#### 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.

#### 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: where is a noisy image and represents the Gaussian random noise with zeros mean and standard deviation .

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:

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 Section 2, 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: where is a clean image and is a noisy image affected by Gaussian random noise with mean zero and standard deviation . is regularization parameter which can balance fidelity term and regularized term in TV model. represents the gradient operator, which is defined as follows: where is the forward discrete derivation operators, which are defined as:

##### 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:

Where , is the fidelity term, and 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 functional is convex. That is to say, for any and , we have:
where .*

*Proof. *The proof of Theorem 1 is given in the appendix A.

Secondly, the uniqueness of the minimum solution of the model (6) can also be proved.

Theorem 2. *If and are two minimize solutions of model (6), then we have .*

*Proof. *The proof of Theorem 2 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, where , , , , , and function and function are convex and lower semicontinuous functions.

By introducing a Lagrangian multiplier vector or dual variable , then the augmented Lagrange function of problem (8) is where is a penalty parameter. By the algorithm of Alternating Direction Method of Multipliers (ADMM), the solution is achieved with the following iteration: where 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:

To solve the problem with the ADMM algorithm, we introduce two new variables: . Apply these two variables, the unconstrained problem (11) is written as the following constrained problem:

Next, we make

Now, we letting

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

According to the structure of the ADMM algorithm, the problem (15) can be rewritten as the following form:

Since the two new variables and in problem (16) are not related to each other, so we can divide two simple minimization subproblem as follows,

First, for the , we can obtain the corresponding Euler-Lagrange equation:

Simplifying the Euler-Lagrange equation above, we can obtain three complicated solutions by the roots formula of the cubic equation:

For equation (19), there is only one positive real number solution (the proof is given in appendix C).

Second, for the , we will apply the shrink operator to obtain the solution . The form of the solution is as follows: where the shrink operator is defined [30] as:

Last, the is a least square problem as follow: where . So the solution of the least square problem (22) equivalent to the solution of the . That is to say, the solution is as follows:

To sum up, the ADMM algorithm can be summarized in Algorithm 1.

#### 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 algorithm 1, the stopping condition is that the solution of two adjacent iterations satisfies: where is the maximum iteration numbers; for algorithm 1. represents the results of the 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;

And the stopping condition (NDR) for finite difference for the new model is as follow

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: where is the clean image and is the restored image. and are the mean intensity of and , respectively. and are the standard deviation of and , respectively. is the covariance of and , and and 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 parameters value in the algorithm 1 for original image in Figure 1. We choose “syn1” and the noise level is . Figure 2 shows that numerical experiments with different parameters in algorithm 1. Figure 3 shows that the different PSNR and SSIM values when different values are used in algorithm 1. Here, we can see that the optimal values of PSNR and SSIM values are at . Therefore, in the following experiment, we choose in algorithm 1.

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##### 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”. Figures 4(a) and 5(a) show the noise images, and the noise deviation are and , 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.

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##### 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 Figure 7 (“lena,” “pirate,” “boat,” “bird,” “house,” and “peppers”).

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Firstly, in the subjective evaluation, we mainly focus on the details of the restored images. The first line of Figure 8 shows the original images (“lena” and “bird”) and the noise images with noise deviation is . 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.

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Secondly, in the objective evaluation, we compare the denoising effects of different models with PSNR and SSIM values. Figures 11, 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.

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##### 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: where is the real ultrasound image and 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.

Figure 15 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.

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#### 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.

#### Appendix

#### A. The Proof of Theorem 3.1.1

*Proof. *Firstly, the function is convex, according to Definition 2.1.2, for any , and , we have
meanwhile, we have

Therefore,

This proof is established.

#### B. The Proof of Theorem 3.1.2

*Proof. *According to Theorem 3.1.1, we have
If , then the above assumption give a contradiction that is not a minimize solution.

#### C. The Proof of Cubic Equation (24) Only One Positive Real Number Solution

*Proof. *For the cubic equation, it has three solutions. Now, we assume are solution of cubic equation (24). According to Vieta theorem, we have
It is easy to know that the cubic equation has at least one real solution and for all Now, let us analyze the three solutions, without loss of generality, we assume , , and are a real solution.

The first case: assume is positive. So we can know and . Meanwhile, are solution of equation ; thus, and .

If , then and are complex numbers. If , then . Thus, and are negative.

The second case: assume is negative. So we can know and . Meanwhile, are solution of equation ; thus, and .

Because of , therefore, and are real number, is positive, and is negative.

This proof is established.

#### Data Availability

The experimental data are obtained by MATLAB R2017a, 2.93 GHz cup, 4 G ram, and windows 7.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Authors’ Contributions

All authors typed, read, and approved the final manuscript.

#### Acknowledgments

This paper is partially supported by the Natural Science Foundation of Guangdong Province (2018A030313364), the Special Innovation Projects of Universities in Guangdong Province (2018KTSCX197), the Science and Technology Planning Project of Shenzhen City (JCYJ20180305125609379), and the China Scholarship Council Project (201508440370).