (g) Bilateral filter, (h) Guided filter, (i) NLM (ED), (j) NLM (SCD), (k) NLM (MD), (l) NLM (CSD), (m) NLM (PD), (n) NLM (BCD), (o) NLM (GMD).

Table 3

Presents a comparative analysis using SNR, PSNR, and MSSIM values of various SR filters and different similarity measure used in NLM framework for denoising B-mode image

Methods	PSNR	SNR	MSSIM
Noisy	8.39	5.59	0.1284
SRAD	8.43	5.62	0.3118
Kaun	8.42	5.61	0.2743
Frost	8.44	5.62	0.2702
Lee	8.43	5.62	0.2743
Guided	8.42	5.61	0.2319
Bilateral	8.41	5.59	0.1374
NLM (ED)	8.42	5.61	0.3589
NLM (SCD)	8.45	5.63	0.3789
NLM (MD)	8.51	5.65	0.3546
NLM (CSD)	8.49	5.64	0.3794
NLM (PD)	8.49	5.64	0.3794
NLM (BCD)	8.52	5.66	0.3832
NLM (GMD)	8.51	5.65	0.3697

4.3 Experiments on field II kidney simulation image

Different denoising methods are also tested on simulated US kidney image. Jensen [36, 37] provided the code to create this image using linear acoustic and Field II program. The pulsed ultrasound fields are assessed by Field II program using Tuphole-Stepanishen method. In this experiment, we have added simulated speckle noise having variance 0.04. After applying all the denoising algorithms on simulated US kidney image, the denoised images are shown in Fig. 4. The comparison of denoising performance between the traditional SR filters and NLM with the proposed distance measures in non-local framework is shown in Table 4.

Fig. 4

Results of experiment III: (a) Noise-free image, (b)Noisy image; results after applying various denoising algorithms, (c) SRAD filter, (d) Kaun filter, (e) Frost filter, (f) Lee filter, (g) Bilateral filter, (h) Guided filter, (i) NLM (ED), (j) NLM (SCD), (k) NLM (MD), (l) NLM (CSD), (m) NLM (PD), (n) NLM (BCD), (o) NLM (GMD).

Table 4

Presents a comparative analysis using SNR, PSNR, and MSSIM values of various SR filters and different similarity measure used in NLM framework for simulated kidney image

Methods	PSNR	SNR	MSSIM
Noisy	14.46	8.60	0.1552
SRAD	15.31	9.33	0.4388
Kaun	15.12	9.17	0.3787
Frost	15.11	9.16	0.3708
Lee	15.12	9.17	0.3787
Bilateral	15.12	9.17	0.3806
Guided	15.24	9.27	0.4426
NLM (ED)	15.27	9.31	0.4696
NLM (SCD)	15.39	9.39	0.4962
NLM (MD)	15.38	9.39	0.4962
NLM (CSD)	15.41	9.41	0.4963
NLM (PD)	15.38	9.39	0.4952
NLM (BCD)	15.38	9.39	0.4962
NLM (GMD)	15.37	9.38	0.4944

4.4 Experiments on real ultrasound images

We have tested different denoising algorithm to the real ultrasound images such as Pancreas and Tarsal Ganglion Cyst. We have downloaded these images from [38]. We have added the speckle noise by setting the parameter γ = 0.5 in Equation (2). In this experiment, the performance of all classical filters and NLM with different distant measures is quantitatively and qualitatively measured on real ultrasound images. We can see in Figs. 5–8 that among all the speckle reduction filters, Guided filter (GF) and Bilateral filter (BF) give the better performance in terms of edge preservation. As far as the NLM is concerned, all the other similarity measures, except the Euclidean distance (ED) have given better speckle noise reduction performance than ED.

Fig. 5

Results of experiment on Pancreas image: (a) Noisy image, results after applying various denoising algorithms, (b) SRAD filter, (c) Kaun filter, (d) Frost filter, (e) Lee filter, (f) Bilateral Filter, (g) Guided Filter, (h) NLM (ED), (i) NLM (SCD), (j) NLM (MD), (k) NLM (CSD), (l) NLM (PD), (m) NLM (BCD), (n) NLM (GMD).

Fig. 6

Residual images of Pancreas: (a) SRAD filter, (b) Kaun filter, (c) Frost filter, (d) Lee filter, (e) Bilateral filter, (f) Guided filter, (g) NLM (ED), (h) NLM (SCD), (i) NLM (MD), (j) NLM (CSD), (k) NLM (PD), (l) NLM (BCD), (m) NLM (GMD).

Fig. 7

Results of experiment on Tarsal Ganglion Cyst image: (a) Noisy image, results after applying various denoising algorithms, (b) SRAD filter, (c) Kaun filter, (d) Frost filter, (e) Lee filter, (f) Bilateral filter, (g) Guided filter, (h) NLM (ED), (i) NLM (SCD), (j) NLM (MD), (k) NLM (CSD), (l) NLM (PD), (m) NLM (BCD), (n) NLM (GMD).

Fig. 8

Residual images of Tarsal Ganglion Cyst image: (a) SRAD filter, (b) Kaun filter, (c) Frost filter, (d) Lee filter, (e) Bilateral filter, (f) Guided filter, (g) NLM (ED), (h) NLM (SCD), (i) NLM (MD), (j) NLM (CSD), (k) NLM (PD), (l) NLM (BCD), (m) NLM (GMD).

In Figs. 5 and Figs. 7, the denoised results of Pancreas and Tarsal Ganglion Cyst US images obtained from all the classical filters (Lee [8], Frost [9], Kaun [10, 11] and SRAD [12]), Bilateral filter [27], Guided filter [16] and all the similarity measures for non-local framework are shown.

In Figs. 6 and Figs. 8, the residual images of all the classical filters (Lee [8], Frost [9], Kaun [10, 11] and SRAD [12]), Bilateral filter [27], Guided filter [16] and all the similarity measures for non-local framework are shown.

CNR and IEF quality measures are used to evaluate the performance of all algorithms quantitatively. In Fig. 9, green patch indicates the background ROI and red patches indicate ROI_1 and ROI_2 (from left to right) in each image.

Fig. 9

Real ultrasound images with ROI.

It is clearly shown in the Tables 5, 6 that Square-chord and Chi-square based distance measures in nonlocal framework performed better than the classical Euclidean distance measure.

Table 5

presents a comparative analysis using CNR values of various SR filters and different similarity measure used in NLM framework for real ultrasound images

Methods	CNR for Pancreas image		CNR for Tarsal Ganglion Cyst image
	ROI_1	ROI_2	ROI_1	ROI_2
SRAD	2.07	1.23	1.15	1.79
Kaun	2.75	1.58	1.28	2.04
Frost	2.67	1.54	1.25	2.01
Lee	2.65	1.52	1.26	2.01
Guided	2.33	1.35	1.19	1.91
Bilateral	2.25	1.29	1.15	1.86
NLM (ED)	2.20	1.31	1.14	1.82
NLM (SCD)	2.23	1.31	1.14	1.83
NLM (MD)	2.21	1.30	1.14	1.83
NLM (CSD)	2.23	1.31	1.15	1.83
NLM (PD)	2.21	1.31	1.14	1.82
NLM (BCD)	2.22	1.29	1.14	1.84
NLM (GMD)	2.22	1.32	1.15	1.84

Table 6

presents a comparative analysis using IEF values of various SR filters and different similarity measure used in NLM framework for real ultrasound images

Methods	IEF for Pancreas image	IEF for Tarsal Ganglion Cyst image
SRAD	0.68	0.80
Kaun	1.17	1.38
Frost	1.19	1.45
Lee	0.97	1.21
Guided	2.66	2.61
Bilateral	3.74	3.76
NLM (ED)	3.85	3.83
NLM (SCD)	4.19	4.00
NLM (MD)	3.29	3.13
NLM (CSD)	4.23	4.01
NLM (PD)	3.40	3.16
NLM (BCD)	2.47	2.64
NLM (GMD)	3.47	3.35

4.5 Computation time of the experiments

Four different size images are used to calculate the computation time of every algorithm. In non-local means filter, the computation time depends on the distance measure. So, the computation time of all seven distance measure is presented in Table 7. Square-chord and Chi-square distance based NLM filters have taken almost the same time as Euclidean distance.

Table 7
presents computation time (in milliseconds) of all distance measures in NLM framework for different image sizes

Method 512×512 256×256 128×128 64×64

NLM (ED) 6697 1781 453 125

NLM (SCD) 6104 1575 391 98

NLM (MD) 13424 3510 844 218

NLM (CSD) 7543 1971 485 125

NLM (PD) 8566 2188 531 141

NLM (BCD) 13877 3519 877 234

NLM (GMD) 5957 1533 378 94

Method	512×512	256×256	128×128	64×64
NLM (ED)	6697	1781	453	125
NLM (SCD)	6104	1575	391	98
NLM (MD)	13424	3510	844	218
NLM (CSD)	7543	1971	485	125
NLM (PD)	8566	2188	531	141
NLM (BCD)	13877	3519	877	234
NLM (GMD)	5957	1533	378	94

5 Conclusion

In this paper, we have investigated empirically the different robust and efficient similarity measures used in non-local framework for weight computation process. The different set of experiments have been performed on synthetic, B-mode, simulated US images and real US images. We have observed that Square chord and Chi-square based distance measures in nonlocal framework have better denoising and structural preservation capability than the classical Euclidean distance measure. We have also seen that in the process of eliminating the speckle noise from ultrasound images, the Square chord and Chi-square-based distance measures in non-local framework have better capability to manage the contrast level. After the visual inspection, we suggest that the performance of the Square chord and Chi-square-based distance measures used in weight computation in non-local framework have better denoising performance than the classical speckle noise reduction approaches.

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