Truncated Fractional-Order Total Variation for Image Denoising under Cauchy Noise

Abstract

In recent years, the fractional-order derivative has achieved great success in removing Gaussian noise, impulsive noise, multiplicative noise and so on, but few works have been conducted to remove Cauchy noise. In this paper, we propose a novel nonconvex variational model for removing Cauchy noise based on the truncated fractional-order total variation. The new model can effectively reduce the staircase effect and keep small details or textures while removing Cauchy noise. In order to solve the nonconvex truncated fractional-order total variation regularization model, we propose an efficient alternating minimization method under the framework of the alternating direction multiplier method. Experimental results illustrate the effectiveness of the proposed model, compared to some previous models.

1. Introduction

Image denoising is the most typical problem in image processing, which aims to recover a clean image from a degraded image with noise. In this paper, we focus on the problem of Cauchy noise removal in images. This noise is commonly found in wireless communication systems, synthetic aperture radar (SAR) images, and biomedical images [1,2,3,4]. Let f be the observed noisy image. The image observation model with Cauchy noise is mathematically expressed as

$$f=u+\tau ,$$

where $\tau$ denotes the Cauchy noise. For the Cauchy noise, a different strategy is needed because of its distribution function. A random variable x follows the Cauchy distribution if it has the following probability density function (PDF):

$$p(x)=\frac{1}{\pi }\frac{\gamma }{{\gamma }^2+{(x-\delta )}^2},$$

where $\gamma >0$ is the scale parameter, and $\delta >0$ is a localization parameter which corresponds to the median of the distribution and may be assumed to be 0.

In recent years, the problem of Cauchy noise removal has drawn significant attention. Achim et al. [5] applied $\alpha$-stable PDF to propose a new statistical model for removing Cauchy noise in the complex wavelet domain. In [6], Wan et al. proposed a novel image segmentation technique for the color image corrupted by Cauchy noise. Based on total variation regularization, Sciacchitano et al. [7] first proposed a variational model for restoring degraded images corrupted by Cauchy noise. The model in [7] can be expressed as follows:

$$\underset{u}{min}{\displaystyle \frac{\lambda }{2}}⟨log({\gamma }^2+{(u-f)}^2),\mathbf{1}⟩+\mu {∥u-f_0∥}_F^2)+{∥\nabla u∥}_1,$$

(1)

where $\lambda >0$ is the regularization parameter, $\mu >0$ is a penalty parameter, $⟨·⟩$ denotes the Frobenius inner product, $\mathbf{1}$ is an n-by-n matrix of ones, and $f_0$ is the predenoising result of the median filter operation for noisy image f. In [7], it is proved that the objective function in (1) is strictly convex when $8\mu {\gamma }^2\ge 1$, and there is a unique solution. However, this model relies greatly on the solution of the last term, namely, the median filter. In order to overcome the shortcomings of Model (1), Mei et al. [8] proposed the following nonconvex variational model:

$$\underset{u}{min}{\displaystyle \frac{\lambda }{2}}⟨log({\gamma }^2+{(u-f)}^2),\mathbf{1}⟩+{∥\nabla u∥}_1.$$

(2)

Due to the nonconvexity and nonsmoothness of Model (2), they proposed a specific alternating direction method of multiplier with convergence guarantees. As is known, although the total variation regularization variational model can preserve sharp edges, it cannot recover the fine details and textures and often produces staircase effects.

In order to overcome of the limitations of total variation regularization, one choice is using high-order total variation as a regularization term, for example, second-order total variation [9,10,11], total generalized variation [12,13,14,15], hybrid high-order total variation [16,17,18], and so on. On the other hand, some fractional-order total variation regularization models [19,20,21] have been proposed for additive and multiplicative noise removal. In all these papers, the fractional-order total variation models can alleviate the staircase effect and preserve sharp edges by choosing the order of derivative properly. Moreover, the fractional-order total variation models can also improve the performance of texture preservation because of “non-local” behavior.

To further improve the performance of the fractional-order variation model in image restoration, recently, Chan and Liang [22,23] proposed a truncated fractional-order total variation model for Gaussian noise removal. Their numerical experiments showed that their proposed truncated fractional-order total variation model performs better for eliminating the staircase effect and preserving textures, compared to the total variation regularization model and fractional-order total variation model. In this paper, we extend the truncated fractional-order total variation regularization to recover Cauchy noise and then propose the following model:

$$\underset{u}{min}{\displaystyle \frac{\lambda }{2}}(⟨log({\gamma }^2+{(u-f)}^2),\mathbf{1}⟩+u\Vert u-u_0{\Vert }_F^2+{∥{\nabla }^{\alpha ,t}u∥}_1,$$

(3)

where ${∥{\nabla }^{\alpha ,t}u∥}_1$ represents the regularization term, which is introduced in Section 2, u₀ denotes the image obtained by applying the median filter to the noisy image f. To our knowledge, the proposed model is the first one that uses the truncated fractional-order total variation regularizer for Cauchy noise removal. Although this seems to be a simple generalization, its optimization is more challenging than Gaussian denoising due to the nonconvexity of the data fidelity term. The main contributions of our work are as follows: (1) We propose a novel minimization model for Cauchy noise removal by adopting the truncated fractional-order total variation regularization. (2) The proposed model can suppress staircasing artifacts better and keep more details. (3) Compared with some previous Cauchy noise models, the restored images obtained by the proposed model not only have higher peak signal-to-noise ratio (PSNR) and structural similarity index measurement (SSIM), but also have better visual effects.

The rest of this paper is organized as follows. In Section 2, we give the definitions of the the truncated fractional-order total variation. In Section 3, under the frame of alternating direction method of multipliers, we develop an efficient alternating minimization algorithm to solve the proposed nonconvex Cauchy noise reduction model. In Section 4, numerical comparisons with existing methods are carried out to confirm the effectiveness of our method. Finally, concluding remarks are given in Section 5.

2. Preliminary

In this section, we briefly review the concept of truncated fractional-order total variation.

Truncated Fractional-Order Total Variation

For an image matrix $u\in {\mathbb{R}}^{n\times n}$, its fractional derivative can be expressed as ${\nabla }^{\alpha }u=({\nabla }_x^{\alpha }u,{\nabla }_y^{\alpha }u)$. The discrete fractional-order gradient at pixel $(i,j)$ is defined as $({({\nabla }_x^{\alpha }u)}_{i,j}$, ${({\nabla }_y^{\alpha }u)}_{i,j})$, where

$$\begin{array}{c}{({\nabla }_x^{\alpha }u)}_{i,j}={\displaystyle \sum _{k=0}^{K-1}}{\omega }_k^{\alpha }{u}_{i-k,j},\hfill \\ {({\nabla }_y^{\alpha }u)}_{i,j}={\displaystyle \sum _{k=0}^{K-1}}{\omega }_k^{\alpha }{u}_{i,j-k},\hfill \\ {\omega }_k^{\alpha }={(-1)}^k\frac{\Gamma (\alpha +1)}{\Gamma (k+1)\Gamma (\alpha -k+1)}=\left\{\begin{array}{cc}1,\hfill & k=0,\hfill \\ {(-1)}^k\frac{\alpha (\alpha -1)\cdots (\alpha -k+1)}{k!},\hfill & k=1,2,\cdots \hfill \end{array}\right.\hfill \end{array}$$

where K is the number of adjacent pixels used to calculate the fractional-order derivative of each pixel. According to the definition of coefficient ${\omega }_k^{\alpha }$, the following recurrence formulas can be obtained:

$${\omega }_0^{\alpha }=1;{\omega }_k^{\alpha }={\omega }_{k-1}^{\alpha }·(1-\frac{\alpha +1}{k}),k=1,2,\cdots .$$

We use the same truncation strategy as in [22,23] to obtain the new coefficient ${\omega }_k^{\alpha ,t}$:

$${\omega }_k^{\alpha ,t}=\left\{\begin{array}{cc}{\omega }_k^{\alpha },\hfill & k=0,1,\cdots ,K-2,\hfill \\ {\displaystyle \sum _{k=K-1}^{\infty }}{\omega }_k^{\alpha },\hfill & \\ 0,\hfill & k=K,K+1,\cdots \hfill \end{array}\right.$$

and then, the truncated fractional-order gradient operators can be defined as follows: ${\nabla }^{\alpha ,t}u=({\nabla }_x^{\alpha ,t}u,{\nabla }_y^{\alpha ,t}u),$ where the discrete truncated fractional-order gradients ${\nabla }_x^{\alpha ,t}u,{\nabla }_y^{\alpha ,t}u$ at pixel $(i,j)$ are defined by

$${({\nabla }_x^{\alpha ,t}u)}_{i,j}=\sum _{k=0}^{K-1}{\omega }_k^{\alpha ,t}{u}_{i-k,j},{({\nabla }_y^{\alpha ,t}u)}_{i,j}=\sum _{k=0}^{K-1}{\omega }_k^{\alpha ,t}{u}_{i,j-k}.$$

Similarly, we also give the divergence operator $di{v}^{\alpha ,t}$ by

$${(di{v}^{\alpha ,t}p)}_{i,j}=\sum _{k=0}^{K-1}(-{\omega }_k^{\alpha ,t})p_{i+k,j}^1+\sum _{k=0}^{K-1}(-{\omega }_k^{\alpha ,t})p_{i,j+k}^2.$$

This divergence satisfies the equation

$$<{\nabla }^{\alpha ,t}u,p>=<u,-di{v}^{\alpha ,t}p>,forp=(p^1,p^2)\in {\mathbb{R}}^{n^2}\times {\mathbb{R}}^{n^2}.$$

Then, the discrete version of the truncated fractional-order total variation can be defined as follows:

$${∥{\nabla }^{\alpha ,t}u∥}_1=\sum _{1\le i,j\le n}|{({\nabla }^{\alpha ,t}u)}_{i,j}|=\sum _{1\le i,j\le n}\sqrt{{({\nabla }_x^{\alpha ,t}u)}_{i,j}^2+{({\nabla }_y^{\alpha ,t}u)}_{i,j}^2},$$

when $\alpha =1$, ${∥{\nabla }^{\alpha ,t}u∥}_1$ and ${∥{\nabla }^{\alpha }u∥}_1$ are equivalent.

3. Proposed Method

In this section, we present an efficient alternating minimization algorithm to solve the proposed Model (3). By introducing two auxiliary variables $v,z$, Model (3) can be transformed into the following constrained problem:

$$\begin{array}{cc}\hfill \underset{v,z}{min}& {\displaystyle \frac{\lambda }{2}}\left(⟨log({\gamma }^2+{(v-f)}^2),\mathbf{1}⟩+\mu {∥v-u_0∥}_F^2\right)+{∥z∥}_1\hfill \\ \hfill s.t.& v=u,z={\nabla }^{\alpha ,t}u.\hfill \end{array}$$

(4)

To deal with the above constrained optimization problem, we give the corresponding augmented Lagrangian function of (4) as follows:

$$\begin{array}{cc}\hfill \mathcal{L}(u,v,z,{\lambda }_1,{\lambda }_2)=& {\displaystyle \frac{\lambda }{2}}\left(⟨log({\gamma }^2+{(v-f)}^2),\mathbf{1}⟩+\mu {∥v-u_0∥}_F^2\right)+{∥z∥}_1+\frac{{\beta }_1}{2}{\parallel v-u\parallel }_2^2\hfill \\ \hfill & -{\lambda }_1^T(v-u)+\frac{{\beta }_2}{2}{\parallel z-{\nabla }^{\alpha ,t}u\parallel }_2^2-{\lambda }_2^T(z-{\nabla }^{\alpha ,t}u),\hfill \end{array}$$

(5)

where ${\beta }_1,{\beta }_2>0$ are penalty parameters and ${\lambda }_1,{\lambda }_2$ are the Lagrangian multipliers. To solve the constrained optimization problem, (4) is equivalent to finding a saddle point of Lagrangian function $\mathcal{L}(u,v,z,{\lambda }_1,{\lambda }_2)$. According to the alternating direction method of multipliers, the problem (5) can be solved by the following iteration scheme:

$$\{\begin{array}{ll}{u}^{k+1}=arg{\displaystyle \underset{u}{min}}\mathcal{L}(u,v^k,z^k,{\lambda }_1^k,{\lambda }_2^k),\hfill & \left(6\mathrm{a}\right)\\ v^{k+1}=arg{\displaystyle \underset{u}{min}}\mathcal{L}(u^{k+1},v,z^k,{\lambda }_1^k,{\lambda }_2^k),\hfill & \left(6\mathrm{b}\right)\\ z^{k+1}=arg{\displaystyle \underset{u}{min}}\mathcal{L}(u^{k+1},v^{k+1},z,{\lambda }_1^k,{\lambda }_2^k),\hfill & \left(6\mathrm{c}\right)\\ {\lambda }_1^{k+1}={\lambda }_1^k-{\beta }_1(v^{k+1}-u^{k+1}).\hfill & \left(6\mathrm{d}\right)\\ {\lambda }_2^{k+1}={\lambda }_2^k-{\beta }_2(z^{k+1}-{\nabla }^{\alpha ,t}{u}^{k+1}).\hfill & \left(6\mathrm{e}\right)\end{array}$$

In the following, we will provide the details to solve (6a)–(6c).

The u subproblem (6b) can be rewritten as

$$u^{k+1}=arg\underset{u}{min}\frac{{\beta }_1}{2}\parallel v^k-u-\frac{{\lambda }_1^k}{{\beta }_1}{\parallel }_2^2+\frac{{\beta }_2}{2}\parallel z^k-{\nabla }^{\alpha ,t}u-\frac{{\lambda }_2^k}{{\beta }_2}{\parallel }_2^2.$$

(7)

The optimality condition of (7) can be given by the following equation:

$${\beta }_1(u-v^k+\frac{{\lambda }_1^k}{{\beta }_1})+{\beta }_2{({\nabla }^{\alpha ,t})}^T({\nabla }^{\alpha ,t}u-z^k+\frac{{\lambda }_2^k}{{\beta }_2})=0.$$

With some simple computation, the above equation can be rewritten as the following equation:

$$({\beta }_1+{\beta }_2{({\nabla }^{\alpha ,t})}^T({\nabla }^{\alpha ,t}))u={\beta }_1{v}^k-{\lambda }_1^k+{({\nabla }^{\alpha ,t})}^T({\beta }_1{z}^k-{\lambda }_2^k).$$

Under the periodic boundary condition of image u, the matrices ${({\nabla }^{\alpha ,t})}^T({\nabla }^{\alpha ,t})$ are block circulant with a circulant block. Therefore, the above equation can be solved efficiently by using fast Fourier transform operation $\mathcal{F}$ and inverse fast Fourier operation ${\mathcal{F}}^{-1}$:

$$u^{k+1}={\mathcal{F}}^{-1}(\frac{\mathcal{F}({\beta }_1{v}^k-{\lambda }_1^k+{({\nabla }^{\alpha ,t})}^T({\beta }_1{z}^k-{\lambda }_2^k))}{\mathcal{F}({\beta }_1+{\beta }_2{({\nabla }^{\alpha ,t})}^T({\nabla }^{\alpha ,t}))}).$$

(8)

The v-subproblems (6a) can be written as:

$$v^{k+1}=arg\underset{v}{min}{\displaystyle \frac{\lambda }{2}}\left(⟨log({\gamma }^2+{(v-f)}^2),\mathbf{1}⟩+\mu {∥v-u_0∥}_F^2\right)+\frac{{\beta }_1}{2}{\parallel v-u-\frac{{\lambda }_1^k}{{\beta }_1}\parallel }_2^2.$$

(9)

Let

$$G\left(v\right)={\displaystyle \frac{\lambda }{2}}\left(⟨log({\gamma }^2+{(v-f)}^2),\mathbf{1}⟩+\mu {∥v-u_0∥}_F^2\right)+\frac{{\beta }_1}{2}{\parallel v-u^{k+1}-\frac{{\lambda }_1^k}{{\beta }_1}\parallel }_2^2,$$

then we can obtain the following gradient and Hessian matrices of

$$\begin{array}{cc}\hfill G^{\prime }\left(v\right)& =\lambda \frac{v-f}{{\gamma }^2+{(v-f)}^2}+\lambda \mu (v-u_0)+{\beta }_1(v-u^{k+1}-\frac{{\lambda }_1^k}{{\beta }_1}),\hfill \\ \hfill G^{\prime \prime }\left(v\right)& =\lambda \frac{{\gamma }^2-{(v-f)}^2}{{({\gamma }^2+{(v-f)}^2)}^2}+({\beta }_1+\lambda \mu )I.\hfill \end{array}$$

Then, the solution of the v-subproblems (6b) can be obtained by the following Newton’s method:

$$v^{k+1,l+1}=v^{k+1,l}-\frac{G^{\prime }(v^{k+1,l})}{G^{\prime \prime }(v^{k+1,l})},$$

(10)

where $v^{k+1,l+1}$ represents the result of the $(l+1)$-th Newton iteration in the $(k+1)$-th outer iteration.

The z-subproblem (6c) can be written as

$$z^{k+1}=arg\underset{z}{min}{∥z∥}_1+\frac{{\beta }_2}{2}\parallel z-{\nabla }^{\alpha ,t}{u}^{k+1}-\frac{{\lambda }_2^k}{{\beta }_2}{\parallel }_2^2.$$

(11)

It is easy to know that its closed-form solution can be given by the following shrinkage formula:

$$z_{i,j}^{k+1}=max\{|{({\nabla }^{\alpha ,t}{u}^{k+1}+\frac{{\lambda }_2^k}{{\beta }_2})}_{i,j}|-\frac{1}{{\beta }_2},0\}·\frac{{({\nabla }^{\alpha ,t}{u}^{k+1}+\frac{{\lambda }_2^k}{{\beta }_2})}_{i,j}}{|{({\nabla }^{\alpha ,t}{u}^{k+1}+\frac{{\lambda }_2^k}{{\beta }_2})}_{i,j}|}.$$

(12)

As a summary, the proposed method to solve the problem (3) is described as follows.

$\mathbf{Proposed}\mathbf{method}\mathbf{for}\mathbf{solving}\mathbf{Model} (3)$

$\mathbf{Input}:$ noise image f, parameters $\lambda ,\alpha ,K,$

$\mathbf{Initialize}:$ $u^0,v^0,z^0,{\lambda }_1^0,{\lambda }_2^0,k=0$, maximum inner iteration N, $ϵ={10}^{-5}$

$\mathbf{While}$  $k<N$ or ${∥u^{k+1}-u^k∥}_F/{∥u^k∥}_F>ϵ$

1. Update $u^{k+1}$ by (8).

2. Update $v^{k+1}$ by (10).

3. Update $z^{k+1}$ by (12).

4. Update ${\lambda }_1^{k+1},{\lambda }_2^{k+1}$ by (6d), (6e).

5. $k=k+1$.

$\mathbf{End}$

$\mathbf{Output}:$ Restoring image u.

4. Numerical Experiments

In this section, we present several numerical results to demonstrate the effectiveness of the proposed method. We compare our method with other two well-known methods: the median filter [24] and Mei’s method [8]. All the test images with the size of $256\times 256$ are shown in Figure 1. All the numerical experiments are implemented under Windows 10 and Matlab R2018a running on a desktop equipped with 2.60 GHz Inter(R) Core(TM) i5-3230M and 4 GB memory. In our experiments, PSNR and SSIM [25] are used to evaluate the image denoising effect. They are defined as follows:

$$\mathrm{PSNR}=10{log}_{10}\frac{{255}^2}{{∥x-y∥}_2^2},$$

$$\mathrm{SSIM}=\frac{(2{\mu }_x{\mu }_y+C_1)(2{\sigma }_{xy}+C_2)}{({\mu }_x^2+{\mu }_y^2+C_1)({\sigma }_x^2+{\sigma }_y^2+C_2)},$$

where ${\mu }_x$ and ${\mu }_y$ represent the mean value of images x and y, ${\sigma }_x$ and ${\sigma }_y$ are the standard variance of images x and y, ${\sigma }_{xy}$ is the covariance of x and y, and $C_1$ and $C_2$ are constants.

Figure 1. Test images.

In this paper, the degraded image contaminated by Cauchy noise can be obtained by the following equation:

$$f=u+\tau =u+\eta \frac{{\xi }_1}{{\xi }_2},$$

where $\eta$ denotes noise level, and ${\xi }_1$ and ${\xi }_2$ follow a Gaussian distribution with a mean of 0 and variance of 1. Empirically, we set $\eta ={\gamma }^2$ for good experimental performance. To make it easier to compare with different methods, the stopping criterion used in all the methods is set to be

$$\frac{{∥u^{k+1}-u^k∥}_F}{{∥u^k∥}_F}<{10}^{-5}.$$

4.1. Parameter Discussion

Throughout all the experiments, we set $K=3$ to be the same as in [22,23], which is the number of pixels to approximate the fractional derivative. In this subsection, we mainly focus on the selection of parameters $\lambda ,\mu ,$ and $\alpha$. The images “Lin”, “Lena” and “Starfish” with the noise level $\eta =0.02$ are used to study the choice of these parameters.

First, we study the selection of the parameters $\lambda ,\mu$ in Model (3). In order to find out how the parameter $\lambda$ impacts the performance of our proposed method, we take the value evenly on the interval $[0.1,2]$, and the step length is set to $0.1$ to test the image recovery quality obtained under different $\lambda$; the remaining parameters are fixed. We use the same strategy to select parameters $\mu$ and $\alpha$. The curves of PSNR and SSIM with respect to $\lambda$ and $\mu$ are shown in Figure 2. It is easy to see that the performance of the proposed method achieves the best results with $\lambda$ in $0.8$ nearby for all the different test images. So, we fix $\lambda =0.8$ in the following experiments. From Figure 2, we can observe that the PSNR and SSIM values are stable when $\mu >20$. However, the calculation time increases as the values of $\mu$ increase. Then, in the following experiments, we set $\mu =20$. Finally, we discuss the selection of the fractional-order parameter $\alpha$. We carry out experiments to compute the values of PSNR and SSIM with respect to $\alpha$. From Figure 2, we can see that our proposed Cauchy noise denoiser can achieve the highest PSNR and SSIM values when $\alpha =1.3$.

Figure 2. (a–f) The values of PSNR and SSIM with respect to the parameters $\lambda ,\mu ,\alpha$.

4.2. Image Denoising

From the above parameter selection discussion, we can find that the fractional order parameter $\alpha$ has an important effect on the quality of the restoration images. In order to further illustrate that the fractional order parameter $\alpha$ we set is reasonable and effective, we fix other parameters and perform numerical experiments, choosing different values of $\alpha$ to examine the denoising effect. We consider the images “Monarch”, “Parrot”, and “Pallons” with a Cauchy noise at $\eta =0.02$, “Starfish” and “Monarch” with a Cauchy noise at $\eta =0.04$, respectively. We carry out our proposed method with different $\alpha$ to remove the Cauchy noise. Figure 3 shows the denoising results obtained by our proposed method with different fractional $\alpha$. We can clearly see that the quality of the restored image is higher when the fractional order parameter $\alpha =1.3$. For these two noise levels, as the fractional parameter $\alpha$ increases, some noise remains in the texture area of the restored image. In Table 1, we show the values of PSNR and SSIM for six different images which are obtained by our proposed method against different values of $\alpha$ and different noise levels. As can be seen, our proposed model can achieve the highest PSNR and SSIM values when $\alpha =1.3$.

Figure 3. The restored image obtained by our proposed method with different $\alpha$.

Table 1. The values of PSNR and SSIM by our proposed method with different fractional order parameter $\alpha$ and different noise level.

$\mathit{\eta }$	Order	Image	Lin	Lena	Starfish	Monarch	Parrots	Pallon
0.02	$\alpha =1$	PSNR (dB)	32.6540	30.8777	29.5223	30.0423	33.6052	31.2158
		SSIM	0.8979	0.8904	0.8851	0.9148	0.8949	0.8952
	$\alpha =1.3$	PSNR (dB)	32.8101	31.0835	29.7979	30.1435	33.6994	31.4119
		SSIM	0.9039	0.8977	0.8920	0.9216	0.8997	0.9011
	$\alpha =1.5$	PSNR (dB)	32.7393	31.0464	29.7474	30.0382	33.6463	31.3735
		SSIM	0.9053	0.8998	0.8919	0.9232	0.9011	0.9026
	$\alpha =1.7$	PSNR (dB)	32.5558	30.8884	29.5741	29.8435	33.5011	31.2102
		SSIM	0.9039	0.8976	0.8889	0.9225	0.9005	0.9015
	$\alpha =1.9$	PSNR (dB)	31.2377	29.8783	28.7559	28.9616	31.9868	30.1515
		SSIM	0.8902	0.8833	0.8758	0.9125	0.8874	0.8876
	$\alpha =2$	PSNR (dB)	30.0010	28.8982	27.9622	28.1182	30.5678	29.1301
		SSIM	0.8784	0.8706	0.8657	0.9019	0.8745	0.8752
0.04	$\alpha =1$	PSNR (dB)	30.5904	28.6729	27.1310	27.6771	29.0138	31.6858
		SSIM	0.8654	0.8428	0.8238	0.8756	0.8622	0.8697
	$\alpha =1.3$	PSNR (dB)	30.6629	28.7953	27.3793	27.8112	29.1444	31.7235
		SSIM	0.8667	0.8482	0.8326	0.8800	0.8644	0.8697
	$\alpha =1.5$	PSNR(dB)	30.0983	28.4428	27.1130	27.2980	28.7271	31.0574
		SSIM	0.8521	0.8376	0.8249	0.8709	0.8508	0.8548
	$\alpha =1.7$	PSNR (dB)	27.4326	26.4269	25.5197	25.5712	26.6072	27.9338
		SSIM	0.8064	0.7998	0.7939	0.8381	0.8088	0.8097
	$\alpha =1.9$	PSNR (dB)	23.4457	22.9880	22.5588	22.5492	23.0828	23.6297
		SSIM	0.7358	0.7383	0.7411	0.7840	0.7439	0.7368
	$\alpha =2$	PSNR(dB)	21.6235	21.3118	21.0121	20.9940	21.3740	21.7438
		SSIM	0.7036	0.7095	0.7147	0.7293	0.7566	0.7143

In order to illustrate the performance of our proposed method for Cauchy noise removal, we compare it with the median filter [24] and Mei’s method [8]. In the first experiment, we added Cauchy noise with $\eta =0.02$ to the test images. In Figure 4 and Figure 5, we present the restored images by different methods with $\eta =0.02$ and $\eta =0.04$, respectively. From Figure 4 and Figure 5, it is easy to see that the median filter can effectively remove Cauchy noise, but it causes the staircase effect and oversmooths the edges. Mei’s method can provide better balance between preserving edges and alleviating the staircase effect; however, the restored images by Mei’s method show residual noise. As a comparison, we can find that our proposed model has better performance in eliminating noise and preserving the edges and details. To quantify performance comparisons, Table 2 and Table 3 list the PSNR and SSIM values of the restored images by different methods at $\eta =0.02$ and $\eta =0.04$, respectively. In comparison with the other two methods, our proposed method can obtain the highest PSNR and SSIM values, which are consistent with the visual comparison.

Figure 4. Denoising results of three methods at $\eta =0.02$.

Figure 5. Denoising results of three methods at $\eta =0.04$.

Table 2. PSNR and SSIM values for different algorithms ($\eta =0.02$).

Algorithm	Median [24]		Mei’s Method [8]		Our Method
Image	PSNR (dB)	SSIM	PSNR (dB)	SSIM	PSNR (dB)	SSIM
Lin	31.2381	0.8563	31.4169	0.8679	32.8101	0.9039
Lena	29.6954	0.8640	30.3413	0.8826	31.0835	0.8977
Starfish	26.2701	0.8630	29.2600	0.8788	29.7979	0.8920
Monarch	29.4393	0.8925	29.7134	0.9043	30.1435	0.9216
Parrot	29.7421	0.8597	30.6362	0.8766	31.4119	0.9011
Pallon	31.9994	0.8539	32.8719	0.8785	33.6994	0.8997
Boats	29.0469	0.8376	30.1942	0.8773	30.5385	0.8818
Elaine	32.1007	0.8852	32.2804	0.8988	33.2166	0.9078

Table 3. PSNR and SSIM values for different algorithms ($\eta =0.04$).

Algorithm	Median [24]		Mei’s Method [8]		Our Method
Image	PSNR (dB)	SSIM	PSNR (dB)	SSIM	PSNR (dB)	SSIM
Lin	28.4978	0.7064	29.3442	0.8326	30.6629	0.8667
Lena	27.3856	0.7540	28.0523	0.8312	28.7953	0.8482
Starfish	26.3759	0.7809	26.4340	0.7926	27.3793	0.8326
Monarch	26.2771	0.7977	27.2914	0.8636	27.8112	0.8800
Parrot	27.3956	0.7293	28.4972	0.8444	29.1444	0.8644
Pallon	28.9130	0.7064	31.5087	0.8668	31.7235	0.8697
Boats	26.8995	0.7392	27.5676	0.8008	28.2589	0.8244
Elaine	28.8692	0.7772	29.7711	0.8490	30.8427	0.8679

4.3. Convergence Analysis

In this subsection, we plot the curves of the PSNR and SSIM values versus the iterations number to verify the convergence of our proposed method. In order to carry out this experiment, we use the images “Lin”, “Lena” and “Starfish” degraded by Cauchy noise with $\eta =0.02$ and $\eta =0.04$, respectively. The PSNR and SSIM curves of our proposed method are presented in Figure 6. It is easy to see that the curves of PSNR and SSIM are flat when the number of iterations exceeds 60.

Figure 6. Top row: Curves of PSNR and SSIM versus iterations number at $\eta =0.02$. Bottom row: Curves of PSNR and SSIM versus iterations number at $\eta =0.04$.

5. Conclusions

Based on the truncated fractional-order total variation regularization, we propose a new variational model for Cauchy noise removal. Under the framework of alternating direction of the multiplier method, we propose an alternating minimization method to solve the proposed model. Extensive experiments demonstrate the superiority of our proposed model and method over some existing methods.