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

School of Electronic Engineering and Automation, Guangxi Key Laboratory of Automatic Detecting Technology and Instruments, Guilin University of Electronic Technology, Guilin 541004, China School of Mathematics and Finance, Hunan University of Humanities, Science and Technology, Loudi 417000, China School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin 541004, China


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]. e mathematical model of image deblurring is usually expressed as where f denotes the observed noisy and blurry image, N imp (·) 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]. e popular model is 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. is TVL2 model is optimal when the measurement noise is Gaussian distributed. However, non-Gaussian noise is more common in practice, and the performance of ℓ 2 -norm based methods may severely degrade. TVL1 model combining TV regularization and ℓ 1 -norm [6][7][8][9] was proposed to deal with impulsive noise (a typical non-Gaussian noise). Its mathematical formulation can be expressed as follows: where ‖u‖ TV is called the TV norm of the variable u, which can take the ℓ 1 -norm and ℓ 2 -norm, i.e., ‖u‖ TV ≔ ‖Du‖ 1 (anisotropic) and ‖u‖ TV ≔ ‖Du‖ 2 (isotropic), 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 primaldual 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], ℓ q -norm (0 < q < 1) [19,20], log-function [21,22], and minimaxconcave penalty (MCP) [23,24].
is 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 ℓ 1 -norm [25][26][27][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: 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]. F log (·) and F scad (·) are defined as where s > 0, c: ≜ (c 1 , c 2 ) > 0 are the threshold parameters. is is a "nonconvex + nonconvex" model, which can have some desirable properties simultaneously. Such penalty of concave functions f s (x) and f c (x) for all elements is nonuniform, which makes f s (x) and f c (x) closer to ℓ 0 -norm than ℓ 1 -norm.
is 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. e core idea is that the Log-function penalty as a regularizer and the SACD-function penalty as a data fidelity term are used. erefore, this model approximates the ℓ 0 -norm more closely than ℓ 1 -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. en, we prove the proposed algorithm that is globally convergent. e rest of this paper is organized as follows. Section 2 gives some notations and preliminaries. e nonconvex LogTVSCAD model and a DCA method with ADMM are shown in Section 3. en, 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.

Notations and Preliminaries
In this section, we first give some notations. en, 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]. 2 Discrete Dynamics in Nature and Society ese conclusions will be used later in the proof of convergence.
For any vector x ∈ R n , x T y or 〈x, y〉 (∀y ∈ R n ) denote their inner product; ‖x‖ q � ( i�1 |x i | q ) 1/q (1 > q > 0) denotes ℓ q -norm; ∇f(x) and zf(x) stand for the gradient and subdifferential of the function f(·) at x, respectively; sign (x) is the signum function. Now, we introduce some properties of the functions f s (x) and f c (x). For fixed s > 0 and c > 0, f s (x) and f c (x) are continuous on R, also increasing, continuously differentiable, and concave on R + , as illustrated in Figure 1. Next, we look at another functions g s (x) and g c (x), which are induced by f s (x) and f c (x): where f s (x) and f c (x) are given by (7). Without loss of generality, we consider the multivariate generalization of the functions g s (x) and g c (x): In fact, the functions G s (v) and G c (v) are continuously differentiable and convex on R n , and then the log function penalty and the SCAD function penalty can be expressed as Note that the functions (10) play an important role in the later algorithm construction. e following useful definitions and properties can be obtained from the literature [34][35][36].

Definition 2.
For an extended-real-valued, proper and lower semicontinuous function f(x), its subdifferential zf at where If f is convex function, then the subdifferential is such that Definition 3. Let f(x) be a proper and lower semicontinuous function; the proximity operator is defined as where η(η > 0) is a penalty parameter.
It is well known that the proximal operator is particularly useful in convex optimization.
is named a KL function if it has the KL property at all points in domzf.

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: i.e., Because Log and SCAD penalty functions are nonconvex, this model is nonconvex. To address this nonconvex model, set j(u) � ‖u‖ TV + μ‖Ku − f‖ 1 , h(u) � G s (Du)+ μG c (Ku − f), and problem (15) can be expressed as the difference of the convex functions j(u) and h(u), i.e., is 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 where v k ∈zh(u k ), 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, where η > 0 is a given proximal parameter. For (15), the next iteration can be expressed as By omitting the constants of formula (19), we have For the above model (20), under certain conditions, its objective function is strongly concave, and u k+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 w � Du, en, we rewrite (20) as a constrained minimization problem: min u,w,z,x Let the augmented Lagrangian function of model (22) be where λ w , λ z , and λ x are Lagrange multipliers and β w > 0, β z > 0, and β x > 0 are penalty parameters. Given and u ≔ u 0 , according to the classical ADMM, the iterative scheme of the problem (23) can be expressed as follows: In this scheme, the ADMM method is directly applied to 2− blocks of variables (w, z, x) and u. Furthermore, via eorem 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 w i+1 and z i+1 . In addition, x i+1 can be computed by a simple projection onto box Ω. Hence, the w, z, x optimizations have a closed-form solution as where sign(·) is the signum function. en, we consider how to solve the u-subproblem of (24). Via the first-order optimality conditions, the corresponding normal equation is where U i � D T D + (β z /β w )K T K + ((β x + η)/(β w ))I 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).

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 parameter η > 0, the sequence u k generated by Algorithm LogTVSCAD satisfies Proof. By the definitions of j(u) and h(u) (28), we obtain  Discrete Dynamics in Nature and Society 13 14 Discrete Dynamics in Nature and Society Following (20), we have We obtain from (30) that F(u k ) − F(u k+1 ) ≥ η‖u k − u k+1 ‖. It is easy to see that ‖Du‖ 1/2 and ‖Ku − f‖ 1 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. en, the following result is obtained.
Proof. It follows from (20) that According to [35], ∇G s and ∇G c are Lipschitz continuous. Moreover, F(u k ) is sufficiently decreasing from Lemma 1. us, we obtain ‖d k+1 ‖ ≤ M‖u k+1 − u k ‖ 2 , where M > 0 is a sufficiently large constant. □ Theorem 1. Let the sequence u k generated by Algorithm LogTVSCAD, and then it converges to a critical point of (5).
Proof. Since u k is bounded, it follows from Lemma 1 that en, k ⟶ ∞, (u k+1 − u k ) ⟶ 0. Let u * be any accumulation point of the sequence u k .
ere exists a subsequence u n k such that us, combining Lemma 2 and 3 and eorem 2.9 [38], the result is right and 0 ∈zF(u * ).

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. e higher PSNR and SNR indicate better quality image; they are defined as follows: where P is the size of the image and x and x k denote the original image and restored image, respectively. x is the mean intensity value of x. e 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. e three types of blurs tested are generated by the MATLAB function fspecial, and they are Gaussian (hsize � 9, std � 10), motion (len � 21, angle � 135 0 ), and average (hsize � 9) blurs, respectively. Two common types of impulsive noise are tested: salt-and-pepper (SP) and randomvalued (RV) noises. It is well known that RV noise is more In LogTVSCAD model, we set the parameters η � 0.001, s � 0.002 * k, c 1 � 0.08/k, and c 2 � max(0.2 * 0.68 k− 1 , 0.002) for the SP noise. For RV noise, we set the parameters s � 0.02 * k, c 1 � 0.08/k and c 2 � max(0.2 * 0.85 k− 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 Figure 17 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.
e 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.
In the end, we compare our method to the recent other nonconvex work (called L0TV) [40]. e 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. e 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. e 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.
e numerical comparison between the two methods is shown in Table 2. It is obvious that the proposed LogTVSCAD also performs well.

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. e global convergence of the proposed algorithm is proved. e 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.

Data Availability
e data used to support the findings of this study are available from the corresponding authors upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.