# A Novel Speckle Noise Removal Algorithm Based on ADMM and Energy Minimization Method

Authors: Bo Chen; Yan Lv; Jinbin Zou; Wensheng Chen; Binbin Pan
Journal: Journal of Function Spaces
 (2020)

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## Abstract

Speckle noise removal in medical ultrasound images is a challenging task. In this paper, a new model is proposed to removal speckle noise, alternating direction method of multipliers algorithm is employed to solve the new energy minimization model. The convexity, existence, and uniqueness of the new energy minimization model’s solution are proved. Series of experiments are designed in this paper. Numerical results show that the new algorithm can reduce the step effect effectively obtain good results in visual effect and quantitative measures by comparing with some traditional models.


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## Body

## 1. Introduction

With the rapid development of science and technology, computer network and image equipment are widely used. There are more and more application fields of digital image, including pattern recognition, medical imaging, video processing, remote sensing, and other fields. At the same time, people have higher and higher requirements for the quality of digital image, and the digital image processing technology has attracted more and more attention of researchers.Image denoising technology is mainly to input a degraded image with noise into the computer for processing, to eliminate the noise in the image, make it closer to the original image, and facilitate the subsequent processing. In the early image denoising work, it can be divided into filter based on convolution methods [1, 2], wavelet based on image denoising methods [3, 4], depth based on learning method, and partial differential equation (PDE) based methods [5–8].Medical ultrasound image reflects the difference of acoustic parameters in media and can obtain information different from optical, X-ray and Y-ray. Ultrasound has a good ability to distinguish human soft tissues, which is helpful to identify micropathological changes in biological tissues. But in the process of medical ultrasound image transmission, speckle noise will be produced, which will lead to a significant decline in the quality of ultrasound image and cover up the damage of some important tissues. How to eliminate speckle noise in ultrasound image while retaining its important features is a challenge.In Ref. [9], Loupas et al. proved by experiments that the noise in medical ultrasound image is no longer the multiplicative noise in the form of Rayleigh distribution, and also proved in the paper that the form of medical ultrasound degraded image can be written:
(1)f=u+un,where f is a noisy image and n represents the Gaussian random noise with zeros mean and standard deviation σ.In the paper [2], Kristian et al. derive the corresponding data fidelity term according to the distribution characteristics of Gaussian noise and the degradation model of medical ultrasonic noise image:
(2)∫Ωf−u2udx,With the passage of time, mathematical research is constantly improving, so stochastic theory [10–14] and PDE [15–20] have been fully developed, and Fractional theory [21, 22], wavelet [23], and statistical information [24] are all employed to deal with intensity inhomogeneity or noise. In this paper, the method of partial differential equation is used to solve speckle noise. In the numerical algorithm, although the use of image denoising is effective, it will not only bring some ladder effect but also low iterative efficiency. Therefore, the optimization algorithm has received great attention in recent years. Many efficient optimization algorithms have been proposed, such as Split Bregman type iterative [25], the Douglas-Rachford method [26], primal-dual algorithm [27], and Alternating Direction Method of Multipliers algorithm (ADMM) [26, 28]. In this paper, we will use the ADMM algorithm to solve the new energy minimization model.The rest of this paper is as follows. In Section2, we introduce the new energy minimization model and the ADMM algorithm. In Section 3, we adopt the ADMM algorithm to solve the new energy minimization model. Section 4 shows some numerical results and compares them with other existing models. The conclusion is drawn in Section 5.


## 2. Background

### 2.1. The TV Model

In 1992, the total variation (TV) model was proposed, which is the classical total variational model. The forms of TV model are generally as follows:(3)min∫Ω∇udx+λ2∫Ωu−u02dx,where u:Ω→R is a clean image and u0 is a noisy image affected by Gaussian random noise with mean zero and standard deviation σ. λ>0 is regularization parameter which can balance fidelity term and regularized term in TV model. ∇ represents the gradient operator, which is defined as follows:
(4)∇u=∂x+u,∂y+u,where ∂x+,∂y+ is the forward discrete derivation operators, which are defined as:
(5)∂x+u=ui,j+1−ui,jif1≤j≤N−10ifj=N∂y+u=ui+1,j−ui,jif1≤i≤M−10ifi=M,


### 2.2. The New Energy Minimization Model

The new energy minimization model is an adaptive total variation model, which is defined by a new regularization term and the variance of speckle noise, as follows:(6)minu∫Ωφ∇udx+α2∫Ωf−u2udx,Whereϕs=slnδ+s, ∫Ωf−u2/udx is the fidelity term, and α represents the regularization parameter which can balance fidelity term and regularization term.Firstly, the energy model (6) is convex, which guarantees the existence of the minimal solution of the model (6).Theorem 1.
The energy functionalEu is convex. That is to say, for any u1,u2∈Ω and t∈0,1, we have:
(7)Etu1+1−tu2≤tEu1+1−tEu2,where Eu=∫Ωφ∇udx+α/2∫Ωf−u2/udx.Proof.
The proof of Theorem1 is given in the appendix A.
Secondly, the uniqueness of the minimum solution of the model (6) can also be proved.Theorem 2.
Ifu1 and u2 are two minimize solutions of model (6), then we have u1=u2.Proof.
The proof of Theorem2 is given in the appendix B.


### 2.3. Alternating Direction Method of Multipliers

Alternating Direction Method of Multipliers is a computational framework for solving optimization problems, which is suitable for solving distributed convex optimization problems. ADMM decomposes a large global problem into several smaller, easy-to-solve local problems, and then solves them alternately. Its essence is the further development of the augmented Lagrange algorithm.For linear constrained minimization problem,(8)minx,yfx+gysuchthatAx+By=b,where x∈Rn, y∈Rm, A∈Rk×n, B∈Rk×m, b∈Rk, and function f:Rn⟶R and function g:Rm⟶R are convex and lower semicontinuous functions.By introducing a Lagrangian multiplier vector or dual variableβ∈Rk, then the augmented Lagrange function of problem (8) is
(9)Lμx,y,β;μ=fx+gy−βTAx+By−b+μ2Ax+By−b22where μ>0 is a penalty parameter. By the algorithm of Alternating Direction Method of Multipliers (ADMM), the solution xk+1,yk+1 is achieved with the following iteration:
(10)xk+1=argminxLμx,yk,βk;μyk+1=argminyLμxk+1,y,βk;μβk+1=βk−κμAxk+1+Byk+1−b,where σ∈0,1+5/2 is a positive parameter. The convergence of the ADMM algorithm was proved in [29].


## 2.1. The TV Model

In 1992, the total variation (TV) model was proposed, which is the classical total variational model. The forms of TV model are generally as follows:(3)min∫Ω∇udx+λ2∫Ωu−u02dx,where u:Ω→R is a clean image and u0 is a noisy image affected by Gaussian random noise with mean zero and standard deviation σ. λ>0 is regularization parameter which can balance fidelity term and regularized term in TV model. ∇ represents the gradient operator, which is defined as follows:
(4)∇u=∂x+u,∂y+u,where ∂x+,∂y+ is the forward discrete derivation operators, which are defined as:
(5)∂x+u=ui,j+1−ui,jif1≤j≤N−10ifj=N∂y+u=ui+1,j−ui,jif1≤i≤M−10ifi=M,


## 2.2. The New Energy Minimization Model

The new energy minimization model is an adaptive total variation model, which is defined by a new regularization term and the variance of speckle noise, as follows:(6)minu∫Ωφ∇udx+α2∫Ωf−u2udx,Whereϕs=slnδ+s, ∫Ωf−u2/udx is the fidelity term, and α represents the regularization parameter which can balance fidelity term and regularization term.Firstly, the energy model (6) is convex, which guarantees the existence of the minimal solution of the model (6).Theorem 1.
The energy functionalEu is convex. That is to say, for any u1,u2∈Ω and t∈0,1, we have:
(7)Etu1+1−tu2≤tEu1+1−tEu2,where Eu=∫Ωφ∇udx+α/2∫Ωf−u2/udx.Proof.
The proof of Theorem1 is given in the appendix A.
Secondly, the uniqueness of the minimum solution of the model (6) can also be proved.Theorem 2.
Ifu1 and u2 are two minimize solutions of model (6), then we have u1=u2.Proof.
The proof of Theorem2 is given in the appendix B.


## 2.3. Alternating Direction Method of Multipliers

Alternating Direction Method of Multipliers is a computational framework for solving optimization problems, which is suitable for solving distributed convex optimization problems. ADMM decomposes a large global problem into several smaller, easy-to-solve local problems, and then solves them alternately. Its essence is the further development of the augmented Lagrange algorithm.For linear constrained minimization problem,(8)minx,yfx+gysuchthatAx+By=b,where x∈Rn, y∈Rm, A∈Rk×n, B∈Rk×m, b∈Rk, and function f:Rn⟶R and function g:Rm⟶R are convex and lower semicontinuous functions.By introducing a Lagrangian multiplier vector or dual variableβ∈Rk, then the augmented Lagrange function of problem (8) is
(9)Lμx,y,β;μ=fx+gy−βTAx+By−b+μ2Ax+By−b22where μ>0 is a penalty parameter. By the algorithm of Alternating Direction Method of Multipliers (ADMM), the solution xk+1,yk+1 is achieved with the following iteration:
(10)xk+1=argminxLμx,yk,βk;μyk+1=argminyLμxk+1,y,βk;μβk+1=βk−κμAxk+1+Byk+1−b,where σ∈0,1+5/2 is a positive parameter. The convergence of the ADMM algorithm was proved in [29].


## 3. The ADMM Algorithm for the Energy Minimization Model

In this subsection, we will describe the ADMM algorithm for solving new energy minimization model. Firstly, the model (6) can be transformed into the following discretized version:
(11)minuϕ∇u,1+α2f−u2u,1,To solve the problem with the ADMM algorithm, we introduce two new variables:z,d=d1,d2. Apply these two variables, the unconstrained problem (11) is written as the following constrained problem:
(12)minuϕd,1+α2f−z2z,1subjectthatz=u,d=∇u,Next, we make(13)A=I00I,x=zdB=−I−∇,y=u,β=β1β2,Now, we letting(14)fx=ϕd,1+α2f−z2z,1andgy=0,So according to constraint problem (8), the constrained problem (12) can be solved by the ADMM algorithm. The augmented Lagrange function of problems (9) is
(15)Lμz,d,u,β1,β2;μ=ϕd,1+α2f−z2z,1−β1Tz−u−β2Td−∇u+μ2z−u22+μ2d−∇u22,According to the structure of the ADMM algorithm, the problem (15) can be rewritten as the following form:
(16)zk+1,dk+1=argminz≥0,dLμz,d,uk,βk1,βk2;μuk+1=argmindLμzk+1,dk+1,u,βk1,βk2;μβk+11=βk1−κμzk+1−uk+1βk+12=βk2−κμdk+1−∇uk+1,Since the two new variablesz and d in problem (16) are not related to each other, so we can divide two simple minimization subproblem as follows,
(17)zk+1=minz≥0α2f−z2z,1−βk1Tz+μ2z−uk22dk+1=mindϕid,1−βk2Td+μ2d−∇uk22,First, for thez−subproblem, we can obtain the corresponding Euler-Lagrange equation:
(18)α21−f2z2−βk1+μz−uk=0,Simplifying the Euler-Lagrange equation above, we can obtain three complicated solutions by the roots formula of the cubic equation:(19)2μz3+α−2βk1−2μukz2−αf2=0,For equation (19), there is only one positive real number solution (the proof is given in appendix C).Second, for thed−subproblem, we will apply the shrink operator to obtain the solution d. The form of the solution d is as follows:
(20)dk+1=mindϕid,1−βk2Td+μ2d−∇uk22=shrinkβk2μ+∇uk,ϕi′dμ,where the shrink operator shrinka,θ is defined [30] as:
(21)shrinka,θ=maxa1−θ,0aa1,Last, theu−subproblem is a least square problem as follow:
(22)uk+1=minuBu+b22,where b=zk+1−βk1/μ,dk+1−βk2/μ. So the solution uk+1 of the least square problem (22) equivalent to the solution of the BTBu=−BTb. That is to say, the solution is as follows:
(23)I+∇T∇u=zk+1−βk1μ+∇Tdk+1−βk2μ,To sum up, the ADMM algorithm can be summarized in Algorithm1.Algorithm 1: ADMM algorithm for the new model.
1:Initializek=0, u0=f, β01=0, β02=02: Given parametersα,μandκ.3:Repeat4:zk+1 is updated by the positive solution of the cubic equation (19);5:dk+1 is updated by the equation (20);6:uk+1 is updated by the equation (23);7:βk+11=βk1−κμzk+1−uk+1;8:βk+12=βk2−κμdk+1−∇uk+1;9: Until a stopping condition is satisfied.10:Final Input:u


## 4. Experimental Results

In this section, we show five sets of experimental results. Test images include synthetic, natural, and real medical ultrasound images. In addition, the denoising effect of the model is compared with that of the existing models, such as TV model [7], ATV model [5], JIN’s model [8], and finite difference for the new model.For the algorithm1, the stopping condition is that the solution of two adjacent iterations satisfies:
(24)uk−uk+12uk2<10−3andk>M,where M is the maximum iteration numbers; M=100 for algorithm 1. uk represents the results of the kth iterations. For the JIN’s model and finite difference for the new model, we calculated the noise deviation reduction (NDR) at each iteration as a convergence condition;
(25)NDR=meanf−ukuk,And the stopping condition (NDR) for finite difference for the new model is as follow(26)NDR−σ<10−1,The effect of image denoising can be evaluated from two aspects. The first is the subjective aspect: it is judged by the subjective consciousness of peoples. The second is the objective aspect: it evaluates the image denoising situation through scientific indicators. In this paper, we evaluate the effect of image denoising by calculating the peak signal-to-noise ratio (PSNR) and structural similarity, which are defined as follows:(27)PSNRu,u¯=10log102552mnu−u¯22,(28)SSIMu,u¯=2μuμu¯+c12σuu¯+c2μu2+μu¯2+c1σu2+σu¯2+c2,where u∈Rm×n is the clean image and u¯∈Rm×n is the restored image. μu and μu¯ are the mean intensity of u and u¯, respectively. σu and σu¯ are the standard deviation of u and u¯, respectively. σuu¯ is the covariance of u and u¯, and c1 and c2 are some constants for stability.Best denoising performance are given in bold.


### 4.1. The Effect of Different Parameters of the Model on Denoising

In the first experiment, we find the optimal parametersδ value in the algorithm 1 for original image in Figure 1. We choose “syn1” and the noise level is σ=2. Figure 2 shows that numerical experiments with different parameters δ in algorithm 1. Figure 3 shows that the different PSNR and SSIM values when different δ values are used in algorithm 1. Here, we can see that the optimal values of PSNR and SSIM values are at δ=50. Therefore, in the following experiment, we choose δ=50 in algorithm 1.Figure 1
Original images. (a)syn1256×256. (b) syn2512×512.
(a)(b)Figure 2
Numerical experiments with different parametersδ with noise standard deviation σ=2. (a) Restored image by δ=1. (b) Restored image by δ=5. (c) Restored image by δ=10. (d) Restored image by δ=20. (e) Restored image by δ=30. (f) Restored image by δ=40. (g) Restored image by δ=50. (h) Restored image by δ=60. (i) Restored image by δ=100. (j)Restored image by δ=200.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Figure 3
The PSNR and SSIM values of differentδ.


### 4.2. Denoising Effect of the ADMM Algorithm

In the second experiment, we mainly evaluate the denoising effect of the ADMM algorithm by testing image “syn1” and “syn2”. Figures4(a) and 5(a) show the noise images, and the noise deviation are σ=2 and σ=3, respectively. Figures 4(b) and 5(b) show the restored image by the JIN’s model. Figures 4(c) and 5(c) show the restored image by finite difference for the new model. Figures 4(d) and 5(d) show the restored image by ADMM for the new model. In addition, Figure 6 shows that the different PSNR and SSIM values when used different models and algorithm. In Figure 6, we can find that our new algorithm has better denoising effect than JIN model and finite difference for the new model.Figure 4
Numerical result of the “syn1” image with noise standard deviationσ=2. (a) Noisy image. (b) Restored image by the JIN’s model. (c) Restored image by finite difference for the new model; (d) Restored image by ADMM for the new model.
(a)(b)(c)(d)Figure 5
Numerical result of the “syn2” image with noise standard deviationσ=3. (a) Noisy image. (b) Restored image by the JIN’s model. (c) Restored image by finite difference for the new model. (d) Restored image by ADMM for the new model.
(a)(b)(c)(d)Figure 6
Folded-line contrast map. (a, b) PSNR and SSIM values of different models with different noise standard deviations (“syn1” image). (c, d) PSNR and SSIM values of different models with different noise standard deviations (“syn2” image).
(a)(b)(c)(d)


### 4.3. Comparison with TV Model, ATV Model, JIN’s Model, and Finite Difference for the New Model

In the third experiment, we compare the denoising effects of the TV model, ATV model, JIN’s model, and finite difference for the new model with ADMM for the new model by subjective and objective evaluation criteria. The test original images are shown in Figure7 (“lena,” “pirate,” “boat,” “bird,” “house,” and “peppers”).Figure 7
Original images: (a) lena (512×512); (b) pirate (512×512); (c) boat (512×512); (d) bird (256×256); (e) house (256×256); (f) peppers (256×256).
(a)(b)(c)(d)(e)(f)Firstly, in the subjective evaluation, we mainly focus on the details of the restored images. The first line of Figure8 shows the original images (“lena” and “bird”) and the noise images with noise deviation is σ=2. The second line is the corresponding detail images in the first line. Figures 9 and 10 show the restoration results of the noisy images (“lena” and “bird”). Figures 9(a), 9(d), 10(a), and 10(d) were restored images of the TV model, JIN’s model, finite difference for the new model, and ADMM for the new model, respectively. Figures 9(e), 9(h), 10(e), and 10(h) are corresponding details images of Figures 9(a), 9(d), 10(a), and 10(d), respectively. By observing at the detailed images in Figures 9 and 10, we can see that the denoising effects of the four models are different. Figures 9(e), 9(f), 10(e), and 10(f)show that the denoising effect of TV model JIN’s model are worse in subjective vision, and Figures 9(g), 9(h), 10(g), and 10(h) show that the denoising effect of finite difference for the new model and ADMM for the new model are well in subjective vision. In addition, we can clearly see the staircase effect in the details images obtained by the TV model and JIN’s model. At the same time, by comparing with finite difference for the new model, the staircase effect of the new model is greatly reduced after it is repaired by the ADMM algorithm. This means that using the ADMM algorithm to solve the new energy minimization model has a better effect than other model algorithms in removing speckle noise.Figure 8
Numerical result of the “lena” and “bird” images with noise standard deviationσ=2. (a, c) Original images. (b, d) Noisy images. (e–h) About the detailed image of (a–d).
(a)(b)(c)(d)(e)(f)(g)(h)Figure 9
Numerical result of the “lena” image with noise standard deviationσ=2. (a) Restored image by the TV model. (b) Restored image by the JIN’s mode. (c) Restored image by finite difference for the new model. (d) Restored image by ADMM for the new model. (e–h) About the detailed image of (a–d), respectively.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 10
Numerical result of the “bird” image with noise standard deviationσ=2. (a) Restored image by the TV model. (b) Restored image by the JIN’s mode. (c) Restored image by finite difference for the new model. (d) Restored image by ADMM for the new model. (e–h) About the detailed image of (a–d), respectively.
(a)(b)(c)(d)(e)(f)(g)(h)Secondly, in the objective evaluation, we compare the denoising effects of different models with PSNR and SSIM values. Figures11, 12, 13, and 14(a) shows the restoration results for images through TV model; Figures 11, 12, 13, and 14(b) shows the restoration results for images through ATV model; Figures 11, 12, 13, and 14(c) show the restoration results for images through JIN’s model; Figures 11, 12, 13 and 14(d) show the restoration results for images through finite difference for the new model; Figures 11, 12, 13, and 14(e) show the restoration results for images through ADMM for the new model. The noise versions of “boat” and “house” and “pirate” and “bird” are obtained by degradation model (1) with standard deviation 3 and 4, respectively. Table 1 shows the PSNR and SSIM values for different test images by using the TV model, ATV model, JIN’s model, finite difference for the new model, and ADMM for the new model. From these values, we observe that the denoising effect of the ADMM for the new model is better than other models and finite difference for the new model.Figure 11
Numerical result of the “pirate” image with noise standard deviationσ=3. (a) Restored image by the TV model; (b) Restored image by the ATV model; (c) Restored image by the JIN’s model; (d) Restored image by finite difference for the new model; (e) Restored image by ADMM for the new model.
(a)(b)(c)(d)(e)Figure 12
Numerical result of the “house” image with noise standard deviationσ=3. (a) Restored image by the TV model. (b) Restored image by the ATV model. (c) Restored image by the JIN’s model. (d) Restored image by finite difference for the new model. (e) Restored image by ADMM for the new model.
(a)(b)(c)(d)(e)Figure 13
Numerical result of the “boat” image with noise standard deviationσ=4. (a) Restored image by the TV model. (b) Restored image by the ATV model. (c) Restored image by the JIN’s model. (d) Restored image by finite difference for the new model. (e) Restored image by ADMM for the new model.
(a)(b)(c)(d)(e)Figure 14
Numerical result of the “peppers” image with noise standard deviationσ=4. (a) Restored image by the TV model. (b) Restored image by the ATV model. (c) Restored image by the JIN’s model. (d) Restored image by finite difference for the new model. (e) Restored image by ADMM for the new model.
(a)(b)(c)(d)(e)Table 1
The PSNR of the restored images by the different model.
ImageσTV (PSNR/SSIM)ATV (PSNR/SSIM)JIN’s (PSNR/SSIM)Finite difference for the new model (PSNR/SSIM)ADMM for the new model (PSNR/SSIM)Lena229.49/0.830629.94/0.873129.98/0.866530.65/0.883130.83/0.9042Bird228.99/0.690730.28/0.784930.36/0.777431.07/0.817331.74/0.8747Pirate228.33/0.897527.22/0.853028.51/0.904228.74/0.906028.81/0.9153House227.45/0.618529.06/0.810128.77/0.687629.74/0.744730.35/0.8157Boat227.26/0.824027.90/0.857428.35/0.862228.64/0.866128.81/0.8808Peppers227.60/0.718328.66/0.837729.46/0.819529.53/0.827429.64/0.8430Lena327.73/0.760927.98/0.814928.49/0.829828.75/0.826628.85/0.8717Bird327.36/0.645127.92/0.693028.48/0.730729.13/0.770129.73/0.8456Pirate326.35/0.839226.38/0.833226.81/0.857826.98/0.858327.05/0.8720House326.08/0.575227.33/0.661127.24/0.655127.82/0.689628.25/0.7846Boat325.79/0.748126.47/0.798626.68/0.802926.82/0.798927.03/0.8218Peppers325.93/0.675826.70/0.740927.13/0.744627.37/0.751627.57/0.8004Lena426.17/0.692526.94/0.812926.82/0.762527.19/0.764127.25/0.8194Bird425.66/0.567926.49/0.658427.25/0.726327.51/0.728227.63/0.7994Pirate424.03/0.765224.33/0.720524.75/0.778725.77/0.815925.79/0.8306House424.63/0.513025.30/0.576026.18/0.639926.43/0.643726.51/0.7385Boat424.47/0.680724.95/0.733825.24/0.737025.51/0.738825.61/0.7745Peppers424.49/0.615325.59/0.715325.63/0.680025.83/0.701126.19/0.7600


### 4.4. Denoising of Real Ultrasound Images

The last experiment is mainly to test the denoising effect of the new algorithm for solving the real ultrasound image and compare it with other denoising models. However, the real ultrasound image does not have the original image, so it cannot be evaluated using the original PSNR and SSIM. In 2005, Buades et al. [31] proposed an evaluation method without original image. The method evaluates the image denoising effect based on the difference and the estimated noise images between the real ultrasound image and the restored image. The difference image and the estimated noise image are defined as:
(29)f−u¯andf−u¯u¯,where f∈Rm×n is the real ultrasound image and u¯∈Rm×n is the restored image. The image denoising effect can be evaluated based on the difference image and the similarity of the estimated noise image, that is, based on the residual image (the difference between the difference image and estimated noise image). If the texture of the residual image is small, the denoising effect is well.Figure15 shows that the experimental results of real ultrasound images by applying JIN’s model, finite difference for the new model, and ADMM for the new model. Figure 16 shows the residual image after applying the JIN model, finite difference for the new model, and ADMM for the new model to restore the real ultrasound image. Visually speaking, the new model has fewer textures of residual images under the new algorithm, which shows that our method is better in denoising effect.Figure 15
Numerical result of the real ultrasound image. (a, e, i) Real ultrasound images. (b, f, j) Restored image by the JIN’s model. (c, g, k) Restored image by finite difference for the new model. (d, h, l) Restored image by ADMM for the new model.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Figure 16
Residual image. (a, d, g) Residual image by the JIN’s model. (b, e, h) Residual image by finite difference for the new model. (c, f, i) Residual image by ADMM for the new model.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


## 4.1. The Effect of Different Parameters of the Model on Denoising

In the first experiment, we find the optimal parametersδ value in the algorithm 1 for original image in Figure 1. We choose “syn1” and the noise level is σ=2. Figure 2 shows that numerical experiments with different parameters δ in algorithm 1. Figure 3 shows that the different PSNR and SSIM values when different δ values are used in algorithm 1. Here, we can see that the optimal values of PSNR and SSIM values are at δ=50. Therefore, in the following experiment, we choose δ=50 in algorithm 1.Figure 1
Original images. (a)syn1256×256. (b) syn2512×512.
(a)(b)Figure 2
Numerical experiments with different parametersδ with noise standard deviation σ=2. (a) Restored image by δ=1. (b) Restored image by δ=5. (c) Restored image by δ=10. (d) Restored image by δ=20. (e) Restored image by δ=30. (f) Restored image by δ=40. (g) Restored image by δ=50. (h) Restored image by δ=60. (i) Restored image by δ=100. (j)Restored image by δ=200.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Figure 3
The PSNR and SSIM values of differentδ.


## 4.2. Denoising Effect of the ADMM Algorithm

In the second experiment, we mainly evaluate the denoising effect of the ADMM algorithm by testing image “syn1” and “syn2”. Figures4(a) and 5(a) show the noise images, and the noise deviation are σ=2 and σ=3, respectively. Figures 4(b) and 5(b) show the restored image by the JIN’s model. Figures 4(c) and 5(c) show the restored image by finite difference for the new model. Figures 4(d) and 5(d) show the restored image by ADMM for the new model. In addition, Figure 6 shows that the different PSNR and SSIM values when used different models and algorithm. In Figure 6, we can find that our new algorithm has better denoising effect than JIN model and finite difference for the new model.Figure 4
Numerical result of the “syn1” image with noise standard deviationσ=2. (a) Noisy image. (b) Restored image by the JIN’s model. (c) Restored image by finite difference for the new model; (d) Restored image by ADMM for the new model.
(a)(b)(c)(d)Figure 5
Numerical result of the “syn2” image with noise standard deviationσ=3. (a) Noisy image. (b) Restored image by the JIN’s model. (c) Restored image by finite difference for the new model. (d) Restored image by ADMM for the new model.
(a)(b)(c)(d)Figure 6
Folded-line contrast map. (a, b) PSNR and SSIM values of different models with different noise standard deviations (“syn1” image). (c, d) PSNR and SSIM values of different models with different noise standard deviations (“syn2” image).
(a)(b)(c)(d)


## 4.3. Comparison with TV Model, ATV Model, JIN’s Model, and Finite Difference for the New Model

In the third experiment, we compare the denoising effects of the TV model, ATV model, JIN’s model, and finite difference for the new model with ADMM for the new model by subjective and objective evaluation criteria. The test original images are shown in Figure7 (“lena,” “pirate,” “boat,” “bird,” “house,” and “peppers”).Figure 7
Original images: (a) lena (512×512); (b) pirate (512×512); (c) boat (512×512); (d) bird (256×256); (e) house (256×256); (f) peppers (256×256).
(a)(b)(c)(d)(e)(f)Firstly, in the subjective evaluation, we mainly focus on the details of the restored images. The first line of Figure8 shows the original images (“lena” and “bird”) and the noise images with noise deviation is σ=2. The second line is the corresponding detail images in the first line. Figures 9 and 10 show the restoration results of the noisy images (“lena” and “bird”). Figures 9(a), 9(d), 10(a), and 10(d) were restored images of the TV model, JIN’s model, finite difference for the new model, and ADMM for the new model, respectively. Figures 9(e), 9(h), 10(e), and 10(h) are corresponding details images of Figures 9(a), 9(d), 10(a), and 10(d), respectively. By observing at the detailed images in Figures 9 and 10, we can see that the denoising effects of the four models are different. Figures 9(e), 9(f), 10(e), and 10(f)show that the denoising effect of TV model JIN’s model are worse in subjective vision, and Figures 9(g), 9(h), 10(g), and 10(h) show that the denoising effect of finite difference for the new model and ADMM for the new model are well in subjective vision. In addition, we can clearly see the staircase effect in the details images obtained by the TV model and JIN’s model. At the same time, by comparing with finite difference for the new model, the staircase effect of the new model is greatly reduced after it is repaired by the ADMM algorithm. This means that using the ADMM algorithm to solve the new energy minimization model has a better effect than other model algorithms in removing speckle noise.Figure 8
Numerical result of the “lena” and “bird” images with noise standard deviationσ=2. (a, c) Original images. (b, d) Noisy images. (e–h) About the detailed image of (a–d).
(a)(b)(c)(d)(e)(f)(g)(h)Figure 9
Numerical result of the “lena” image with noise standard deviationσ=2. (a) Restored image by the TV model. (b) Restored image by the JIN’s mode. (c) Restored image by finite difference for the new model. (d) Restored image by ADMM for the new model. (e–h) About the detailed image of (a–d), respectively.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 10
Numerical result of the “bird” image with noise standard deviationσ=2. (a) Restored image by the TV model. (b) Restored image by the JIN’s mode. (c) Restored image by finite difference for the new model. (d) Restored image by ADMM for the new model. (e–h) About the detailed image of (a–d), respectively.
(a)(b)(c)(d)(e)(f)(g)(h)Secondly, in the objective evaluation, we compare the denoising effects of different models with PSNR and SSIM values. Figures11, 12, 13, and 14(a) shows the restoration results for images through TV model; Figures 11, 12, 13, and 14(b) shows the restoration results for images through ATV model; Figures 11, 12, 13, and 14(c) show the restoration results for images through JIN’s model; Figures 11, 12, 13 and 14(d) show the restoration results for images through finite difference for the new model; Figures 11, 12, 13, and 14(e) show the restoration results for images through ADMM for the new model. The noise versions of “boat” and “house” and “pirate” and “bird” are obtained by degradation model (1) with standard deviation 3 and 4, respectively. Table 1 shows the PSNR and SSIM values for different test images by using the TV model, ATV model, JIN’s model, finite difference for the new model, and ADMM for the new model. From these values, we observe that the denoising effect of the ADMM for the new model is better than other models and finite difference for the new model.Figure 11
Numerical result of the “pirate” image with noise standard deviationσ=3. (a) Restored image by the TV model; (b) Restored image by the ATV model; (c) Restored image by the JIN’s model; (d) Restored image by finite difference for the new model; (e) Restored image by ADMM for the new model.
(a)(b)(c)(d)(e)Figure 12
Numerical result of the “house” image with noise standard deviationσ=3. (a) Restored image by the TV model. (b) Restored image by the ATV model. (c) Restored image by the JIN’s model. (d) Restored image by finite difference for the new model. (e) Restored image by ADMM for the new model.
(a)(b)(c)(d)(e)Figure 13
Numerical result of the “boat” image with noise standard deviationσ=4. (a) Restored image by the TV model. (b) Restored image by the ATV model. (c) Restored image by the JIN’s model. (d) Restored image by finite difference for the new model. (e) Restored image by ADMM for the new model.
(a)(b)(c)(d)(e)Figure 14
Numerical result of the “peppers” image with noise standard deviationσ=4. (a) Restored image by the TV model. (b) Restored image by the ATV model. (c) Restored image by the JIN’s model. (d) Restored image by finite difference for the new model. (e) Restored image by ADMM for the new model.
(a)(b)(c)(d)(e)Table 1
The PSNR of the restored images by the different model.
ImageσTV (PSNR/SSIM)ATV (PSNR/SSIM)JIN’s (PSNR/SSIM)Finite difference for the new model (PSNR/SSIM)ADMM for the new model (PSNR/SSIM)Lena229.49/0.830629.94/0.873129.98/0.866530.65/0.883130.83/0.9042Bird228.99/0.690730.28/0.784930.36/0.777431.07/0.817331.74/0.8747Pirate228.33/0.897527.22/0.853028.51/0.904228.74/0.906028.81/0.9153House227.45/0.618529.06/0.810128.77/0.687629.74/0.744730.35/0.8157Boat227.26/0.824027.90/0.857428.35/0.862228.64/0.866128.81/0.8808Peppers227.60/0.718328.66/0.837729.46/0.819529.53/0.827429.64/0.8430Lena327.73/0.760927.98/0.814928.49/0.829828.75/0.826628.85/0.8717Bird327.36/0.645127.92/0.693028.48/0.730729.13/0.770129.73/0.8456Pirate326.35/0.839226.38/0.833226.81/0.857826.98/0.858327.05/0.8720House326.08/0.575227.33/0.661127.24/0.655127.82/0.689628.25/0.7846Boat325.79/0.748126.47/0.798626.68/0.802926.82/0.798927.03/0.8218Peppers325.93/0.675826.70/0.740927.13/0.744627.37/0.751627.57/0.8004Lena426.17/0.692526.94/0.812926.82/0.762527.19/0.764127.25/0.8194Bird425.66/0.567926.49/0.658427.25/0.726327.51/0.728227.63/0.7994Pirate424.03/0.765224.33/0.720524.75/0.778725.77/0.815925.79/0.8306House424.63/0.513025.30/0.576026.18/0.639926.43/0.643726.51/0.7385Boat424.47/0.680724.95/0.733825.24/0.737025.51/0.738825.61/0.7745Peppers424.49/0.615325.59/0.715325.63/0.680025.83/0.701126.19/0.7600


## 4.4. Denoising of Real Ultrasound Images

The last experiment is mainly to test the denoising effect of the new algorithm for solving the real ultrasound image and compare it with other denoising models. However, the real ultrasound image does not have the original image, so it cannot be evaluated using the original PSNR and SSIM. In 2005, Buades et al. [31] proposed an evaluation method without original image. The method evaluates the image denoising effect based on the difference and the estimated noise images between the real ultrasound image and the restored image. The difference image and the estimated noise image are defined as:
(29)f−u¯andf−u¯u¯,where f∈Rm×n is the real ultrasound image and u¯∈Rm×n is the restored image. The image denoising effect can be evaluated based on the difference image and the similarity of the estimated noise image, that is, based on the residual image (the difference between the difference image and estimated noise image). If the texture of the residual image is small, the denoising effect is well.Figure15 shows that the experimental results of real ultrasound images by applying JIN’s model, finite difference for the new model, and ADMM for the new model. Figure 16 shows the residual image after applying the JIN model, finite difference for the new model, and ADMM for the new model to restore the real ultrasound image. Visually speaking, the new model has fewer textures of residual images under the new algorithm, which shows that our method is better in denoising effect.Figure 15
Numerical result of the real ultrasound image. (a, e, i) Real ultrasound images. (b, f, j) Restored image by the JIN’s model. (c, g, k) Restored image by finite difference for the new model. (d, h, l) Restored image by ADMM for the new model.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Figure 16
Residual image. (a, d, g) Residual image by the JIN’s model. (b, e, h) Residual image by finite difference for the new model. (c, f, i) Residual image by ADMM for the new model.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


## 5. Conclusion

In this paper, we introduce the ADMM algorithm to optimize the denoising effect of the model for the new speckle noise recovery model based on adaptive variational method. The new algorithm is a computational framework for solving optimization problems, which is suitable for solving distributed convex optimization problems. The numerical experiments results show the effectiveness of the new method. In addition, by comparing with some existing methods, the experiment results show the high efficiency of the new method in image restoration. In recent years, nonconvex model denoising methods have received more and more attention, so it is possible to study nonconvex models in the next work.



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*Source: 3416958-2020-08-12.xml*