Improved RPCA method via non‐convex regularisation for image denoising

IET Signal ProcessingVolume 14, Issue 5 p. 269-277 Research ArticleFree Access Improved RPCA method via non-convex regularisation for image denoising Sijie Wang, Sijie Wang orcid.org/0000-0002-2912-0773 School of Electronics and Information Engineering, Hebei University of Technology, Tianjin, People's Republic of ChinaSearch for more papers by this authorKewen Xia, Corresponding Author Kewen Xia kwxia@hebut.edu.cn School of Electronics and Information Engineering, Hebei University of Technology, Tianjin, People's Republic of ChinaSearch for more papers by this authorLi Wang, Li Wang School of Electronics and Information Engineering, Hebei University of Technology, Tianjin, People's Republic of ChinaSearch for more papers by this authorJiangnan Zhang, Jiangnan Zhang School of Electronics and Information Engineering, Hebei University of Technology, Tianjin, People's Republic of ChinaSearch for more papers by this authorHuaijin Yang, Huaijin Yang School of Electronics and Information Engineering, Hebei University of Technology, Tianjin, People's Republic of ChinaSearch for more papers by this author Sijie Wang, Sijie Wang orcid.org/0000-0002-2912-0773 School of Electronics and Information Engineering, Hebei University of Technology, Tianjin, People's Republic of ChinaSearch for more papers by this authorKewen Xia, Corresponding Author Kewen Xia kwxia@hebut.edu.cn School of Electronics and Information Engineering, Hebei University of Technology, Tianjin, People's Republic of ChinaSearch for more papers by this authorLi Wang, Li Wang School of Electronics and Information Engineering, Hebei University of Technology, Tianjin, People's Republic of ChinaSearch for more papers by this authorJiangnan Zhang, Jiangnan Zhang School of Electronics and Information Engineering, Hebei University of Technology, Tianjin, People's Republic of ChinaSearch for more papers by this authorHuaijin Yang, Huaijin Yang School of Electronics and Information Engineering, Hebei University of Technology, Tianjin, People's Republic of ChinaSearch for more papers by this author First published: 01 July 2020 https://doi.org/10.1049/iet-spr.2019.0365Citations: 12AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract The traditional robust principal component analysis (RPCA) model is based on the nuclear norm, which usually underestimates the singular values of the low-rank matrix. As a consequence, the restoration image experiences serious interference by Gaussian noise, and the image quality degenerates during the denoising process. Therefore, an improved RPCA method via non-convex regularisation terms is proposed to remedy the above shortcomings. First, in order to estimate the singular value of the low-rank matrix more accurately, the authors employ the non-convex penalty function and add a weight vector to it. Then, the regularisation with non-convex penalty function and its weighted version are used to replace the nuclear norm and entry-wise norm in original RPCA, respectively, to establish an improved model. Finally, an optimal solution algorithm is derived by developing the alternating direction multiplier method. Experimental results show that the proposed method has better performance in terms of both quantitative measurement and visual perception quality than other several state-of-the-art image denoising methods. 1 Introduction Restoring a clean image from its noisy version is a hot research issue in the field of low-level computer vision because the noise can destroy important details in the image and reduce its quality. Image denoising is a broad research subject in the past few decades [1-20]. The extensively studied image denoising algorithms can be divided into two main categories: sparse representation and low-rank matrix recovery (LRMR). Sparse representation [1, 4, 12, 16-19, 21, 22] methods usually use the sparse linear combinations of atoms to restore an image. However, learning a dictionary containing these atoms requires a large number of computation and operation time. To avoid this issue, the robust principal component analysis (RPCA) [23-25], a primary model for LRMR, has recently been applied to image denoising and made remarkable improvements. Specifically, we express the image denoising problem as (1)where represents an original noisy image. L notes a clean image obtained after denoising. S is a sparse noise, mainly including impulse noise, deadline, band noise, and mixed noise. And N is the dense Gaussian noise with zero-mean. The purpose of RPCA is to estimate the low-rank matrix from its observability matrix by resolving the optimisation problem formulated as (2)where notes the nuclear norm, used as a low-rank regularisation term. notes the entry-wise norm to induce sparsity of sparse matrix. notes Frobenius norm, and are trade-off parameters to balance these three norm terms to achieve data fidelity. In (2), the nuclear norm is used to approximate the minimum rank of a low-rank matrix, also known as nuclear norm minimisation (NNM). Cai et al. [26] proved that the nuclear norm could effectually recover the low-rank matrix under certain conditions, and a singular value thresholding (SVT) model is proposed to solve NNM. Thanks to this pioneering work, a large number of improved models and optimisation algorithms have been put forward for RPCA. Furthermore, with the increase of attention to RPCA, it has been applied to images reconstruction, video foreground/background separation, and to data prediction [27-30]. Although the nuclear norm is the most popular low-rank regularisation term and an effective convex relaxation method for the original rank minimisation problem, it is not the tightest convex relaxation of the rank. The nuclear norm always underestimates the rank of a matrix, making the restoration performance sub-optimal. Recent studies have shown that a non-convex regulariser is capable of evaluating the singular values of the matrix more exactly, including -norm [31] and truncated nuclear norm [32]. Parekh and Selesnick [33] proposed using a non-convex surrogate function to estimate low-rank matrixes. And showed how to set correlation parameters to ensure the objective function is strictly convex. Subsequently, Parekh and Selesnick [34] further proposed employing the non-convex penalty function as the low-rank and sparsity-inducing regularisers simultaneously. With the increasing attention, more and more theories and applications have proved the superiority of non-convex optimisation [15, 17, 31, 35-40]. In addition, the traditional nuclear norm regularises each singular value equally, giving the same amount of contraction for different singular values. Such treatment violates the physical meaning of the singular value and weakens its ability of image denoising. In order to improve the flexibility and feasibility of the nuclear norm, Zhang et al. [41] proposed a truncated nuclear norm regularisation (TNNR) method. Assume that the number of singular values and ranks are N and r, respectively. TNNR minimises N - r singular values while keeping r singular values constant. Although TNNR improves the accuracy of matrix rank estimation, r is difficult to calculate, which limits the application of TNNR. In addition, the texture structure and features of an image have strong similarity, which is called non-local self-similarity [13] (NSS). Gu et al. [14] segmented the whole image into several non-local image patches and adopted the idea of block-matching to group these patches. Wang et al. [42] proved that the matrix aggregated by non-local similar patches has a strong low-rank characteristic. On the basis of making full use of NSS to group image patches, Gu et al. [14] proposed the weighted nuclear norm minimisation (WNNM). Compared with TNNR, WNNM assigns different weights to each singular value, which leads to an estimation of matrix rank more accurately. Therefore, WNNM has enough flexibility to deal with a variety of practical problems. Inspired by the non-convex penalty function and WNNM, we formulate a weight vector to the non-convex penalty function, and then we introduce the non-convex penalty function and its weighted version to RPCA-based image denoising model. The main contributions of this paper are summarised as follows: (i) We propose a novel weighted non-convex penalty function for low-rank induced regularisation. It is capable of handling different singular value components discriminatively, so as to achieve a better low-rank approximation. (ii) We replace the entry-wise norm with a non-convex penalty function as the sparsity induced regularisation. (iii) An improved RPCA model (the proposed model is abbreviated as GRPCA, because we use to represent the non-convex regularisation terms) is formulated for low-rank matrix recovery. (iv) An algorithm on the strength of alternating direction method of multipliers (ADMMs) is developed to solve GRPCA, and its convergence analysis has been provided. (v) Experiments on synthetic data and lung CT images prove that GRPCA outperforms several state-of-the-art image denoising methods in terms of both quantitative measure and visual perception quality. The remaining of this paper is organised as follows. Section 2 introduces the non-convex penalty functions and their proximity operators. In Section 3, the weight vector is added to the non-convex penalty function and a novel GRPCA model is formulated, whilst Section 4 develops an iterative algorithm based on the ADMM framework to solve GRPCA and provides some theorems to demonstrate the convergence of the algorithm. Section 5 represents numerical and real CT images of experimental results. Finally, this paper ends with a brief concluding remarks in Section 6. 2 Preliminary 2.1 Notations We use upper and lower case letters to denote matrices and vectors, respectively. For a matrix , denotes the largest i th singular value. And denotes the ordered singular values matrices. The nuclear norm of is defined as . The norm of is defined as . The Frobenius norm of is defined as . 2.2 Non-convex penalty functions In order to compensate for the deviation of the nuclear norm approach, we employ non-convex penalty functions parameterised by the parameter [21, 43, 44] to estimate the minimum rank of a matrix. Below, we provide the definition of such non-convex penalty functions. Assumption 1.The non-convex penalty function satisfies the following criteria [34]: (i) is continuous on . (ii) is twice differentiable on . (iii) (iv) . (v) . (vi) . (vii) . (viii) Examples of non-convex penalty functions [17, 45, 46] satisfying Assumption 1 are the logarithmic penalty function (3)the rational penalty function (4)the arctangent penalty function, (5)and the partly quadratic penalty function (6)In particular, while , these four penalty functions are approximated as the absolute value function, i.e. . They are illustrated in Fig. 1 with a different parameter a. Fig. 1Open in figure viewerPowerPoint Penalty functions satisfying Assumption 1 (a) , (b) , (c) 2.3 Proximity operators The mathematical model for recovering the low-rank matrix from a noisy matrix is (7)where W is additive white Gaussian noise (AWGN). Then, the low-rank matrix approximation problem with non-convex penalty function is defined as [33] (8)where , and the second term is low-rank regulariser. is the i th singular value of matrix , is the non-convex penalty function satisfying Assumption 1. Definition 1.Let be a non-convex penalty function satisfying Assumption 1. The proximal operator of , , is defined as [34, 47, 48]: (9) If a parameter a in the above formula satisfies , then is a non-linear continuous threshold function with as the threshold, i.e. (10)Let be the singular value decomposition (SVD) of the matrix , then the global minimiser of (8) can be defined as (11)Note that as , the problem (8) becomes the standard NNM problem. In other words, NNM is actually a special case of (8). And the solution to the NNM problem is SVT, i.e. (12)where represents the soft-threshold function, which equally cuts off the singular values of . It can be found that the soft-threshold function is the case as with . 3 Formulation of the proposed method 3.1 Weighted non-convex penalty function For convenience, we rewrite the non-convex penalty function (shown in (8)) as a regularisation term used to approach singular values of the low-rank matrix (see (1) and (2)) as follows: (13)However, the function is essentially a threshold function that gives the same threshold processing for different singular values of the matrix . According to the physical meaning of singular values, the larger singular value corresponds to the main information of the image, and the similarity with the image is higher, so a smaller shrinkage threshold should be given. While a smaller singular value should be processed with a larger threshold. Inspired by WNNM, we adopt weighting factor to balance the threshold . is defined as the reciprocal of the singular value of matrix , i.e. (14)where is the constant that prevents the denominator from being zero. Then (13) can be represented in a weighted form as follows: (15)According to Section 2.3, it can be further deduced that the solution of is formulated as (16)where is the singular value matrix of which is shown in (1). And is a threshold function with a variable threshold . 3.2 Improved RPCA model The weighted non-convex penalty function performs matrix recovery by calculating singular values, and has a specific solution formula. Hence, the can supersede the nuclear norm as a regulariser in the RPCA model (2). On the other hand, it is observed that (2) uses a well-known norm to induce sparsity of the matrix . In fact, the nuclear norm has the following relationship with norm: . So we also employ the non-convex penalty function to approach non-zero values of the sparse matrix , expressed as (17)where notes the entry in row i and column j of a matrix . Then the global minimiser solution of (17) can be defined as (18)where is a threshold function with a threshold . Considering the NSS characteristics of the observed image itself, performing a block-matching operation on the entire image is advantageous for ensuring the low-rank of image. Specifically, we set the search window with size , the current image patch size is set as , and the Euclidean distance is used as the matching criterion. Many image patches similar to the current patch can be found in the search window, and then these image patches can form patch groups, so that we can segment the whole image into multiple groups of patches. On the basis of guaranteeing strong low-rank, we propose the following improved RPCA model with non-convex penalty function (abbreviated as GRPCA): (19) 4 Algorithm for GRPCA In this section, we use ADMM to obtain an algorithm for the solution to the GRPCA model. Hong et al. [37] provided certain lemmas and assumptions to guarantee that the ADMM framework can be adopted to solve the non-convex optimisation problem, and the solution is able to converge to a stationary point iteratively. 4.1 Updating rules By enforcing the constraint with the Lagrange multiplier , (19) is converted into an unconstrained optimisation problem: (20)where is the scalar augmented Lagrangian parameter. Based on variable separation and iterative rules of ADMM [49, 50], formula (19) can be divided into the following three sub-problems for optimal solution: (21)For term : This is a minimisation problem of the entry-wise non-convex penalty function, which can be effectively solved through the proximal operator, expressed as (22)where represents a threshold function with a threshold value . And it is applied to the entry of the matrix . is a parameter satisfying For term : It is essentially a problem of minimising the weighted non-convex penalty function. Specifically, we first compute the SVD of the matrix as follows: (23)Next, we employ a weight vector to give different degrees of contraction for every singular value, it can be defined as (24)Then, using a proximity operator , we get the reconstructed matrix : (25)where represents a threshold function with a threshold value . And is a parameter satisfying For term : The solution of the Gaussian noise matrix is actually a square optimisation problem, we indicate it as , i.e. (26)Then, we calculate the partial derivative of and let the result of the equation be zero, i.e. (27)Thus, the closed solution is derived as follows: (28)Next, the Lagrangian multiplier needs to be iteratively updated as follows: (29)where is the iterative step-size. It determines the speed of iteration, but has no effect on the performance of the algorithm itself. The solution of the GRPCA model (see (19)) can be derived by alternately solving each of the above sub-problems and iterating until the maximum iteration is reached or the stop condition is satisfied. Our proposed algorithm is summarised in Fig. 2. Fig. 2Open in figure viewerPowerPoint Proposed algorithm for GRPCA 4.2 Convergence analysis The convergence of ADMM to the global minimum is guaranteed when the objective function is the sum of several convex functions. The following theorems ensure that the objective function (20) is strictly convex. In particular, we show the designate region of , and the condition on the value of . Theorem 1.Let be a parameterised non-convex penalty function satisfying Assumption 1. The function defined as [34]: (30)is strictly convex if (31)It is well known the convexity of Frobenius norm. If and satisfy (31), function (20) is strictly convex because of being a sum of three convex functions. Theorem 2.When the parameters satisfy the conditions: and , the ADMM has convergence [17]. In Algorithm 1, we set and to satisfy the above theorem, the algorithm can reach iterative convergence. In addition, we set the values of and to be positively correlated with AWGN to facilitate the improvement of the signal-to-noise ratio of the proposed algorithm, specifically expressed as: and . For GRPCA model, describes the contribution degree of the function . We set it as based on the conclusion of [24]. constrained Gaussian noise term can be expressed equivalently as . Therefore, we believe that is closely related to the variance of Gaussian noise, so we set it as . 5 Experimental results In this section, we compared the performance of RPCA and GRPCA models on singular value estimation using a synthetic data test, and then apply the proposed GRPCA model and its solving algorithm to lung CT image denoising. The experimental analysis is done on simulation software Matlab R2015a, and the operating environment is the Windows10 64-bit operating system platform with 4.00 GB RAM and Intel Core CPU 2.40 GHz. 5.1 Synthetic data test We generate a synthetic matrix of rank r, such that (32)where and are random matrices with the i.i.d normal distribution. Let be a matrix with rank . We add the AWGN to the matrix to create the noisy matrices, and the noise standard deviations are set as 5, 10, and 40, respectively. Under different intensities of noise, RPCA and GRPCA are adopted to estimate singular values of noisy matrices, and the experimental results are shown in Fig. 3. Fig. 3Open in figure viewerPowerPoint Singular value estimation results with different s (a) , (b) , (c) It can be seen from Fig. 3, under the influence of low noise intensity, the estimation results of the number of singular values by RPCA and GRPCA are both close to the real quantity, but the latter is more accurate. In the degree of numerical approximation of singular value, the estimated result of RPCA is about 100 less than the true value on average. While the estimated result of GRPCA is always close to the true value, showing an obvious accuracy advantage. With increased noise intensity, there is a large error in the estimation of the number of singular values by RPCA. When the noise intensity is as , the number of singular values estimated by RPCA is only half of the real quantity, the difference between the estimated results of singular value and true value gets larger and larger, and the difference increases to about 160. In comparison, the approximation performance of GRPCA is more stable. Under the influence of different noise intensities, GRPCA is always closer to the true value, whether it is the estimation of singular values or numerical size of values. In summary, the results of the synthetic data test show that RPCA has a serious problem of insufficient estimation of singular values; the proposed GRPCA is able to overcome this disadvantage. The experimental results prove that GRPCA performs better robustness and accuracy. 5.2 Lung CT image denoising In this subsection, we demonstrate the excellent performance of proposed GRPCA through a medical CT image denoising experiment. The experimental images are obtained from an international public lung CT image database LIDC/IDRI. The LIDC database contains a total of 1018 patient cases, each of which stores the lung CT images of each patient (image size is ), and an XML comment file containing lung nodule information. The annotation information was diagnosed by four radiologists with extensive experience in CT cases and provides diagnostic information and evaluation information of the medical signs of the lung nodule [51, 52]. The specific experimental processing is shown in Fig. 4. To facilitate the observation and analysis of the denoising performance, we superimpose the AWGN with noise intensities of 10, 20, 30, 40, 50, and 60 on the original lung CT image. Then, the experimental results are compared with the state-of-the-art image denoising methods, such as RPCA [23], K-SVD [1], BCD [40], BM3D [2], GHP [6], WNNM [14], WSNM [53], PGPD [54] and ISLR [34]. Peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) are used to quantitatively measure denoising images of different methods. Fig. 4Open in figure viewerPowerPoint Lung CT image denoising experiment process The size of the original lung CT image is . When the noise intensity is , we set the search window size to . And for the rest of the situation of noise intensities, we set the search windows size to , which can speed up the operation. The size of the image patch is set to , , and for , , and , respectively. For the sake of fairness, other denoising methods used for comparison all take the parameters that have been set in source codes. Figs. 5 and 6 show the original input images and noisy images, respectively. Figs. 7-16 show the image denoising results of different methods under . The PSNR and SSIM values under various noise intensities are shown in Table 1. Fig. 5Open in figure viewerPowerPoint Original input images (a) Original image, (b) Partially enlarged image Fig. 6Open in figure viewerPowerPoint Noisy images (a) Noisy image, (b) Partially enlarged image Fig. 7Open in figure viewerPowerPoint Denoising results by RPCA (a) Denoising image, (b) Partially enlarged image Fig. 8Open in figure viewerPowerPoint Denoising results by K-SVD (a) Denoising image, (b) Partially enlarged image Fig. 9Open in figure viewerPowerPoint Denoising results by BCD (a) Denoising image, (b) Partially enlarged image Fig. 10Open in figure viewerPowerPoint Denoising results by BM3D (a) Denoising image, (b) Partially enlarged image Fig. 11Open in figure viewerPowerPoint Denoising results by GHP (a) Denoising image, (b) Partially enlarged image Fig. 12Open in figure viewerPowerPoint Denoising results by WNNM (a) Denoising image, (b) Partially enlarged image Fig. 13Open in figure viewerPowerPoint Denoising results by WSNM (a) Denoising image, (b) Partially enlarged image Fig. 14Open in figure viewerPowerPoint Denoising results by PGPD (a) Denoising image, (b) Partially enlarged image Fig. 15Open in figure viewerPowerPoint Denoising results by ISLR (a) Denoising image, (b) Partially enlarged image Fig. 16Open in figure viewerPowerPoint Denoising results by GRPCA (a) Denoising image, (b) Partially enlarged image Table 1. PSNR (dB) and SSIM by different methods RPCA PSNR 30.2359 26.1362 23.6078 22.5431 21.5289 20.5623 SSIM 0.9543 0.9103 0.9061 0.8792 0.8652 0.8465 K-SVD PSNR 32.0636 28.1406 25.9628 24.4266 23.2258 22.2615 SSIM 0.9676 0.9314 0.9059 0.8820 0.8591 0.8372 BCD PSNR 32.2415 28.9728 27.0573 25.5561 23.9995 23.4028 SSIM 0.9612 0.9259 0.9157 0.9055 0.8725 0.8575 BM3D PSNR 32.6954 29.0607 27.2933 26.0547 24.8785 24.0944 SSIM 0.9694 0.9353 0.9214 0.9109 0.8965 0.8893 GHP PSNR 32.7003 28.9057 27.0649 26.0825 24.9824 24.0941 SSIM 0.9692 0.9351 0.9214 0.9111 0.8990 0.8891 WNNM PSNR 32.7053 29.0557 27.1684 26.1191 25.0336 24.1615 SSIM 0.9709 0.9367 0.9226 0.9122 0.8993 0.8895 WSNM PSNR 32.7042 29.0952 27.1271 26.1211 25.0809 26.1647 SSIM 0.9713 0.9374 0.9221 0.9120 0.8998 0.8890 PGPD PSNR 32.7122 29.1580 27.2489 26.1896 25.0446 24.1764 SSIM 0.9714 0.9371 0.9229 0.9126 0.8995 0.8909 ISLR PSNR 32.7108 29.0648 27.2958 26.0561 24.8799 24.0970 SSIM 0.9706 0.9364 0.9220 0.9115 0.8988 0.8902 GRPCA PSNR 32.7326 29.2223 27.4309 26.2035 25.1127 24.2845 SSIM 0.9722 0.9392 0.9245 0.9133 0.9004 0.8918 From Figs. 5-16, it can be seen that RPCA, K-SVD, BCD, BM3D, GHP, and WNNM have weak removal effect on dense Gaussian noise, and there is some noise interference in the background after denoising. In the process of image restoration, the noise is confused with lung tissue structure, and the noise is mistakenly restored to the main information part of the image. Meanwhile, after denoising by WSNM, PGPD, and ISLR, the marginal information of lung tissue is slightly blurred, and there is a certain degree of ‘oil painting’ phenomenon. In contrast, denoising by GRPCA, the useful information of an image is significantly distinguished from the background. GRPCA effectively removes the noise information, completely retains the lung parenchyma, and the recovery effect of the edge portion is strong. Through subjective visual evaluations, it can be determined that GRPCA outperforms other methods. Next, we do a quantitative analysis. Compared with the RPCA, our proposed GRPCA has an average increase of 3.42 dB and 0.301 in terms of PSNR and SSIM, with increasing rates of 14.61 and 3.397%, respectively. With increasing noise intensity, the denoising effect of RPCA gradually deteriorates, and the performance under high noise intensity is not satisfactory. Compared with RPCA, the PSNR and SSIM values of other methods are improved to some extent, but the effect of GRPCA is still better than these methods. We can conclude that under the objective evaluation, the proposed GRPCA also has the best denoising performance. 6 Conclusion In this paper, the non-convex penalty function is weighted, and the nuclear norm and entry-wise norm in standard RPCA model are replaced by weighted non-convex function and non-convex function as the low-rank regularisation and sparsity regularisation, respectively. A novel GRPCA model is established, and an iterative algorithm for the proposed model is derived by developing the ADMM framework. The synthetic data experiment proves that the proposed GRPCA has a more accurate non-zero singular value estimation ability than RPCA. Especially under high noise intensity, GRPCA performs better robustness. Experimental results of lung CT image denoising show the advantages of the GRPCA method more intuitively. The restored image quality is excellent, the edge structure is clear, and the objective evaluations are superior to other state-of-the-art denoising methods. 7 Acknowledgments This work was supported by the National Natural Science Foundation of China (grant no. U1813222), Tianjin Natural Science Foundation (grant no. 18JCYBJC16500) and Key Research and Development Project from Hebei Province (grant no. 19210404D). 8 References 1Aharon, M., Elad, M., Bruckstein, A.: ‘K-SVD: an algorithm for designing overcomplete dictionaries for sparse representation’, IEEE Trans. Signal Process., 2006, 54, (11), pp. 4311– 4322 2Dabov, K., Foi, A., Katkovnik, V. et al: ‘Image denoising by sparse 3-D transform-domain collaborative filtering’, IEEE Trans Image Process, 2007, 16, (8), pp. 2080– 2095 3Beck, A., Teboulle, M.: ‘Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems’, IEEE Trans Image Process, 2009, 18, (11), pp. 2419– 2434 4Dong, W., Li, X., Zhang, L. et al: ‘Sparsity-based image denoising via dictionary learning and structural clustering’. IEEE Conf. on Computer Vision and Pattern Recognition, Colorado Springs, CO, USA, 2011, pp. 457– 464 5Wang, Y., Guo, J., Chen, W. et al: ‘Image denoising using modified Perona–Malik model based on directional Laplacian’, Signal Process., 2013, 93, (9), pp. 2548– 2558 6Zuo, W., Zhang, L., Song, C. et al: ‘Gradient histogram estimation and preservation for texture enhanced image denoising’, IEEE Trans Image Process, 2014, 23, (6), pp. 2459– 2472 7Xie, C., Chang, J., Xu, W.: ‘Medical image denoising by generalised Gaussian mixture modelling with edge information’, IET Image Process., 2014, 8, (8), pp. 464– 476 8Sethunadh, R., Thomas, T.: ‘Spatially adaptive image denoising using inter-scale dependence in directionlet domain’, IET Image Process., 2015, 9, (2), pp. 142– 152 9Wang, H., Cen, Y., He, Z. et al: ‘Reweighted low-rank matrix analysis with structural smoothness for image denoising’, IEEE Trans Image Process, 2017, 27, (4), pp. 1777– 1792 10Liu, Z., Yu, L., Sun, H.: ‘Image denoising via nonlocal low rank approximation with local structure preserving’, IEEE Access, 2019, 7, (1), pp. 7117– 7132 11Chen, G., Wang, J., Zhang, F. et al: ‘Image denoising in impulsive noise via weighted Schatten p-norm regularization’, J. Electron. Imaging, 2019, 28, (1), pp. 1– 10 12Lyu, Q., Guo, M., Ma, M. et al: ‘External prior learning and internal mean sparse coding for image denoising’, J. Electron. Imaging, 2019, 28, (3), pp. 1– 10 13Wu, Z., Wang, Q., Wu, Z. et al: ‘Total variation-regularized weighted nuclear norm minimization for hyperspectral image mixed denoising’, J. Electron. Imaging, 2016, 25, (1), pp. 1– 10 14Gu, S., Zhang, L., Zuo, W. et al: ‘Weighted nuclear norm minimization with application to image denoising’. IEEE Conf. on Computer Vision and Pattern Recognition, Columbus, OH, USA, 2014, pp. 2862– 2869 15Li, C., Li, Y., Zhao, Z. et al: ‘A mixed noise removal algorithm based on multi-fidelity modeling with nonsmooth and nonconvex regularization’, Multimedia Tools Appl., 2019, 76, (1), pp. 1– 24 16Elad, M., Aharon, M.: ‘Image denoising via sparse and redundant representations over learned dictionaries’, IEEE Trans Image Process, 2006, 15, (12), pp. 3736– 3745 17Hong, M., Luo, Z. Q.: ‘On the linear convergence of the alternating direction method of multipliers’, Math. Program., 2017, 162, (1–2), pp. 165– 199 18Zarmehi, N., Marvasti, F.: ‘Removal of sparse noise from sparse signals’, Signal Process., 2019, 158, (1), pp. 91– 99 19Liu, J., Osher, S.: ‘Block matching local SVD operator based sparsity and TV regularization for image denoising’, J. Sci. Comput., 2019, 78, (1), pp. 607– 624 20Çalışkan, A., Çevik, U.: ‘An efficient noisy pixels detection model for CT images using extreme learning machines’, Teh. Vjesn., 2018, 25, (3), pp. 679– 686 21Selesnick, I. W., Bayram, I.: ‘Sparse signal estimation by maximally sparse convex optimization’, IEEE Trans. Signal Process., 2014, 62, (5), pp. 1078– 1092 22Sun, Y., Chen, H., Tao, J.: ‘Sparse signal recovery via minimax-concave penalty and ℓ1-norm loss function’, IET Signal Process., 2018, 12, (9), pp. 1091– 1098 23Wright, J., Ganesh, A., Rao, S. et al: ‘Robust principal component analysis: exact recovery of corrupted low-rank matrices via convex optimization’. Neural Inf. Process. Syst., Vancouver, Canada, 2009, pp. 2080– 2088 24Candès, E. J., Li, X., Ma, Y. et al: ‘Robust principal component analysis?’, J. ACM, 2011, 58, (3), pp. 1– 11 25Shi, J., Zheng, X., Wei, Z. et al: ‘Survey on algorithms of low-rank matrix recovery’, Comput. Appl. Res., 2013, 30, (6), pp. 1601– 1605 26Cai, J. F., Candès, E. J., Shen, Z.: ‘A singular value thresholding algorithm for matrix completion’, SIAM J. Optim., 2010, 20, (4), pp. 1956– 1982 27Bouwmans, T., Sobral, A., Javed, S. et al: ‘Decomposition into low-rank plus additive matrices for background/foreground separation: A review for a comparative evaluation with a large-scale dataset’, Comput. Sci. Rev., 2017, 23, (1), pp. 1– 71 28Bouwmans, T., Javed, S., Zhang, H. et al: ‘On the applications of robust PCA in image and video processing’, Proc. IEEE, 2018, 106, (8), pp. 1427– 1457 29Yi, X., Park, D., Chen, Y. et al: ‘Fast algorithms for robust PCA via gradient descent’. Neural Inf. Process. Syst., Barcelona, Spain, 2016, pp. 4152– 4160 30Succurro, M., Arcuri, G., Costanzo, G.D.: ‘A combined approach based on robust PCA to improve bankruptcy forecasting’, Rev. Account. Finance, 2019, 18, (2), pp. 296– 320 31Quach, K. G., Duong, C. N., Luu, K. et al: ‘Non-convex online robust PCA: enhance sparsity via ℓp-norm minimization’, Comput. Vis. Image Underst., 2017, 158, (1), pp. 126– 140 32Hu, Y., Zhang, D., Ye, J. et al: ‘Fast and accurate matrix completion via truncated nuclear norm regularization’, IEEE Trans Pattern Anal Mach Intell, 2012, 35, (9), pp. 2117– 2130 33Parekh, A., Selesnick, I. W.: ‘Enhanced low-rank matrix approximation’, IEEE Signal Process. Lett., 2016, 23, (4), pp. 493– 497 34Parekh, A., Selesnick, I. W.: ‘Improved sparse low-rank matrix estimation’, Signal Process., 2017, 139, (1), pp. 62– 69 35Kopriva, I., Shi, F., Chen, X.: ‘Enhanced low-rank + sparsity decomposition for speckle reduction in optical coherence tomography’, J. Biomed. Opt., 2016, 21, (7), pp. 1– 10 36Zhu, Z., Li, Q., Tang, G. et al: ‘Global optimality in low-rank matrix optimization’, IEEE Trans. Signal Process., 2018, 66, (13), pp. 3614– 3628 37Hong, M., Luo, Z. Q., Razaviyayn, M.: ‘Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems’, SIAM J. Optim., 2016, 26, (1), pp. 337– 364 38Chen, W.: ‘Simultaneously sparse and low-rank matrix reconstruction via nonconvex and nonseparable regularization’, IEEE Trans. Signal Process., 2018, 66, (20), pp. 5313– 5323 39Wen, F., Ying, R., Liu, P. et al: ‘Robust PCA using generalized nonconvex regularization’, IEEE Trans. Circuits Syst. Video Technol., 2019, doi: https://doi.org/10.1109/TCSVT.2019.2908833, to appear 40Wen, F., Ying, R., Liu, P. et al: ‘Nonconvex regularized robust PCA using the proximal block coordinate descent algorithm’, IEEE Trans. Signal Process., 2019, 67, (20), pp. 5402– 5416 41Zhang, D., Hu, Y., Ye, J. et al: ‘Matrix completion by truncated nuclear norm regularization’. IEEE Conf. on Computer Vision and Pattern Recognition, Providence, RI, USA, 2012, pp. 2192– 2199 42Wang, S., Zhang, L., Liang, Y.: ‘Nonlocal spectral prior model for low-level vision’. Asian Conf. on Computer Vision, Daejeon, Republic of Korea, 2012, pp. 231– 244 43Chengshi, L., Zhigang, Z., Qiang, L.: ‘Improvement of low-rank representation denoising model based on nonconvex penalty function’. Proc. of the 4th Int. Conf. on Communication and Information Processing, Qingdao, People's Republic of China, 2018, pp. 299– 304 44Parekh, A.: ‘ Bypassing the Limits of Ll Regularization: Convex Sparse Signal Processing Using Non-Convex Regularization’. PhD Thesis, New York University, 2016 45Bayram, I.: ‘On the convergence of the iterative shrinkage/thresholding algorithm with a weakly convex penalty’, IEEE Trans. Signal Process., 2015, 64, (6), pp. 1597– 1608 46Yang, Z., Jacob, M.: ‘Non-local regularization of inverse problems: a unified variational framework’, IEEE Trans Image Process, 2012, 22, (8), pp. 3192– 3203 47Combettes, P. L., Pesquet, J. C.: ‘Proximal thresholding algorithm for minimization over orthonormal bases’, SIAM J. Optim., 2007, 18, (4), pp. 1351– 1376 48Combettes, P. L., Pesquet, J. C.: ‘Proximal splitting methods in signal processing’. Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Alberta, Canada, 2009, pp. 185– 212 49Boyd, S., Parikh, N., Chu, E. et al: ‘Distributed optimization and statistical learning via the alternating direction method of multipliers’, Found. Trends® Mach. Learn., 2011, 3, (1), pp. 1– 122 50Li, W., Zhao, L., Xu, D. et al: ‘Efficient completion for corrupted low-rank images via alternating direction method’, J. Electron. Imaging, 2014, 23, (3), pp. 1– 10 51Wu, P., Xia, K., Yu, H.: ‘Relevance vector machine based pulmonary nodule classification’, J. Med. Imaging. Health. Inform., 2016, 6, (1), pp. 163– 169 52Wu, P., Xia, K., Yu, H.: ‘Correlation coefficient based supervised locally linear embedding for pulmonary nodule recognition’, Comput. Methods Programs Biomed., 2016, 136, (1), pp. 97– 106 53Xie, Y., Gu, S., Liu, Y. et al: ‘Weighted Schatten p-norm minimization for image denoising and background subtraction’, IEEE Trans Image Process, 2016, 25, (10), pp. 4842– 4857 54Xu, J., Zhang, L., Zuo, W. et al: ‘Patch group based nonlocal self-similarity prior learning for image denoising’. IEEE Int. Conf. on Computer Vision, Santiago, Chile, 2015, pp. 244– 252 Citing Literature Volume14, Issue5July 2020Pages 269-277 FiguresReferencesRelatedInformation