# A LogTVSCAD Nonconvex Regularization Model for Image Deblurring in the Presence of Impulse Noise

Authors: Zhijun Luo; Zhibin Zhu; Benxin Zhang
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

This paper proposes a nonconvex model (called LogTVSCAD) for deblurring images with impulsive noises, using the log-function penalty as the regularizer and adopting the smoothly clipped absolute deviation (SCAD) function as the data-fitting term. The proposed nonconvex model can effectively overcome the poor performance of the classical TVL1 model for high-level impulsive noise. A difference of convex functions algorithm (DCA) is proposed to solve the nonconvex model. For the model subproblem, we consider the alternating direction method of multipliers (ADMM) algorithm to solve it. The global convergence is discussed based on Kurdyka–Lojasiewicz. Experimental results show the advantages of the proposed nonconvex model over existing models.


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## Body

## 1. Introduction

Image deblurring is a hot research topic of digital image processing, which is widely used in engineering and medicine fields [1]. In this paper, we focus on how to recover an image degraded by blur and impulsive noise. In this paper, we focus on how to recover an image degraded by blur and impulsive noise. Image blur noise may result from inaccurate focus, object relative movement, and optical degradation in the process of digital image acquisition and transmission. Impulse noise, such as salt-and-pepper noise (SP) and random-value noise (RV), is caused in the storage and transmission process dues to low-quality sensors or electromagnetic interference [2]. The mathematical model of image deblurring is usually expressed as(1)f=NimpKu,where f denotes the observed noisy and blurry image, Nimp⋅ represents the formation mechanism of the impulsive noise, and K and u denote a bounded blurring operator and the original image, respectively. For given the blurring operator K, our goal is to recover the original image u from the observation f. In general, the operator matrix K is often ill-conditioned, which cannot recover the original image u from f by direct inversion. To stabilize the recovery of u, one popular approach is the variational method, which includes a data fitting term and a regularization term. Rudin, Osher and Fatemi [3] first proposed total variation (TV) regularization model, which is widely used, for it can better keep the object boundaries information of the signal [4, 5]. The popular model is(2)minuuTV+μKu−f22,where u and f are the original image and the observed image, respectively; K denotes the linear blurring; μ>0 is the regularization parameter used to balance the regularization term and data-fitting term. This TVL2 model is optimal when the measurement noise is Gaussian distributed. However, non-Gaussian noise is more common in practice, and the performance of ℓ2-norm based methods may severely degrade. TVL1 model combining TV regularization and ℓ1-norm [6–9] was proposed to deal with impulsive noise (a typical non-Gaussian noise). Its mathematical formulation can be expressed as follows:(3)minuuTV+μKu−f1,where uTV is called the TV norm of the variable u, which can take the ℓ1-norm and ℓ2-norm, i.e.,(4)uTV≔Du1anisotropicanduTV≔Du2isotropic,where D is a finite difference operator. Generally, the TVL1 model with anisotropic TV norm can express a linear system, which is easier to deal with than the isotropic one. However, the isotropic TV norm is more realistic and more effective [10]. Numerically, some efficient algorithms have been proposed to solve the TVL1 model (3), such as the split Bregman [11, 12], the primal-dual method [7, 8, 13], and the alternating direction method of multipliers [9, 14, 15].However, the classical TV norm regularization model often underestimates the amplitudes of signal discontinuities [16, 17]. For high-level impulsive noise, the solution of TVL1 model (3) is biased, because the penalties of data fitting term for all data are equal [15]. In order to improve the performance of image restoration, nonconvex approaches are considered, e.g., the smoothly clipped absolute deviation (SCAD) [18], ℓq-norm 0<q<1 [19, 20], log-function [21, 22], and minimax-concave penalty (MCP) [23, 24]. This nonconvex technique plays an increasingly important role in solving image restoration problems. Because the nonconvex model can obtain a better approximation solution, it can improve the bias problem of the ℓ1-norm [25–28]. In [26], the authors have developed a nonconvex model called TVSCAD with the SCAD penalty function as data fitting term. In this model, they suggested that if the observation data is not severely damaged, data fitting should be enforced; otherwise, less or null penalize those data. Based on this work, the authors [27] very recently proposed a TV-Log model via using the TV as regularizer and Log penalty function as data fitting.In this paper, we continue to study the problem of image restoration with impulse noise. Our goal is to obtain a higher quality recovery solution through the newly constructed nonconvex model. Using the Log-function penalty as a nonconvex regularizer and the SACD-function penalty as a data fitting term, a nonconvex model is proposed:(5)minuFlogDu+μFscadKu−f,0≤u≤1,where μ>0 is the parameter. Note that here we add the bound constraint 0≤u≤1, which can improve the image recovery quality [26]. Flog⋅ and Fscad⋅ are defined as(6)Flogν=∑fsνi,Fscadν=∑fγνi,(7)fsx=1slog1+sx,fγx=x,x<γ1,2γ2x−x2−γ122γ2−γ1,γ1≤x<γ2,γ1+γ22,x≥γ2,where s>0,γ:≜γ1,γ2>0 are the threshold parameters. This is a “nonconvex + nonconvex” model, which can have some desirable properties simultaneously. Such penalty of concave functions fsx and fγx for all elements is nonuniform, which makes fsx and fγx closer to ℓ0-norm than ℓ1-norm. This result can be easily seen in Figure 1. Since the proposed model is nonconvex and nonsmooth, it is difficult to find an effective algorithm. To solve the proposed nonconvex model, we combine the difference-of-convex algorithm (DCA) [29, 30] with the proximal splitting method [31, 32]. In summary, the main contributions of this article are as follows:(1)
A new “nonconvex + nonconvex” model for image restoration with impulsive noise is proposed. The core idea is that the Log-function penalty as a regularizer and the SACD-function penalty as a data fidelity term are used. Therefore, this model approximates theℓ0-norm more closely than ℓ1-norm and is useful in image restoration with impulse noise.(2)
To solve the nonconvex model, we consider the DCA method with ADMM, which has been efficiently used in many nonconvex optimization problems. Then, we prove the proposed algorithm that is globally convergent.(3)
Numerical examples show the effectiveness of the proposed LogTVSCAD method, and we compare it with other recovery algorithms.Figure 1
Plot ofx1, x0, SCAD function fscad and Log-function flog.The rest of this paper is organized as follows. Section2 gives some notations and preliminaries. The nonconvex LogTVSCAD model and a DCA method with ADMM are shown in Section 3. Then, in Section 4, we prove that the proposed algorithm converges to a stationary point. Section 5 presents the experimental results, which illustrate the effectiveness of the new nonconvex model. Finally, some conclusions are given.


## 2. Notations and Preliminaries

In this section, we first give some notations. Then, some properties of the Log-function and SCAD-function penalty are given. Next, we show the definitions of the subdifferentials and basic properties of the Kurdyka–Lojasiewicz functions [33]. These conclusions will be used later in the proof of convergence.For any vectorx∈Rn, xTy or x,y (∀y∈Rn) denote their inner product; xq=∑i=1xiq1/q1>q>0 denotes ℓq-norm; ∇fx and ∂fx stand for the gradient and subdifferential of the function f⋅ at x, respectively; signx is the signum function. Now, we introduce some properties of the functions fsx and fγx. For fixed s>0 and γ>0, fsx and fγx are continuous on R, also increasing, continuously differentiable, and concave on R+, as illustrated in Figure 1. Next, we look at another functions gsx and gγx, which are induced by fsx and fγx:(8)gsx=x−fsx,gγx=x−fγx,where fsx and fγx are given by (7). Without loss of generality, we consider the multivariate generalization of the functions gsx and gγx:(9)Gsv=∑i=1ngsvi,Gγv=∑i=1ngγvi,v∈Rn.In fact, the functionsGsv and Gγv are continuously differentiable and convex on Rn, and then the log function penalty and the SCAD function penalty can be expressed as(10)Flogv=v1−Gsv,Fscadv=v1−Gγv.Note that the functions (10) play an important role in the later algorithm construction. The following useful definitions and properties can be obtained from the literature [34–36].Definition 1.
Iffx0≤liminfx⟶x0fx, then the function fx is lower semicontinuous at a point x0. A function fx is said to be lower semicontinuous in its domain of definition if it is lower semicontinuous at all x∈domf≔x∈Rm:fx<+∞.Definition 2.
For an extended-real-valued, proper and lower semicontinuous functionfx, its subdifferential ∂f at x∈domf is defined as(11)∂fx≔v∈Rm:∃xk⟶x,fxk⟶fx,vk∈∂^fxk⟶v,k⟶+∞,where ∂^fxk≔liminfz≠xk,z⟶xk1/z−xkfz−fxk−vk,z−xk.Iff is convex function, then the subdifferential is such that(12)∂fx≔v∈Rm:fy≥fx+v,y−x,∀y∈Rm.Furthermore, iffx is continuously differentiable at x0, then ∂fx0=∇fx0. If the point x∗∈Rm is a minimizer of fx, a necessary condition is 0∈∂fx∗, which is named a stationary or critical point of fx.Definition 3.
Letfx be a proper and lower semicontinuous function; the proximity operator is defined as(13)proxf,ηt=argminxfx+η2x−t2,where ηη>0 is a penalty parameter.It is well known that the proximal operator is particularly useful in convex optimization.Definition 4.
A functionfx is called to possess the Kurdyka−Lojasiewicz (KL) property at a point x0∈dom∂f if there exist η>0, a neighborhood U of x0 and a continuous concave function g:0,η⟶ℝ+ such that(i)
g0=0, gx is continuously differentiable and g′x>0 for all x∈0,η(ii)
∀x∈U satisfying fx0<fx<fx0+η; it holds that g′fx−fx0dist0,∂fx≥1
A proper closed functionfx is named a KL function if it has the KL property at all points in dom∂f.


## 3. Model and Algorithm

In this section, we first propose a nonconvex model for image restoration and then use DC programming to give the ADMM algorithm to solve the model.In the definitions of (10), we consider replacing v with gradient Dx and Ku−f respectively, which leads to our definition of the LogTVSCAD model as follows:(14)minuFlogDu+μFscadKu−f,0≤u≤1.i.e.,(15)minuuTV−GsDu+μKu−f1−GγKu−f,0≤u≤1.Because Log and SCAD penalty functions are nonconvex, this model is nonconvex. To address this nonconvex model, setju=uTV+μKu−f1, hu=GsDu+μGγKu−f, and problem (15) can be expressed as the difference of the convex functions ju and hu, i.e.,(16)minuFu=ju−hu,0≤u≤1.This is a DC programming problem, which has been efficiently used in many nonconvex optimization problems; for more details, please see [30, 37]. According to the classic DC algorithm (DCA) iteration, for (16), we have(17)uk+1=minuju−huk−vkTu−uk,0≤u≤1,where vk∈∂huk,k=0,1,2,…. To obtain a more accurate solution, we adopt the suggestion of the literature [26] and add a proximal term in our DCA iterations,(18)uk+1=minuju−huk−vkTu−uk+η2u−uk2,0≤u≤1,where η>0 is a given proximal parameter. For (15), the next iteration can be expressed as(19)uk+1=minuuTV+μKu−f1−DT∇GsDuk+μKT∇GγKuk−fTu−uk+η2u−uk2,0≤u≤1.By omitting the constants of formula (19), we have(20)uk+1=minuuTV+μKu−f1−∇GsDukTDu−μ∇GγKuk−fTKu−f+η2u−uk2,0≤u≤1.For the above model (20), under certain conditions, its objective function is strongly concave, and uk+1 has a unique solution in every step. To solve the problem of (20), we first introduce auxiliary variables w,z,p,q,x and define(21)w=Du,pk=∇GγKuk−f,x=uz=Ku−f,qk=∇GsDuk.Then, we rewrite (20) as a constrained minimization problem:(22)minu,w,z,xw2+μz1−μpkTz−qkTw+η2u−uk2s.t.w=Du,z=Ku−f,x=u,x∈Ω=u|0≤u≤1.Let the augmented Lagrangian function of model (22) be(23)ℒβu,w,z,x,λw,λz,λx=w2−λwTw−Du+βw2w−Du2−qkTw+μz1−μpkTz−λzTz−Ku−f+βz2z−Ku−f2+η2u−uk2−λxTx−u+βw2x−u2,where λw,λz,andλx are Lagrange multipliers and βw>0,βz>0,andβx>0 are penalty parameters. Given λw,λz,λx≔λw0,λz0,λx0 and u≔u0, according to the classical ADMM, the iterative scheme of the problem (23) can be expressed as follows:(24)wi+1,zi+1,xi+1=argminℒβw,z,x,uk,λwi,λzi,λxi|w,z,x∈Ω,ui+1=argminℒβwi+1,zi+1,xi+1,u,λwi,λzi,λxi,λwi+1=λwi−βwwi+1−Dui+1,λzi+1=λzi−βzzi+1−Kui+1−f,λxi+1=λxi−βxxi+1−ui+1.In this scheme, the ADMM method is directly applied to2−blocks of variables w,z,x and u. Furthermore, via Theorem 3.2 in [9], we can ensure that the proposed ADMM (24) for solving the subproblem is convergent. In fact, w,z, and x in (24) are separable from each other, so this optimization problem can be performed in parallel. Moreover, via the definition of the proximity operator, we can get the explicit solution of wi+1 and zi+1. In addition, xi+1 can be computed by a simple projection onto box Ω. Hence, the w,z,x optimizations have a closed-form solution as(25)wi+1=maxw^i2−1βw,0w^iw^i2,where,w^i=Dui+λwi+qiβw,(26)zi+1=signz^imaxz^i−μβz,0,where,z^i=Kui−f+λzi+μpiβz,(27)xi+1=maxminui+λxiβx,1,0,where sign⋅ is the signum function. Then, we consider how to solve the u-subproblem of (24). Via the first-order optimality conditions, the corresponding normal equation is(28)U^iui+1=DTwi+1−λwiβw+βzβwKTzi+1−λziβz+βzβwKTf+βxβwxi+1−λxiβx+ηβwui.where U^i=DTD+βz/βwKTK+βx+η/βwI is nonsingular under certain conditions. For problem (28), there is an efficient solution by an inverse fast Fourier transforms [14].Finally, we propose an ADMM algorithm for solving the proposed LogTVSCAD model (5).


### 3.1. Algorithm (LogTVSCAD)

Step 0 Initialization and date:Input parametersμ,s,γ1,γ2,βw,βz,βx,α,η>0, the tolerance ε>0. Given u0, let k≔0;Step 1 Givenλw0, λz0, λx0 and uk, compute the new iterate by (29);Fori≔0;(29)wi+1=maxw^i2−1βw,0w^iw^i2,zi+1=signz^imaxz^i−μβz,0,xi+1=maxminuk+λxiβx,1,0,ui+1=Ui^−1U˜i,λwi+1=λwi−βwwi+1−Dui+1,λzi+1=λzi−βzzi+1−Kui+1−f,λxi+1=λxi−βxxk+1−ui+1.whereU˜i=DTwi+1−λwi/βw+βz/βwKTzi+1−λzi/βz+βz/βwKTf+βx/βwxi+1−λxi/βx+η/βwui, w^i,z^i,and Ui^ are given by (25), (26) and (28), respectively. If ui−ui+12/1+ui≤ε, break,endStep 2 Setuk+1=ui+1, if uk−uk+12≤ε, STOP; otherwise let k=:k+1. Go back to step 1.Note: the algorithm is composed of inner and outer loops. The inner loop uses the ADMM algorithm to solve the subproblems, and the outer loop uses the DC programming framework to solve the nonconvex model.


## 3.1. Algorithm (LogTVSCAD)

Step 0 Initialization and date:Input parametersμ,s,γ1,γ2,βw,βz,βx,α,η>0, the tolerance ε>0. Given u0, let k≔0;Step 1 Givenλw0, λz0, λx0 and uk, compute the new iterate by (29);Fori≔0;(29)wi+1=maxw^i2−1βw,0w^iw^i2,zi+1=signz^imaxz^i−μβz,0,xi+1=maxminuk+λxiβx,1,0,ui+1=Ui^−1U˜i,λwi+1=λwi−βwwi+1−Dui+1,λzi+1=λzi−βzzi+1−Kui+1−f,λxi+1=λxi−βxxk+1−ui+1.whereU˜i=DTwi+1−λwi/βw+βz/βwKTzi+1−λzi/βz+βz/βwKTf+βx/βwxi+1−λxi/βx+η/βwui, w^i,z^i,and Ui^ are given by (25), (26) and (28), respectively. If ui−ui+12/1+ui≤ε, break,endStep 2 Setuk+1=ui+1, if uk−uk+12≤ε, STOP; otherwise let k=:k+1. Go back to step 1.Note: the algorithm is composed of inner and outer loops. The inner loop uses the ADMM algorithm to solve the subproblems, and the outer loop uses the DC programming framework to solve the nonconvex model.


## 4. Convergence Analysis

In this section, we analyze the convergence of Algorithm LogTVSCAD. First, we present the following lemma, which is the basis for proving the global convergence.Lemma 1.
For given parameterη>0, the sequence uk generated by Algorithm LogTVSCAD satisfies Fuk−Fuk+1≥ηuk−uk+1.Proof.
By the definitions ofju and hu (28), we obtain(30)juk−juk+1≥∂juk+1,uk−uk+1,huk+1−huk≥∂huk,uk+1−uk.
Following (20), we have(31)μKT∇GγKuk+1−f+ηuk−uk+1∈∂juk+1,DT∇GsDuk+μKT∇GγKuk−f∈∂huk.
We obtain from (30) that Fuk−Fuk+1≥ηuk−uk+1. It is easy to see that Du1/2 and Ku−f1 are semialgebraic from their definitions. We also know from the literature [26, 27] that the penalty functions SCAD and the Log enjoy the KL property. Then, the following result is obtained.Lemma 2.
For given parameterη>0, the function Fu is a KL function.Lemma 3.
Letdk∈∂Fuk, for a sufficiently large constant M>0, and then(32)dk+1≤Muk+1−uk2.Proof.
It follows from (20) that(33)dk+1=DT∇GsDuk+μKT∇GγKuk−f−μKT∇GγKuk+1−f−ηuk−uk+1∈∂Fuk+1.
According to [35], ∇Gs and ∇Gγ are Lipschitz continuous. Moreover, Fuk is sufficiently decreasing from Lemma 1. Thus, we obtain dk+1≤Muk+1−uk2, where M>0 is a sufficiently large constant.Theorem 1.
Let the sequenceuk generated by Algorithm LogTVSCAD, and then it converges to a critical point of (5).Proof.
Sinceuk is bounded, it follows from Lemma 1 that(34)∑k=0∞uk+1−uk2≤Fu0η.
Then,k⟶∞, uk+1−uk⟶0. Let u∗ be any accumulation point of the sequence uk. There exists a subsequence unk such that(35)unk⟶u∗,Funk⟶Fu∗,k⟶∞.
Thus, combining Lemma2 and 3 and Theorem 2.9 [38], the result is right and 0∈∂Fu∗.


## 5. Numerical Results

In this section, we evaluate the performance of the proposed LogTVSCAD method through some numerical experiments. We provide the results of classical TVL1 [9] for reference purpose, and we also compare with TVSCAD nonconvex model [26] and TVLog nonconvex model [27]. All of our test experiments are performed on MATLAB R2015a on the PC with Intel(R) Core(TM) 2.2 GHz CPU and 8.0 GB RAM.To assess the quality of the recovered image, we use the peak signal-to-noise ratio (PSNR) and the signal-to-noise ratio (SNR) as the evaluation indicators, because they are widely used in image processing kinds of literature. The higher PSNR and SNR indicate better quality image; they are defined as follows:(36)PSNR=10lgPxk−x2,SNR=20lgx−x¯xk−x,where P is the size of the image and x and xk denote the original image and restored image, respectively. x¯ is the mean intensity value of x. The structural similarity (SSIM) is another effective evaluation index for comparing the image quality, and the SSIM value closer to the one showing better structure preservation; for more details, please see [39].In our experiments, we choose two images of House (256) and Peppers (512) as the test images, because they are widely used in the academic research work of image processing. The three types of blurs tested are generated by the MATLAB functionfspecial, and they are Gaussian (hsize =9, std =10), motion (len =21, angle =1350), and average (hsize =9) blurs, respectively. Two common types of impulsive noise are tested: salt-and-pepper (SP) and random-valued (RV) noises. It is well known that RV noise is more difficult to remove than SP noise. In the experiments, we tested with various noise levels: 60%, 90% SP noise, and 70% RV noise.The selection of parameters in models and algorithms is an open question. If we use the same parameters, the performances of different methods may have opposite results. The best parameters selection of TVL1, TVSCAD, and TVLog adopts the recommendations of the literatures [26, 27]. For the regularization parameter μ, we swept over(37)μ∈1,5,10,15,20,25,30,50,100,200,400,450,500.The penalty parametersβw,βz,βx of (24) are chosen from(38)βw,βz,βx∈1,1.5,5,10,20,25,50,100,200,1000,2000.In LogTVSCAD model, we set the parametersη=0.001, s=0.002∗k, γ1=0.08/k, and γ2=max0.2∗0.68k−1,0.002 for the SP noise. For RV noise, we set the parameters s=0.02∗k, γ1=0.08/k and γ2=max0.2∗0.85k−1,0.02, where k denotes the number of iterations. From all the tested cases, we can see that the performance of the three nonconvex models (TVSCAD, TVLog, and LogTVSCAD) outperforms that of the TVL1 model. Figures 2–4 show that the three nonconvex models have approximately the same effect at low noise level. In Figures 5–10, we show the restored results by TVL1, TVSCAD, TVLog, and LogTVSCAD on test images House (256) and Peppers (512) with Gaussian blur, Motion blur, and Average blur, and the SP noise level is 90%, respectively. These experimental results show that our LogTVSCAD is slightly better than the other methods at the high noise level. For the 70% RV noise, the higher PSNR values of Figures 11–16 indicate that the LogTVSCAD method is also better than the other three methods. Further, all the test results including CPU time, SNR, and SSIM are listed in Table 1. Through the table, we note that the LogTVSCAD model achieves higher PSNR and SSIM values in various kinds of cases. The experimental results show that the LogTVSCAD model has a better performance for both low and high noise levels. The results in Table 1 also show that the TVL1 model has superiority in CPU time cost.Figure 2
The recovery results (House 256) by Gaussian blur with 60% SP noise. (a) Original. (b) TVL1, PSNR = 24.6266. (c) TVSCAD, PSNR = 30.3598. (d) Gaussian blur (lev = 0.6). (e) TVLog, PSNR = 30.3535. (f) LogTVSCAD, PSNR = 30.3635.
(a)(b)(c)(d)(e)(f)Figure 3
The recovery results (House 256) by Motion blur with 60% SP noise. (a) Original. (b) TVL1, PSNR = 23.739. (c) TVSCAD, PSNR = 29.7605. (d) Motion blur (lev = 0.6). (e) TVLog, PSNR = 29.7593. (f) LogTVSCAD, PSNR = 29.7678.
(a)(b)(c)(d)(e)(f)Figure 4
The recovery results (House 256) by Average blur with 60% SP noise. (a) Original. (b) TVL1, PSNR = 24.5554. (c) TVSCAD, PSNR = 30.4234. (d) Average blur (lev = 0.6). (e) TVLog, PSNR = 30.3966. (f) LogTVSCAD, PSNR = 30.4302.
(a)(b)(c)(d)(e)(f)Figure 5
The recovery results (House 256) by Gaussian blur with 90% SP noise. (a) Original. (b) TVL1, PSNR = 19.0225. (c) TVSCAD, PSNR = 27.0674. (d) Gaussian blur (lev = 0.9). (e) TVLog, PSNR = 24.362. (f) LogTVSCAD, PSNR = 27.2301.
(a)(b)(c)(d)(e)(f)Figure 6
The recovery results (House 256) by Motion blur with 90% SP noise. (a) Original. (b) TVL1, PSNR = 18.7106. (c) TVSCAD, PSNR = 25.0094. (d) Motion blur (lev = 0.9). (e) TVLog, PSNR = 24.2055. (f) LogTVSCAD, PSNR = 25.8013.
(a)(b)(c)(d)(e)(f)Figure 7
The recovery results (House 256) by Average blur with 90% SP noise. (a) Original. (b) TVL1, PSNR = 18.9538. (c) TVSCAD, PSNR = 26.9624. (d) Average blur (lev = 0.9). (e) TVLog, PSNR = 25.0969. (f) LogTVSCAD, PSNR = 27.1779.
(a)(b)(c)(d)(e)(f)Figure 8
The recovery results (Peppers 512) by Gaussian blur with 90% SP noise. (a) Original. (b) TVL1, PSNR = 18.8935. (c) TVSCAD, PSNR = 24.1732. (d) Gaussian blur (lev = 0.9). (e) TVLog, PSNR = 24.7241. (f) LogTVSCAD, PSNR = 25.6992.
(a)(b)(c)(d)(e)(f)Figure 9
The recovery results (Peppers 512) by Motion blur with 90% SP noise. (a) Original. (b) TVL1, PSNR = 19.4711. (c) TVSCAD, PSNR = 25.3923. (d) Motion blur (lev = 0.9). (e) TVLog, PSNR = 25.5329. (f) LogTVSCAD, PSNR = 25.7575.
(a)(b)(c)(d)(e)(f)Figure 10
The recovery results (Peppers 512) by Average blur with 90% SP noise. (a) Original. (b) TVL1, PSNR = 19.2008. (c) TVSCAD, PSNR = 24.3041. (d) Average blur (lev = 0.9). (e) TVLog, PSNR = 25.2903. (f) LogTVSCAD, PSNR = 25.7948.
(a)(b)(c)(d)(e)(f)Figure 11
The recovery results (House 256) by Gaussian blur with 70% RV noise. (a) Original. (b) TVL1, PSNR = 20.9161. (c) TVSCAD, PSNR = 26.9073. (d) Gaussian blur (lev = 0.7). (e) TVLog, PSNR = 27.6482. (f) LogTVSCAD, PSNR = 29.0782.
(a)(b)(c)(d)(e)(f)Figure 12
The recovery results (House 256) by Motion blur with 70% RV noise. (a) Original. (b) TVL1, PSNR = 20.4153. (c) TVSCAD, PSNR = 25.1866. (d) Motion blur (lev = 0.7). (e) TVLog, PSNR = 25.9254. (f) LogTVSCAD, PSNR = 27.1259.
(a)(b)(c)(d)(e)(f)Figure 13
The recovery results (House 256) by Average blur with 70% RV noise. (a) Original. (b) TVL1, PSNR = 20.6806. (c) TVSCAD, PSNR = 26.7422. (d) Average blur (lev = 0.7). (e) TVLog, PSNR = 27.9585. (f) LogTVSCAD, PSNR = 28.5088.
(a)(b)(c)(d)(e)(f)Figure 14
The recovery results (Peppers 512) by Gaussian blur with 70% RV noise. (a) Original. (b) TVL1, PSNR = 19.1797. (c) TVSCAD, PSNR = 26.0516. (d) Gaussian blur (lev = 0.7). (e) TVLog, PSNR = 26.4081. (f) LogTVSCAD, PSNR = 27.9344.
(a)(b)(c)(d)(e)(f)Figure 15
The recovery results (Peppers 512) by Motion blur with 70% RV noise. (a) Original. (b) TVL1, PSNR = 19.1338. (c) TVSCAD, PSNR = 25.9661. (d) Motion blur (lev = 0.7). (e) TVLog, PSNR = 26.1602. (f) LogTVSCAD, PSNR = 26.8849.
(a)(b)(c)(d)(e)(f)Figure 16
The recovery results (Peppers 512) by Average blur with 70% RV noise. (a) Original. (b) TVL1, PSNR = 19.1807. (c) TVSCAD, PSNR = 25.9819. (d) Average blur (lev = 0.7). (e) TVLog, PSNR = 26.4268. (f) LogTVSCAD, PSNR = 27.6354.
(a)(b)(c)(d)(e)(f)Table 1
Performance comparison of different methods for SP and RV.
ImageLevelMethodMotionAverageGaussianCPUSNRSSIMCPUSNRSSIMCPUSNRSSIMHouse0.6 (SP)TVL15.18759.37290.72724.968810.84850.74845.48455.31560.6282TVSCAD46.875615.94180.844247.156316.71650.848951.812516.61460.8465TVLog47.046815.94010.844346.734416.68970.848743.484316.56660.8451LogTVSCAD46.218715.95010.844346.562516.72330.849048.156216.62080.8466House0.9 (SP)TVL15.23443.13350.63155.20315.24700.63465.48455.31560.6282TVSCAD47.656310.85270.764446.140613.25560.794348.937513.3600.7935TVLog46.93759.91730.755145.218711.39000.770945.906310.65510.7661LogTVSCAD47.343711.70850.765544.781213.47100.797244.640613.52320.7958House0.7 (RV)TVL15.29696.70840.66035.64066.97380.67605.10937.20920.6770TVSCAD51.90611.47980.767150.390613.03530.794644.062513.20040.8120TVLog49.7512.21860.797150.859314.25160.819244.046913.20040.8232LogTVSCAD50.218813.41900.802651.093714.80190.821044.390615.37130.8287Peppers0.9 (SP)TVL130.37506.94050.609331.59375.59030.628230.60936.63470.6255TVSCAD146.625012.86170.7461146.171911.22060.7616148.359312.30830.7656TVLog145.578113.00230.7621146.843712.48570.7755148.062512.21560.7723LogTVSCAD147.984313.22690.7674148.546812.96620.7756148.906212.76090.7746Peppers0.7 (RV)TVL129.98436.60320.629030.03126.65010.643330.42186.64910.6453TVSCAD284.87513.43550.7837285.734313.45120.7904286.578113.52090.7924TVLog297.781213.62950.8039304.640613.89620.7925288.906213.87750.7921LogTVSCAD286.343714.35420.8043290.093715.10480.8130287.87515.40380.8136The bold values indicate that the SNR and SSIM results obtained by LogTVSCAD are better than the other three methods.Figure17 shows the convergence curves of the three methods of LogTVSCAD, TVLog, and TVSCAD. For the Peppers image with 90% SP noise, Figures 17(a)–17(c) show the plots of SNR versus iteration number under Gaussian blur, Motion blur, and Average blur, respectively. It is easy to see that the SNR values increase as the number of iterations increases. The curves of LogTVSCAD are slightly higher than those of TVLog and TVSCAD. From Figures 17(d)–17(f), for 70% RV noise, we can see that the proposed LogTVSCAD has better performance than TVSCAD and TVLog. Hence, the proposed new model is effective and desirable.Figure 17
SNR versus iteration number for the Peppers image. (a) Gaussian, 90% SP noise. (b) Motion, 90% SP noise. (c) Average, 90% SP noise. (d) Gaussian, 70% RV noise. (e) Motion, 70% RV noise. (f) Average, 70% RV noise.
(a)(b)(c)(d)(e)(f)In the end, we compare our method to the recent other nonconvex work (called L0TV) [40]. The authors reformulated the L0TV model as an equivalent biconvex mathematical program with equilibrium constraints and then solved it using the proximal ADMM. In the experiments, we chose three other typical images to test, i.e., satellite (256), macaws (512), and boat (1024), which are shown in Figure 18. The blur types still adopt the motion, average, and Gaussian blurs. In the experiments, we tested 60% SP noise, 90% SP noise, and 70% RV noise. In terms of parameter settings, satellite (256) and macaws (512) images adopt the same settings as before. For boat (1024) image, we consider selecting μ=30, and other parameters remain unchanged. The parameters selection of L0TV method adopt the recommendations of the literature [40]. Figures 19–21 show some visual comparisons of image restorations. We observe that the L0TV and LogTVSCAD can restore the blur and noise images well. In Figure 21, the PSNR values of both methods are greater than 30, which shows that both methods are effective for low-level noise. The numerical comparison between the two methods is shown in Table 2. It is obvious that the proposed LogTVSCAD also performs well.Figure 18
Original images. (a) Satellite (256). (b) Macaws (512). (c) Boat (1024).
(a)(b)(c)Figure 19
The recovery results (satellite256×256) by Average blur with 60% SP noise. (a) Average blur (lev = 0.6). (b) L0TV, PSNR = 24.4399. (c) LogTVSCAD, PSNR = 25.5613.
(a)(b)(c)Figure 20
The recovery results (macaws512×512) by Motion blur with 90% SP noise. (a) Motion blur (lev = 0.9). (b) L0TV, PSNR = 27.1022. (c) LogTVSCAD, PSNR = 27.5358.
(a)(b)(c)Figure 21
The recovery results (boat1024×1024) by Gaussian blur with 30% RV noise. (a) Gaussian blur (lev = 0.3). (b) L0TV, PSNR = 31.4702. (c) LogTVSCAD, PSNR = 34.3214.
(a)(b)(c)Table 2
Performance comparison of L0TV and LogTVSACD methods.
ImageLevelMethodMotionAverageGaussianPSNRSNRSSIMPSNRSNRSSIMPSNRSNRSSIMSatellite0.6(SP)L0TV24.439113.28260.862524.439813.28350.865324.400013.24350.8648LogTVSCAD25.736514.58000.906325.561414.40490.898725.430714.27420.8951Macaws0.9(SP)L0TV26.40279.78990.706627.023010.41920.715527.000010.39620.7157LogTVSCAD27.573310.96950.734827.666711.06290.732427.485710.88290.7310Boat0.3(RV)L0TV30.303515.51680.819031.486616.69990.840831.470216.68340.8408LogTVSCAD33.702018.91520.896534.383619.59690.899634.321419.53470.8984


## 6. Conclusions

In this paper, we proposed a new LogTVSCAD model for image restoration with impulsive noise. To solve the nonconvex model, we firstly apply the DCA to reformulate the nonconvex problem and then use the ADMM method to solve the subproblem. The global convergence of the proposed algorithm is proved. The experimental results on recovering images show that the proposed LogTVSCAD model is an effective approach in impulsive noise and is competitive with TVL1, TVSCAD, and TVLog. In future work, we will consider the application of this method in other fields and investigate other nonconvex reconstruction methods.



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