# Weightedlp Norm Sparse Error Constraint Based ADMM for Image Denoising

Authors: Jiucheng Xu; Nan Wang; Zhanwei Xu; Keqiang Xu
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In the process of image denoising, the accurate prior knowledge cannot be learned due to the influence of noise. Therefore, it is difficult to obtain better sparse coefficients. Based on this consideration, a weightedlp norm sparse error constraint (WPNSEC) model is proposed. Firstly, the suitable setting of powerp in the lp norm is made a detailed analysis. Secondly, the proposed model is extended to color image denoising. Since the noise of RGB channels has different intensities, a weight matrix is introduced to measure the noise levels of different channels, and a multichannel weighted lp norm sparse error constraint algorithm is proposed. Thirdly, in order to ensure that the proposed algorithm is tractable, the multichannel WPNSEC model is converted into an equality constraint problem solved via alternating direction method of multipliers (ADMM) algorithm. Experimental results on gray image and color image datasets show that the proposed algorithms not only have higher peak signal-to-noise ratio (PSNR) and feature similarity index (FSIM) but also produce better visual quality than competing image denoising algorithms.


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## Body

## 1. Introduction

In computer vision and image processing, one of the most fundamental problems is the influence of noise. To overcome this problem, image denoising has attracted more and more attention. Image denoising is designed for image quality enhancement, aiming to remove noiseN from the noisy observationY and recover the clean imageX.N is often assumed to be AWGN (additive white Gaussian noise) with a standard deviation σn. Recently, a number of image denoising methods have been reported, including sparse representation [1–3], nonlocal self-similarity [4–7], dictionary learning [1, 8], and deep learning [9, 10].The purpose of the sparse representation methods is to express most of the original signals with fewer fundamental signals. The previous models are based on the pixel level primarily, such as the TV (total variation) method [11], and Paul et al. [12] proposed an efficient minimization method for a generalized total variation functional. Since this method damages the detailed features of the images, patch-based sparse representation methods have been proposed. Transforming the image patches into a sparse linear combination in a dictionary is one of the most representative methods [13–16]. Liu et al. [17] proposed an augmented Lagrangian approach to general dictionary learning. However, the patch-based sparse representation model often assumes that the image patches are independent, ignoring the correlation among nonlocal similar patches. In addition, the time complexity is high for dictionary learning. In recent years, a large number of nonlocal self-similarity works have been proposed [18–20]. Gu et al. [21] proposed the weighted nuclear norm minimization and applied it to low-level vision, which has favorable reconstruction performance. Owing to the l0 norm being NP-hard, the existing sparse representation methods for image denoising often replace l0 norm convex optimization with l1 norm [22], and Zhao et al. [23] proposed a l1 norm low-rank matrix factorization. But it is difficult to obtain accurate sparse results by using the l1 norm convex optimization in some inverse problems. Therefore, Xie et al. [24] proposed a weighted Schatten p norm minimization. Zha et al. [25] proposed a nonconvex lp norm minimization based similar group sparse representation model. And Wang et al. [26] proposed a nonconvex weighted lp norm minimization based group sparse representation framework. Thus, it is not trivial to utilize the nonlocal similarity property of images. In addition, learning dictionary from noise images ignores the impact of noise and reduces the accuracy of sparse coefficients. And the powerp in the lp norm is set artificially, which may be not sufficient to reflect the effectiveness of the algorithm.At present, the most existing image denoising algorithms are devised for gray images. Since the color information plays an important role in image understanding and object recognition, color image denoising is crucial. The color image includes RGB channels with strong correlation. In [27, 28], the gray image denoising algorithms were applied to three channels of the color image, respectively, whereas the key issue of this method fails to consider the correlation among the channels. It not only increases the time consumption but also generates some wrong colors and artifacts. In [29, 30], the image patches of RGB channels stitched into one vector were treated equally, therefore, their defect overlooks the distinctive noise statistics among the channels. Nam et al. [31] proposed a cross-channel real image denoising algorithm, improving the performance of color image denoising.In view of the above problems, a weightedlp norm sparse error constrain (WPNSEC) model is proposed, which constrains the sparse error based on the nonconvex weighted lp norm. The proposed model reduces the influence of noise on dictionary learning. At the same time, the sparse error is decreased and the accuracy of the sparse coefficient is promoted. Then, the WPNSEC model is extended to color image denoising and a cross-channel color image denoising algorithm is proposed. The proposed algorithm makes use of the nonlocal self-similarity of images and decreases the time consumption. Since the noise of the RGB channels has different intensities, a weight matrix is introduced to measure the noise levels of different channels.The rest of this paper is structured as follows. In Section2, the cross-channel similar patch group and a nonconvex weighted lp norm model are introduced. In Section 3, first, the suitability setting of powerp is discussed in detail. Second, a sparse error constraint is added to the weight lp norm and the WPNSEC model is constructed. Third, the multichannel WPNSEC algorithm is proposed and the model is solved by using the ADMM algorithm. In Section 4, the experimental results of gray image denoising and color image denoising are presented. Section 5 concludes this paper.


## 2. Related Work

### 2.1. Cross-Channel Similar Patch Group

The patch group with nonlocal self-similarity has achieved a great success for image denoising [21]. The purpose of image denoising is to reconstruct a clear imageX=Y-N from the observed degradation noise imageY, whereN represents noise. For a noise imageY, the imageY is partitionedn overlapping image patches yi∈Rl2(i=1,2,…,n) and the size of each image patch is l×l. IfY is a color image, the cross-channel color image patches of size l×l×3 denote yic=[yrTygTybT]T∈R3l2, where yr,yg,yb∈Rl2 are the patches in R, G, and B channels. The standard deviations of noise in RGB channels are σc={σr,σg,σb}. For each local image patch yic, the most similar image patches yicm are extracted exploiting Euclidean distance in a search window around it. All similar patches are put into a matrix Yi column by column. The matrix with all similar noise patches is denoted as noise similar patch group Yi={yic1,yic2,…,yicm}, where yicm represents them-th similar patch of thei-th similar patch group inc channel. The SVD of Yi is Yi=UiΣiVi, and the nonzero elements in Σi are expressed as λi. The dictionary Di=UiViTDi=d1,d2,…,di,…,dK,di=uiviT, andK represents the number of dictionary atoms.


### 2.2. Weightedlp Norm Minimization

The norm minimization problem aims to estimate the true sparse result under certain constraint conditions. Traditional patch-based sparse coding is resolved by using thel1 norm and the weighted l1 norm, regularly. However, for some image inverse problems, such as image deblurring, image denoising, and other image restoration problems, the convex regularization does not get accurate sparse results. The reason is that the sparsity of an image cannot be measured by any benchmark. In [26], in order to improve the accuracy of sparse representation consequence, the convex l1 norm was substituted by the nonconvex lp norm. For image denoising, the weighted lp0<p<1 penalty function is extended to the sparse representation based patch group, and the weighted lp norm minimization (WPNM) can be represented as(1)α^i=argminαi∑i=1nYi-XiF2+wi·αipwhere Xi denotes the clean patch group: Xi=Diαi. αi is the sparse coefficient of thei-th patch group. p represents nonconvex lp norm. · denotes the dot product between vectors and wi is weight and its updating formula is wi=c∗22σ2/σi.c is a constant and σi is the estimated variance of αi.


## 2.1. Cross-Channel Similar Patch Group

The patch group with nonlocal self-similarity has achieved a great success for image denoising [21]. The purpose of image denoising is to reconstruct a clear imageX=Y-N from the observed degradation noise imageY, whereN represents noise. For a noise imageY, the imageY is partitionedn overlapping image patches yi∈Rl2(i=1,2,…,n) and the size of each image patch is l×l. IfY is a color image, the cross-channel color image patches of size l×l×3 denote yic=[yrTygTybT]T∈R3l2, where yr,yg,yb∈Rl2 are the patches in R, G, and B channels. The standard deviations of noise in RGB channels are σc={σr,σg,σb}. For each local image patch yic, the most similar image patches yicm are extracted exploiting Euclidean distance in a search window around it. All similar patches are put into a matrix Yi column by column. The matrix with all similar noise patches is denoted as noise similar patch group Yi={yic1,yic2,…,yicm}, where yicm represents them-th similar patch of thei-th similar patch group inc channel. The SVD of Yi is Yi=UiΣiVi, and the nonzero elements in Σi are expressed as λi. The dictionary Di=UiViTDi=d1,d2,…,di,…,dK,di=uiviT, andK represents the number of dictionary atoms.


## 2.2. Weightedlp Norm Minimization

The norm minimization problem aims to estimate the true sparse result under certain constraint conditions. Traditional patch-based sparse coding is resolved by using thel1 norm and the weighted l1 norm, regularly. However, for some image inverse problems, such as image deblurring, image denoising, and other image restoration problems, the convex regularization does not get accurate sparse results. The reason is that the sparsity of an image cannot be measured by any benchmark. In [26], in order to improve the accuracy of sparse representation consequence, the convex l1 norm was substituted by the nonconvex lp norm. For image denoising, the weighted lp0<p<1 penalty function is extended to the sparse representation based patch group, and the weighted lp norm minimization (WPNM) can be represented as(1)α^i=argminαi∑i=1nYi-XiF2+wi·αipwhere Xi denotes the clean patch group: Xi=Diαi. αi is the sparse coefficient of thei-th patch group. p represents nonconvex lp norm. · denotes the dot product between vectors and wi is weight and its updating formula is wi=c∗22σ2/σi.c is a constant and σi is the estimated variance of αi.


## 3. The Proposed Image Denoising Algorithm

### 3.1. The Setting of Powerp

A classical image denoising model often includes a fidelity term and a regularization term based on image prior knowledge. Some recent works have proven that the denoising algorithm is very efficient by using the nonconvex sparse coding based image prior. However, in the low rank matrix approximation problem, the singular values obtained by the weighted Schattenp norm minimization are overshrunk, which reduce the accuracy of the sparse coefficient. The weighted Schattenp norm minimization problem can be represented as(2)X^i=arg⁡minXi12Yi-XiF2+λXiw,SpThe SVD of the matrixX is X=U∑‍VT,∑‍=diag⁡σi1≤i≤r. In this paper, we use the GST algorithm to solve the nonconvex sparse coding problem. In order to get better denoising effects, the GST algorithm needs to converge to a better minimum. The soft-thresholding operator is(3)DiX=UiDλ∑iViTThe nonconvex weighted lp norm minimization problem is(4)A^i=argminAiYi-XiF2+λXiw,pBecause of Xi=DiAi,Yi=DiBi, (4) can be rewritten:(5)α^i=argminαiαi-α~i22+λα~iw,pwhereαi and α~i are the vectors of matrices Ai and Bi and the soft-thresholding operator is(6)α^i=softαi,λ(7)Xi=DiAi=∑j=1nsoftαi,j,λui,jvi,jT=UiDλ∑iViTIt can be proven that the nonconvex weightedlp norm is equivalent to the weighted Schattenp minimization problem. In [24], when the powerp gets larger and larger, the singular value of the clear image patches matrix is more different from the singular value of the matrix solved by the WSNM algorithm. Therefore, more high rank components of the singular values matrix solved by the WSNM algorithm will become 0 as the powerp increases, while the low rank components will get closer to the singular values of the clear similar patches. In other words, the overshrunk problem also exists in the nonconvex weighted lp norm minimization. However, it assigns the values of the powerp of the nonconvex weighted lp norm minimization manually. In order to achieve more robust denoising performance, the suitable value of powerp is set through experiments. It reduces the uncertainty of manual parameters setting. There are 20 images selected randomly from the Berkeley Segmentation Dataset [32]. For different levels of noise, the image denoising performance (PSNR) is tested by using the WPNSEC algorithm with different powerp. The value of powerp changes from 0.05 to 1 in steps of 0.05, gradually. As shown in Figure 1, six noise levels are 20, 30, 40, 50, 60, and 80, respectively. For each noise level, the optimal value of the powerp will be directly applied to the experimental part of the next section.Figure 1
The average PSNR values under different powerp and noise levels on 20 images.In Figure1, the ordinate denotes the average value of the PSNR on 20 images, and the abscissa is the value ofp. The best value ofp is higher when dealing with lower levels of noise. Figure 1 shows that the noise levels σ are 20 and 30 and the optimal values of powerp are 0.9 and 0.8, respectively. When the noise levels σ are 40 and 50, the optimal value of powerp is 0.75. As the level of the noise increases, more rank elements in the matrix are contaminated by noise. And the higher the rank, the greater the effect. Therefore, when the level of noise is high, the best value of powerp is small. Meanwhile, when the noise levels σ are 60 and 80, the optimal values of powerp are 0.5 and 0.2, respectively. In short, the best value of powerp is inversely proportional to the noise level and the best value of powerp is smaller when the noise levels become stronger.


### 3.2. Weightedlp Norm Sparse Error Constraint Model

From Section2.2, we can learn that the sparse coefficients are obtained from the noise image directly. Since the detail information of the noise image has been destroyed by noise, the sparse coefficient α of the denoised patch group is not accurate. Assume that the sparse coefficient of the clear image patch group Xi is α~. There are errors between the noise sparse coefficient α and the clear sparse coefficient α~. Thus, it is extremely important to reduce the sparse error α-α~ and raise the accuracy of α. The noise sparse coefficient α is expected to be as close as possible to α~. The sparse error constraint is added to the weighted lp norm, and the weighted lp norm sparse error constraint model (WPNSEC) is proposed. However, the clear imageX is unknown. The noise imageY will be preoperated using the BM3D algorithm, and a great estimate of the sparse coefficient α~ is obtained. Since the distribution of sparse error is closer to the Laplacian, the sparse error is solved by using the l1 norm. The WPNSEC model is represented as(8)α^i=argminαi∑i=1nYi-DiαiF2+wiαip+γαi-α~i1where γ denotes regularization parameters. To represent following the formulas clearly, we omit the number of overlapping image patchesi.Since the formulawαp is a nonconvex lp norm and γα-α~1 is a convex l1 norm, the two formulas cannot be solved at the same time. Therefore, (8) is divided into two subproblems: the sparse coefficient α of the weighted lp norm minimization is solved by GST algorithm first; then, the sparse coefficient α~ of the sparse error constraint is solved by surrogate algorithm. Finally, the average value between α and α~ is set as the final sparse coefficient. Equation (8) is rewritten as(9)α^=arg⁡minαY-DαF2+12w·αp+γα-α~1The description of WPNSEC model is shown in Algorithm 1.Algorithm 1:Image denoising by WPNSEC.
Require: Noisy image Y. Ensure: Denoised image X^. 1: Initialization:X^0=Y, c, δ, γ; 2:For  t = 0, 1, ⋯, K1 do 3:   SetYt+1=X^t+δ(Y-X^t); 4:   Extract local image patchesyi (i=1,2,⋯,n) from the noise imageYt+1; 5:For each local image patch yi do 6:     The nonlocal similar patchesyi are searched by Euclidean distance; 7:     Stack similar patchesyim form a noisy similar patch groupYit+1; 8:     Updateαit+1 by GST algorithm; 9:     Updateα~it+1 by surrogate algorithm; 10:    Updateα^t+1 by Eq. (9); 11:End for 12:  AggregateXit+1 to form the clean image X^t+1; 13:End for 14:Return The denoised image X^;


### 3.3. The Multichannel WPNSEC Model

At present, the most existing image denoising algorithms are devised for gray images. Since the color images play a crucial role in real life, it is necessary to improve color image quality. Thus, the WPNSEC model is extended to color image denoising and a multichannel WPNSEC color image denoising algorithm is proposed. For the color image denoising, it is unreasonable to apply the gray image denoising algorithm to the cross-channel noise image patch groupYi directly. To remove the noise of color images effectively, the strength of noise in different color channels should be taken into account. So, a weight matrixW [33] is introduced to balance the noise of the RGB channels. The weightW is a diagonal matrix and the diagonal elements are σc:(10)W=σr-1I000σg-1I000σb-1IThe multichannel WPNSEC minimization is as follows:(11)α^=arg⁡minαWY-DαF2+12wαp+γα-α~1Since (11) is a large-scale nonconvex optimization problem, it is very difficult to solve it. The nonconvex optimization problem is easily solved under the ADMM framework. Updating procedures of the ADMM method are shown in Algorithm 2. First, utilizing variable splitting scheme, (11) is transformed into an equality-constrained problem. Then the multichannel WPNSEC model is broken down into two variables by introducing an auxiliary variablez. Equation (11) is reformulated as(12)α^=arg⁡minz,αWY-DαF2+12wαp+γα-α~1s.t.z=Dα.Algorithm 2:Solve multichannel WPNSEC via ADMM.
Require: Noisy image Y and weight W, μ > 1, K2. Ensure: Sparse coefficient α and auxiliary variable z. 1: Initialization:t, ρ > 0, z, α and b; 2:Repeat 3:   Updatez as zt+1=argminz1/2W(Y-z)F2+ρ/2z-Dαt-bt22; 4:   Updateα as αt+1=argminα1/2(w⋅αp+γα-α~1)+ρ/2zt+1-Dα-bt22; 5:   Updateb as bt+1=bt-(zt+1-Dαt+1); 6:   Updateρ as ρt+1=μ∗ρt; 7:t←t+1; 8:Until 9: The convergence condition is satisfied ort≥K2.The Lagrangian function of (12) is(13)Lα,z,b,ρ=WY-zF2+12wαp+γα-α~1+b,z-Dα+ρ2z-DαF2whereb is the Lagrangian multiplier, ρ is the penalty parameter, and ρ>0. In thet-th (t = 1, 2, ⋯) iteration, the Lagrangian multiplierb, the penalty parameter ρ, the optimization variablesz, and α are represented as bt,ρt,zt,αt, respectively. Equation (13) based on the ADMM is transformed as(14)zt+1=argminzLραt+1,z,bt(15)αt+1=argminαLρα,zt,bt(16)bt+1=bt+ραt+1-zt+1Next, the effective solution for each subquestion will be introduced. The specific updating is as follows.(1) According to (14), updatez when fixing α andb:(17)zt+1=argminzWY-zF2+ρ2z-Dαt+1-bt22Obviously, (17) has a closed-form solution. So we have(18)zt+1=WTW+ρ2I-1WTWY+ρ2Dαt+1-bt(2) According to (15), update α when fixingz andb:(19)αt+1=argminα12wαp+γα-α~1+ρ2zt+1-Dα-bt22In the following proof, in order to avoid parameter confusion, the iteration numbert is omitted. Equation (19) is simplified as(20)α=minα12ρwαp+γα-α~1+12z-Dα-b22LetR=z-b; (20) is rewritten as(21)α=minα12R-Dα22+12ρwαp+γ2ρα-α~1Considering that 1/2ρwαp is nonconvex lp norm, it is difficult to solve (21). To obtain a tractable solution, Theorem 3 in [25] is employed: (22)α=minα12R-Dα22+12ρwαp+γ2ρα-α^1=minαi⁡∑i=1n12Ri-DiαiF2+12ρwiαip+γ2ραi-α^i1where R=Dλ. Since D=UVT, then(23)Ri-DiαiF2=Diλi-DiαiF2=Diλi-αiF2=Uidiag⁡λi-αiViF2=traceUidiag⁡λi-αiViViTdiag⁡λi-αiUiT=traceUidiag⁡λi-αidiag⁡λi-αiUiT=tracediag⁡λi-αiUiUiTdiag⁡λi-αi=tracediag⁡λi-αidiag⁡λi-αi=λi-αi22So (22) is rewritten as(24)α=minαi∑i=1n12λi-αi22+12ρwiαip+γ2ραi-α^i1Hence, (19) is simplified as (24), and the generalized soft threshold (GST) algorithm [34] is exploited to solve (24) effectively. We have the following equation:(25)α=SpGSTλ,τpGSTw,p,Jwhere SpGST denotes generalized soft-thresholding operator;J is the iteration number of GST algorithm. If λ<τpGST(w), the global minimum is α=0, where the threshold τpGST(w)=(2w(1-p))1/2-p+wp(2w(1-p))p-1/2-p. Otherwise, the optimal value will be at a nonzero point.(3) According to (16), updateb when fixingz and α:(26)bt+1=bt-zt+1-Dαt+1(4) Update the value ofρ:(27)ρt+1=μ∗ρt,μ>1Since the weightedlp norm is nonconvex, the unbounded sequence of ρ can sustain the convergence in Algorithm 2. By solving the above subproblems separately, an effective solution is obtained. The multichannel WPNSEC denoising algorithm is summarized in Algorithm 3.Algorithm 3:Colour image denoising by multichannel WPNSEC.
Require: Noisy image Y, {σr,σg,σb} and K3. Ensure: Denoised image X^. 1: Initialization:X^0=Y; 2:For  t=0,1,⋯,K3 do 3:   SetYt+1=X^t; 4:   Extract a local image patchesyic from the noise image Yt+1; 5:For each local patch yic do 6:     The non-local similar patchesyic are searched by Euclidean distance; 7:     Stack similar patchesyicm form a cross-channel noisy patch group Yit+1; 8:     Updateα^t+1 by WPNSEC model; 9:     Apply Eq. (4) to solve Yit+1 and obtain the estimation Xit+1=Dit+1α^it+1; 10:End for 11:  AggregateXit+1 to form the clean image X^t+1; 12:End for 13:Return The denoised image X^;


## 3.1. The Setting of Powerp

A classical image denoising model often includes a fidelity term and a regularization term based on image prior knowledge. Some recent works have proven that the denoising algorithm is very efficient by using the nonconvex sparse coding based image prior. However, in the low rank matrix approximation problem, the singular values obtained by the weighted Schattenp norm minimization are overshrunk, which reduce the accuracy of the sparse coefficient. The weighted Schattenp norm minimization problem can be represented as(2)X^i=arg⁡minXi12Yi-XiF2+λXiw,SpThe SVD of the matrixX is X=U∑‍VT,∑‍=diag⁡σi1≤i≤r. In this paper, we use the GST algorithm to solve the nonconvex sparse coding problem. In order to get better denoising effects, the GST algorithm needs to converge to a better minimum. The soft-thresholding operator is(3)DiX=UiDλ∑iViTThe nonconvex weighted lp norm minimization problem is(4)A^i=argminAiYi-XiF2+λXiw,pBecause of Xi=DiAi,Yi=DiBi, (4) can be rewritten:(5)α^i=argminαiαi-α~i22+λα~iw,pwhereαi and α~i are the vectors of matrices Ai and Bi and the soft-thresholding operator is(6)α^i=softαi,λ(7)Xi=DiAi=∑j=1nsoftαi,j,λui,jvi,jT=UiDλ∑iViTIt can be proven that the nonconvex weightedlp norm is equivalent to the weighted Schattenp minimization problem. In [24], when the powerp gets larger and larger, the singular value of the clear image patches matrix is more different from the singular value of the matrix solved by the WSNM algorithm. Therefore, more high rank components of the singular values matrix solved by the WSNM algorithm will become 0 as the powerp increases, while the low rank components will get closer to the singular values of the clear similar patches. In other words, the overshrunk problem also exists in the nonconvex weighted lp norm minimization. However, it assigns the values of the powerp of the nonconvex weighted lp norm minimization manually. In order to achieve more robust denoising performance, the suitable value of powerp is set through experiments. It reduces the uncertainty of manual parameters setting. There are 20 images selected randomly from the Berkeley Segmentation Dataset [32]. For different levels of noise, the image denoising performance (PSNR) is tested by using the WPNSEC algorithm with different powerp. The value of powerp changes from 0.05 to 1 in steps of 0.05, gradually. As shown in Figure 1, six noise levels are 20, 30, 40, 50, 60, and 80, respectively. For each noise level, the optimal value of the powerp will be directly applied to the experimental part of the next section.Figure 1
The average PSNR values under different powerp and noise levels on 20 images.In Figure1, the ordinate denotes the average value of the PSNR on 20 images, and the abscissa is the value ofp. The best value ofp is higher when dealing with lower levels of noise. Figure 1 shows that the noise levels σ are 20 and 30 and the optimal values of powerp are 0.9 and 0.8, respectively. When the noise levels σ are 40 and 50, the optimal value of powerp is 0.75. As the level of the noise increases, more rank elements in the matrix are contaminated by noise. And the higher the rank, the greater the effect. Therefore, when the level of noise is high, the best value of powerp is small. Meanwhile, when the noise levels σ are 60 and 80, the optimal values of powerp are 0.5 and 0.2, respectively. In short, the best value of powerp is inversely proportional to the noise level and the best value of powerp is smaller when the noise levels become stronger.


## 3.2. Weightedlp Norm Sparse Error Constraint Model

From Section2.2, we can learn that the sparse coefficients are obtained from the noise image directly. Since the detail information of the noise image has been destroyed by noise, the sparse coefficient α of the denoised patch group is not accurate. Assume that the sparse coefficient of the clear image patch group Xi is α~. There are errors between the noise sparse coefficient α and the clear sparse coefficient α~. Thus, it is extremely important to reduce the sparse error α-α~ and raise the accuracy of α. The noise sparse coefficient α is expected to be as close as possible to α~. The sparse error constraint is added to the weighted lp norm, and the weighted lp norm sparse error constraint model (WPNSEC) is proposed. However, the clear imageX is unknown. The noise imageY will be preoperated using the BM3D algorithm, and a great estimate of the sparse coefficient α~ is obtained. Since the distribution of sparse error is closer to the Laplacian, the sparse error is solved by using the l1 norm. The WPNSEC model is represented as(8)α^i=argminαi∑i=1nYi-DiαiF2+wiαip+γαi-α~i1where γ denotes regularization parameters. To represent following the formulas clearly, we omit the number of overlapping image patchesi.Since the formulawαp is a nonconvex lp norm and γα-α~1 is a convex l1 norm, the two formulas cannot be solved at the same time. Therefore, (8) is divided into two subproblems: the sparse coefficient α of the weighted lp norm minimization is solved by GST algorithm first; then, the sparse coefficient α~ of the sparse error constraint is solved by surrogate algorithm. Finally, the average value between α and α~ is set as the final sparse coefficient. Equation (8) is rewritten as(9)α^=arg⁡minαY-DαF2+12w·αp+γα-α~1The description of WPNSEC model is shown in Algorithm 1.Algorithm 1:Image denoising by WPNSEC.
Require: Noisy image Y. Ensure: Denoised image X^. 1: Initialization:X^0=Y, c, δ, γ; 2:For  t = 0, 1, ⋯, K1 do 3:   SetYt+1=X^t+δ(Y-X^t); 4:   Extract local image patchesyi (i=1,2,⋯,n) from the noise imageYt+1; 5:For each local image patch yi do 6:     The nonlocal similar patchesyi are searched by Euclidean distance; 7:     Stack similar patchesyim form a noisy similar patch groupYit+1; 8:     Updateαit+1 by GST algorithm; 9:     Updateα~it+1 by surrogate algorithm; 10:    Updateα^t+1 by Eq. (9); 11:End for 12:  AggregateXit+1 to form the clean image X^t+1; 13:End for 14:Return The denoised image X^;


## 3.3. The Multichannel WPNSEC Model

At present, the most existing image denoising algorithms are devised for gray images. Since the color images play a crucial role in real life, it is necessary to improve color image quality. Thus, the WPNSEC model is extended to color image denoising and a multichannel WPNSEC color image denoising algorithm is proposed. For the color image denoising, it is unreasonable to apply the gray image denoising algorithm to the cross-channel noise image patch groupYi directly. To remove the noise of color images effectively, the strength of noise in different color channels should be taken into account. So, a weight matrixW [33] is introduced to balance the noise of the RGB channels. The weightW is a diagonal matrix and the diagonal elements are σc:(10)W=σr-1I000σg-1I000σb-1IThe multichannel WPNSEC minimization is as follows:(11)α^=arg⁡minαWY-DαF2+12wαp+γα-α~1Since (11) is a large-scale nonconvex optimization problem, it is very difficult to solve it. The nonconvex optimization problem is easily solved under the ADMM framework. Updating procedures of the ADMM method are shown in Algorithm 2. First, utilizing variable splitting scheme, (11) is transformed into an equality-constrained problem. Then the multichannel WPNSEC model is broken down into two variables by introducing an auxiliary variablez. Equation (11) is reformulated as(12)α^=arg⁡minz,αWY-DαF2+12wαp+γα-α~1s.t.z=Dα.Algorithm 2:Solve multichannel WPNSEC via ADMM.
Require: Noisy image Y and weight W, μ > 1, K2. Ensure: Sparse coefficient α and auxiliary variable z. 1: Initialization:t, ρ > 0, z, α and b; 2:Repeat 3:   Updatez as zt+1=argminz1/2W(Y-z)F2+ρ/2z-Dαt-bt22; 4:   Updateα as αt+1=argminα1/2(w⋅αp+γα-α~1)+ρ/2zt+1-Dα-bt22; 5:   Updateb as bt+1=bt-(zt+1-Dαt+1); 6:   Updateρ as ρt+1=μ∗ρt; 7:t←t+1; 8:Until 9: The convergence condition is satisfied ort≥K2.The Lagrangian function of (12) is(13)Lα,z,b,ρ=WY-zF2+12wαp+γα-α~1+b,z-Dα+ρ2z-DαF2whereb is the Lagrangian multiplier, ρ is the penalty parameter, and ρ>0. In thet-th (t = 1, 2, ⋯) iteration, the Lagrangian multiplierb, the penalty parameter ρ, the optimization variablesz, and α are represented as bt,ρt,zt,αt, respectively. Equation (13) based on the ADMM is transformed as(14)zt+1=argminzLραt+1,z,bt(15)αt+1=argminαLρα,zt,bt(16)bt+1=bt+ραt+1-zt+1Next, the effective solution for each subquestion will be introduced. The specific updating is as follows.(1) According to (14), updatez when fixing α andb:(17)zt+1=argminzWY-zF2+ρ2z-Dαt+1-bt22Obviously, (17) has a closed-form solution. So we have(18)zt+1=WTW+ρ2I-1WTWY+ρ2Dαt+1-bt(2) According to (15), update α when fixingz andb:(19)αt+1=argminα12wαp+γα-α~1+ρ2zt+1-Dα-bt22In the following proof, in order to avoid parameter confusion, the iteration numbert is omitted. Equation (19) is simplified as(20)α=minα12ρwαp+γα-α~1+12z-Dα-b22LetR=z-b; (20) is rewritten as(21)α=minα12R-Dα22+12ρwαp+γ2ρα-α~1Considering that 1/2ρwαp is nonconvex lp norm, it is difficult to solve (21). To obtain a tractable solution, Theorem 3 in [25] is employed: (22)α=minα12R-Dα22+12ρwαp+γ2ρα-α^1=minαi⁡∑i=1n12Ri-DiαiF2+12ρwiαip+γ2ραi-α^i1where R=Dλ. Since D=UVT, then(23)Ri-DiαiF2=Diλi-DiαiF2=Diλi-αiF2=Uidiag⁡λi-αiViF2=traceUidiag⁡λi-αiViViTdiag⁡λi-αiUiT=traceUidiag⁡λi-αidiag⁡λi-αiUiT=tracediag⁡λi-αiUiUiTdiag⁡λi-αi=tracediag⁡λi-αidiag⁡λi-αi=λi-αi22So (22) is rewritten as(24)α=minαi∑i=1n12λi-αi22+12ρwiαip+γ2ραi-α^i1Hence, (19) is simplified as (24), and the generalized soft threshold (GST) algorithm [34] is exploited to solve (24) effectively. We have the following equation:(25)α=SpGSTλ,τpGSTw,p,Jwhere SpGST denotes generalized soft-thresholding operator;J is the iteration number of GST algorithm. If λ<τpGST(w), the global minimum is α=0, where the threshold τpGST(w)=(2w(1-p))1/2-p+wp(2w(1-p))p-1/2-p. Otherwise, the optimal value will be at a nonzero point.(3) According to (16), updateb when fixingz and α:(26)bt+1=bt-zt+1-Dαt+1(4) Update the value ofρ:(27)ρt+1=μ∗ρt,μ>1Since the weightedlp norm is nonconvex, the unbounded sequence of ρ can sustain the convergence in Algorithm 2. By solving the above subproblems separately, an effective solution is obtained. The multichannel WPNSEC denoising algorithm is summarized in Algorithm 3.Algorithm 3:Colour image denoising by multichannel WPNSEC.
Require: Noisy image Y, {σr,σg,σb} and K3. Ensure: Denoised image X^. 1: Initialization:X^0=Y; 2:For  t=0,1,⋯,K3 do 3:   SetYt+1=X^t; 4:   Extract a local image patchesyic from the noise image Yt+1; 5:For each local patch yic do 6:     The non-local similar patchesyic are searched by Euclidean distance; 7:     Stack similar patchesyicm form a cross-channel noisy patch group Yit+1; 8:     Updateα^t+1 by WPNSEC model; 9:     Apply Eq. (4) to solve Yit+1 and obtain the estimation Xit+1=Dit+1α^it+1; 10:End for 11:  AggregateXit+1 to form the clean image X^t+1; 12:End for 13:Return The denoised image X^;


## 4. Experimental Results and Analysis

### 4.1. Parameters Selection

In gray image denoising experiment, the noise standard deviationσ is 10, 20, 30, 40, 50, 60, and 80. When the noise standard deviation σ≤20, the size of the overlapping block is 6 × 6; when the noise standard deviation is 20<σ≤50, the size of the overlapping block is 7 × 7; when the noise standard deviation is 50<σ≤70, the size of the overlapping block is 8 × 8; when the noise standard deviation is 70<σ≤80, the size of the overlapping block is 9 × 9. The size of the searching window for selecting similar patches is 40 × 40, and the number of similar patches of each window is 70. When σ≤30, c,ζ,λ is (0.2, 0.18, 0.67); otherwise, c,ζ,λ is (0.3, 0.22, 0.67). The simulation experiment carried out with the gray test images is shown in Figure 2 (from left to right and from top to bottom: airplane, barbara, boat, cameraman, couple, foreman, gold hill, house, leaves, lena, lin, monarch, parrots, peppers, girl, and man).Figure 2
The partial test images for gray images denoising.The parameters of the multichannel WPNSEC algorithm for color image denoising are set as follows in detail: the size of searching window is40×40, and the size of each image patch isl=6. The number of the nonlocal similar patches ism=60. The updating parameter is μ=1.001 and the initial penalty parameter is ρ0=3. The numbers of iterations in Algorithms 2 and 3 are K2=10 and K3=8, respectively. In addition, all experiments are performed under the Matlab-R2016 environment on a machine with AMD Athlon(tm) II X4 645 Processor CPU 4GB RAM.


### 4.2. Advantages of the Weightedlp Norm Sparse Error Constraint

This section demonstrates the advantages of the proposed algorithm. Figure3 shows the denoised results of nonconvex weighted lp norm minimization model and weighted lp norm sparse error constraint method on test imageparrot. In Figure 3, an image patch is cropped randomly from the denoised image (marked by a red box). In order to observe clearly, this small patch is enlarged and pasted into the lower left corner of the denoised image.Figure 3
Denoising results on imageparrot with σ =20. (a) Nonconvex weighted lp norm minimization, PSNR=32.58dB. (b) WPNSEC, PSNR=33.03dB.
(a) (b)Figure3(a) reflects that the edge information around the parrot’s eye is well recovered, but it is too smooth in the flat area to lose some detailed information. The reason is that the detailed information of the noise image has been destroyed by noise. The sparse coefficient obtained from the noise image will decrease the denoising performance of the algorithm. The proposed algorithm introduces a sparse error constraint, which reduces the influence of noise on sparse coefficients and improves the denoising performance. In Figure 3(b), the edge information around the parrot’s eyes is retained well and the texture information of the smooth area is better restored.To observe that the WPNSEC algorithm outperforms the WPNM algorithm intuitively, a line graph is shown in Figure4. Figure 4 reflects that the PSNR of the proposed algorithm and WPNM algorithm decreases gradually when the noise standard deviation increases. And the PSNR of the WNPSEC algorithm is higher than competing method under different noise standard deviations σ.Figure 4
The average PSNR (dB) results of nonconvex weightedlp norm minimization and WPNSEC algorithms on 16 gray images.


### 4.3. Results on Gray Noisy Images Denoising

The proposed WPNSEC algorithm is compared with several related classical algorithms, including BM3D [35], EPLL [36], NCSR [37], WNNM [20], and WSNM [24]. In order to evaluate the performance of the WPNSEC algorithm in gray image denoising, the peak signal-to-noise ratio (PSNR) and feature similarity index (FSIM) are used as the standard of the experimental results. The PSNR values of the proposed algorithm and all competing algorithms on 10 test images are listed in Table 1, and the FSIM values of the proposed algorithm and all competing algorithms on 10 test images are displayed in Table 2. The noise level is σ = 10, σ = 20, σ = 30, and σ = 50 from top to bottom. An overall impression can be observed from Table 1; when the noise level increases from 20 to 50, the improvements of WPNSEC increase 0.60dB, 0.44dB, and 0.52dB on average, respectively. And the average PSNR of proposed algorithm increases by 0.96 dB, 1.32 dB, 0.66 dB, 0.63 dB, and 0.56 dB over the BM3D, EPLL, NCSR, WNNM, and WSNM. In summary, although the PSNR value decreases with the increase of the standard deviation, most of the PSNR values of the proposed algorithm are higher than those in other related algorithms and the average PSNR is higher than that in all competing algorithms. In order to observe the comparison between our proposed algorithm intuitively and the competing methods on PSNR and FSIM, a line graph of the correlation algorithm is shown in Figure 5, which is drawn by average PSNR (dB) and FSIM of different denoising algorithms on 16 gray images with standard deviation σ=20,30,40,50,60,80. At the same time, in order to reflect the superiority of the WPNSEC algorithm, the average running time of all algorithms is shown in Table 3.Table 1
Denoising PSNR (dB) results of different denoising algorithms under different standard deviationsσ (the highest PSNR values are bold).
σ=10 Image BM3D EPLL NCSR WNNM WSNM WPNSEC Monarch 33.24 33.17 33.21 33.40 33.51 34.93 Couple 32.17 32.08 32.50 32.84 32.93 35.81 House 34.42 34.21 35.16 35.42 35.57 36.83 Lena 32.89 33.16 34.31 34.62 34.81 35.49 Parrot 32.59 32.21 33.65 34.44 34.49 35.75 C. man 32.52 32.55 33.17 33.48 33.62 34.24 Boat 32.57 32.64 32.91 32.89 32.97 33.70 Leaves 31.84 32.17 33.04 33.11 33.35 34.95 Peppers 33.24 33.29 33.27 33.34 33.47 34.67 Airplane 33.32 33.21 33.41 33.52 33.74 34.61 Average 32.88 32.87 33.46 33.71 33.85 35.10  σ=20  Image BM3D EPLL NCSR WNNM WSNM WPNSEC Monarch 30.35 30.48 30.69 31.10 31.13 32.37 Couple 30.76 30.54 30.56 30.82 30.83 32.63 House 33.77 32.98 33.97 34.01 34.05 34.08 Lena 31.60 32.61 32.92 33.12 33.13 32.86 Parrot 29.96 29.97 32.16 30.19 30.21 33.03 C. man 30.28 30.34 30.48 30.75 30.77 31.47 Boat 29.69 30.66 30.74 31.00 31.02 31.26 Leaves 29.08 30.76 30.59 30.74 30.96 31.12 Peppers 31.14 31.17 31.26 31.53 31.55 31.86 Airplane 30.44 32.41 32.58 32.82 32.82 33.06 Average 30.71 31.19 31.60 31.65 31.74 32.26  σ=30  Image BM3D EPLL NCSR WNNM WSNM WPNSEC Monarch 28.36 28.35 28.46 28.91 28.93 28.96 Couple 30.08 28.61 28.57 28.98 29.02 30.68 House 32.08 31.22 32.07 32.52 32.54 32.69 Lena 29.55 31.41 29.43 29.83 31.48 30.08 Parrot 29.13 28.07 30.38 28.33 28.33 30.61 C. man 28.63 28.36 28.58 28.80 28.83 29.31 Boat 29.11 28.89 28.94 29.24 29.26 29.52 Leaves 27.82 27.38 28.14 28.61 28.62 28.64 Peppers 29.28 28.35 31.11 29.48 29.54 30.56 Airplane 27.56 30.41 30.70 30.87 30.92 31.21 Average 29.16 29.11 29.64 29.56 29.75 30.23  σ=50  Image BM3D EPLL NCSR WNNM WSNM WPNSEC Monarch 25.51 25.77 25.78 26.18 26.30 26.32 Couple 26.46 26.23 26.19 27.45 26.71 28.30 House 29.69 28.76 29.62 30.23 30.31 30.29 Lena 27.07 26.42 27.12 27.16 27.28 27.40 Parrot 25.89 25.83 27.88 26.00 26.10 28.23 C. man 26.13 26.03 26.15 26.42 26.44 26.62 Boat 26.78 26.65 26.67 26.97 27.01 26.88 Leaves 24.75 24.36 24.96 25.49 25.54 25.69 Peppers 26.68 26.62 28.07 26.81 26.94 28.77 Airplane 25.10 27.88 28.18 28.44 28.49 28.46 Average 26.41 26.46 27.06 27.12 27.11 27.70Table 2
Denoising FSIM results of different denoising algorithms under different standard deviationsσ (the highest FSIM values are bold).
σ=10 Image BM3D EPLL NCSR WNNM WSNM WPNSEC Monarch 0.9578 0.9564 0.9624 0.9550 0.9567 0.9661 Couple 0.9794 0.9781 0.9818 0.9815 0.9843 0.9897 House 0.9511 0.9462 0.9601 0.9589 0.9590 0.9592 Lena 0.9653 0.9633 0.9819 0.9820 0.9831 0.9872 Parrot 0.9367 0.9364 0.9618 0.9549 0.9563 0.9621 C. man 0.9553 0.9561 0.9553 0.9530 0.9541 0.9590 Boat 0.9723 0.9726 0.9813 0.9814 0.9832 0.9879 Leaves 0.9492 0.9512 0.9730 0.9628 0.9642 0.9743 Peppers 0.9587 0.9573 0.9611 0.9530 0.9543 0.9612 Airplane 0.9396 0.9468 0.9563 0.9515 0.9532 0.9591 Average 0.9565 0.9564 0.9675 0.9634 0.9648 0.9706  σ=20  Image BM3D EPLL NCSR WNNM WSNM WPNSEC Monarch 0.9298 0.9294 0.9320 0.9176 0.9187 0.9389 Couple 0.9573 0.9461 0.9555 0.9532 0.9541 0.9583 House 0.9190 0.9173 0.9201 0.9177 0.9182 0.9202 Lena 0.9541 0.9551 0.9598 0.9601 0.9603 0.9357 Parrot 0.9345 0.9302 0.9348 0.9212 0.9237 0.9356 C. man 0.9097 0.9095 0.9086 0.9096 0.9125 0.9175 Boat 0.9551 0.9537 0.9553 0.9561 0.9574 0.9621 Leaves 0.9527 0.9493 0.9469 0.9296 0.9326 0.9503 Peppers 0.9322 0.9314 0.9327 0.9149 0.9161 0.9341 Airplane 0.8995 0.9172 0.9185 0.9116 0.9142 0.9218 Average 0.9344 0.9339 0.9367 0.9292 0.9308 0.9375  σ=30  Image BM3D EPLL NCSR WNNM WSNM WPNSEC Monarch 0.9013 0.8994 0.9082 0.8898 0.8965 0.9183 Couple 0.9279 0.9273 0.9285 0.9323 0.9344 0.9311 House 0.8991 0.8987 0.9000 0.8930 0.8952 0.9023 Lena 0.9563 0.9531 0.9446 0.9452 0.9449 0.9453 Parrot 0.9194 0.9175 0.9208 0.8999 0.9051 0.9279 C. man 0.8632 0.8651 0.8707 0.8737 0.8739 0.8746 Boat 0.9351 0.9293 0.9309 0.9346 0.9351 0.9390 Leaves 0.9266 0.9185 0.9279 0.9041 0.9184 0.9284 Peppers 0.8996 0.9042 0.9126 0.8897 0.8931 0.9115 Airplane 0.8791 0.8788 0.8897 0.8865 0.8895 0.8967 Average 0.9108 0.9092 0.9130 0.9049 0.9086 0.9175  σ=50  Image BM3D EPLL NCSR WNNM WSNM WPNSEC Monarch 0.8601 0.8552 0.8615 0.8517 0.8573 0.8821 Couple 0.8746 0.8765 0.8800 0.8913 0.8928 0.8922 House 0.8573 0.8493 0.8675 0.8706 0.8741 0.8834 Lena 0.8921 0.8896 0.9191 0.9177 0.9182 0.9210 Parrot 0.8931 0.8779 0.8926 0.8855 0.8860 0.8994 C. man 0.7922 0.7869 0.8091 0.8279 0.8292 0.8350 Boat 0.8876 0.8815 0.8847 0.8954 0.8953 0.8939 Leaves 0.8564 0.8475 0.8865 0.8754 0.8791 0.9011 Peppers 0.8693 0.8531 0.8685 0.8489 0.8504 0.8777 Airplane 0.8360 0.8256 0.8378 0.8420 0.8432 0.8483 Average 0.8619 0.8543 0.8707 0.8705 0.8726 0.8834Table 3
Average run time(s) of all methods on the BSD68 dataset.
Algorithms BM3D EPLL NCSR WNNM WSNM WPNSEC Times 2.67 103.78 374.01 169.32 174.21 89.64Figure 5
The average PSNR (dB) and FSIM results of different denoising algorithms on BSD68 dataset.
(a) (b)In order to display the visual quality of the proposed denoising algorithm, this part uses theleaves test image with σ = 20 to carry out a simulation experiment, and the experimental results are shown in Figure 6. It can be clearly seen from the highlighted red window in Figure 6 that the WPNSEC algorithm recovers the texture of the leaves well, but other competing denoising algorithms contain more noise and produce many artifacts. In Figure 6(c), since the EPLL algorithm ignores the nonlocal self-similarity, the details of the texture and structure edges are seriously lost, which affects the visual perception. The WPNSEC algorithm that is based on similar patches group preserves the image features better while effectively removing the image noise. According to the above analysis, the proposed algorithm has robust denoising performance, which can not only obtain higher PSNR and FSIM indices but also produce better visual quality.Figure 6
Denoising results on imageleaves by different methods (σ = 20). (a) Truth. (b) Noise. (c) BM3D, PSNR=29.08dB. (d) NCSR, PSNR=30.76dB. (e) WNNM, PSNR=30.59dB. (f) EPLL, PSNR=30.74dB. (g) WSNM, PSNR=30.96dB. (h) WPNSEC, PSNR=31.12dB.
(a) (b) (c) (d) (e) (f) (g) (h)In Algorithm1, the size of an image patch is l2×m, wherel represents the size of the image patch yi andm is the number of similar patches in a similar patches group. The constructing dictionary is the main computation in each iteration (Step 5 in Algorithm 1), and it costs O(min{l2m2,l4m}). The generalized soft threshold algorithm (Step 8 in Algorithm 1) only costs O(Km)), whereK represents the number of iterations in the generalized soft threshold algorithm. Therefore, the cost for the proposed WPNSEC image denoising algorithm is K1×S×O(min{l2m2,l4m}+Km), K1 represents the number of iterations of Algorithm 1, andS represents the number of image patches in a sliding window.


### 4.4. Results on Color Noisy Images Denoising

In this subsection, the multichannel WPNSEC method is compared with other competing denoising methods, including CBM3D [38], MLP [39], DnCNN [40], TNRD[41], and MC-WNNM [33], on the 24 color images displayed in Figure 7 (from left to right and from top to bottom: gate, red door, caps, bikes, sailing 1, flower buildings, sailing 4, couple, stream, rapids, girl, ocean, lighthouse 1, house, painted house, parrots, plane, woman hat, sailing 2, sailing 3, statue, woman, and lighthouse 2). In order to obtain the color noise images, additive white Gaussian noise is added to each channel of the clear image, respectively: σr=40, σg=20, and σb=30.Figure 7
The 24 color images from the Kodak PhotoCD Dataset (http://r0k.us/graphics/kodak/).The PSNR values of competing methods on the Kodak PhotoCD dataset are reported in Table4 and the PSNR average values of all methods on the CBSD68 dataset are reflected in Table 5. The highest PSNR values for each color image are bold. Table 4 shows that most of the PSNR values of our proposed algorithm are higher than the PSNR values of other competing methods and that the average value of the PSNR is the highest. The improvements of multichannel WPNSEC achieve 1.84dB, 0.43dB, 0.29dB, 8.39dB, and 0.27dB over CBM3D, MLP, TNDR, DnCNN, and MC-WNNM on average, respectively. At the same time, in order to reflect the high efficiency of the proposed algorithm, the average running time of each algorithm is shown in Table 6.Table 4
Denoising PSNR (dB) results of different denoising algorithms on color images (the highest PSNR values are bold).
σ r = 40 σ g = 20 σ b = 30 Image CBM3D MLP TNRD DnCNN MC-WNNM Ours Gate 25.24 25.70 25.74 20.47 25.83 26.13 Red door 28.27 30.12 30.21 20.47 29.75 30.19 Caps 28.81 31.19 31.49 20.53 31.54 31.64 Woman hat 27.95 29.88 29.86 20.47 29.94 30.37 Bikes 25.03 26.00 26.18 20.52 25.74 26.52 Sailing 1 26.24 26.84 26.90 20.66 27.00 27.42 Flower 27.88 30.28 30.40 20.52 30.53 30.47 Buildings 25.05 25.59 25.83 20.57 25.95 26.32 Sailing 2 28.44 30.75 30.81 20.50 30.95 31.04 Sailing 3 28.27 30.38 30.57 20.52 30.52 30.83 Sailing 4 26.95 28.00 28.14 20.52 28.23 28.43 Couple 28.76 30.87 31.05 20.60 31.12 31.41 Stream 23.76 23.95 23.99 20.52 24.01 24.63 Rapids 26.02 26.97 27.11 20.51 27.03 27.21 Girl 28.38 30.15 30.44 20.71 30.52 30.76 Ocean 27.75 28.82 28.87 20.52 29.00 29.27 Statue 27.90 29.57 29.80 20.56 29.60 29.76 Woman 25.77 26.40 26.41 20.53 26.52 26.73 Lighthouse 1 27.30 28.67 28.81 20.53 28.89 29.34 Plane 28.96 30.40 30.76 21.44 30.80 30.92 Lighthouse 2 26.54 27.53 27.60 20.51 27.59 27.91 House 27.05 28.17 28.27 20.51 28.00 28.27 Parrots 29.14 32.31 32.51 20.54 32.06 32.19 Painted house 25.75 26.41 26.53 20.59 26.64 27.42 Average 27.13 28.54 28.68 20.58 28.70 28.97Table 5
The average PSNR (dB) results of different denoising algorithms on the CBSD68 dataset.
Algorithms CBM3D MLP TNRD DnCNN MC-WNNM Ours Average 26.28 27.69 27.93 19.74 28.05 28.22Table 6
Average run time(s) of all methods on the CBSD68 dataset.
Algorithms CBM3D MLP TNRD DnCNN MC-WNNM Ours Times 3.69 3.92 - 190.73 1127.91 132.47The denoised images of our proposed algorithm and all competing methods on color imagebuildings are shown in Figure 8. In order to observe the details of the image more clearly, a red window is used to select a partial block of the image and zoom in it. Figure 8 shows that the denoising quality of the multichannel WPNSEC algorithm is better than other algorithms. DnCNN, MLP, and CBM3D can preserve as much edge information as possible when removing the noise, but these algorithms produce some distortion and artifacts. For example, in Figures 8(c), 8(d), and 8(f), there are a lot of noises on the windows of the house and a large amount of artifacts in the smooth area. TNRD method reduces noise and artifacts, so it has better denoising effect than DnCNN, MLP, and CBM3D. Due to the oversmoothing effect in the TNRD method, some boundaries and details are lost. For instance, in Figure 8(e), the red mark in the lower left corner of the red window is unclear. MC-WNNM is a state-of-the-art image denoising algorithm, but it is constructed based on convex optimization, so the sparse coefficients are a lack of robustness. Although the MC-WNNM algorithm performs well on denoising quality, it is not as good as the multichannel WPNSEC algorithm. As can be seen from Figure 8(g), the details of the letters and the red mark in the lower left window are slightly lost. In Figure 8(h), the edge and detail information can be well protected, the background is smoother, and it gets a better visual experience. Although our proposed algorithm does not achieve a clear representation of all the details (such as the red mark), it shows the best visual effect in the competing methods.Figure 8
Denoising results on imagebuildings under different methods. (a) Truth. (b) Noise. (c) CBM3D, PSNR=25.03dB. (d) MLP, PSNR=26.00dB. (e) TNRD, PSNR=26.18dB. (f) DnCNN, PSNR=20.52dB. (g) MC-WNNM, PSNR=26.95dB. (h) Ours, PSNR=26.52dB.
(a) (b) (c) (d) (e) (f) (g) (h)In Algorithm2, the size of a color image patch is 3l2m, wherel represents the size of the image patch yic andm is the number of similar patches in a similar patches group. UpdatingZ costs O(max{l4m,m3}) and updating α costs O(l4m+m3). The cost for updatingb and ρ can be ignored. Therefore, the cost of the ADMM algorithm is O((l4m+m3)K2), where K2 represents the number of iterations of Algorithm 2. In Algorithm 3, for multichannel color image denoising, the overall cost is O((l4m+m3)K2K3S), where K3 represents the number of iterations of Algorithm 3 andS represents the number of image patches in a sliding window.


## 4.1. Parameters Selection

In gray image denoising experiment, the noise standard deviationσ is 10, 20, 30, 40, 50, 60, and 80. When the noise standard deviation σ≤20, the size of the overlapping block is 6 × 6; when the noise standard deviation is 20<σ≤50, the size of the overlapping block is 7 × 7; when the noise standard deviation is 50<σ≤70, the size of the overlapping block is 8 × 8; when the noise standard deviation is 70<σ≤80, the size of the overlapping block is 9 × 9. The size of the searching window for selecting similar patches is 40 × 40, and the number of similar patches of each window is 70. When σ≤30, c,ζ,λ is (0.2, 0.18, 0.67); otherwise, c,ζ,λ is (0.3, 0.22, 0.67). The simulation experiment carried out with the gray test images is shown in Figure 2 (from left to right and from top to bottom: airplane, barbara, boat, cameraman, couple, foreman, gold hill, house, leaves, lena, lin, monarch, parrots, peppers, girl, and man).Figure 2
The partial test images for gray images denoising.The parameters of the multichannel WPNSEC algorithm for color image denoising are set as follows in detail: the size of searching window is40×40, and the size of each image patch isl=6. The number of the nonlocal similar patches ism=60. The updating parameter is μ=1.001 and the initial penalty parameter is ρ0=3. The numbers of iterations in Algorithms 2 and 3 are K2=10 and K3=8, respectively. In addition, all experiments are performed under the Matlab-R2016 environment on a machine with AMD Athlon(tm) II X4 645 Processor CPU 4GB RAM.


## 4.2. Advantages of the Weightedlp Norm Sparse Error Constraint

This section demonstrates the advantages of the proposed algorithm. Figure3 shows the denoised results of nonconvex weighted lp norm minimization model and weighted lp norm sparse error constraint method on test imageparrot. In Figure 3, an image patch is cropped randomly from the denoised image (marked by a red box). In order to observe clearly, this small patch is enlarged and pasted into the lower left corner of the denoised image.Figure 3
Denoising results on imageparrot with σ =20. (a) Nonconvex weighted lp norm minimization, PSNR=32.58dB. (b) WPNSEC, PSNR=33.03dB.
(a) (b)Figure3(a) reflects that the edge information around the parrot’s eye is well recovered, but it is too smooth in the flat area to lose some detailed information. The reason is that the detailed information of the noise image has been destroyed by noise. The sparse coefficient obtained from the noise image will decrease the denoising performance of the algorithm. The proposed algorithm introduces a sparse error constraint, which reduces the influence of noise on sparse coefficients and improves the denoising performance. In Figure 3(b), the edge information around the parrot’s eyes is retained well and the texture information of the smooth area is better restored.To observe that the WPNSEC algorithm outperforms the WPNM algorithm intuitively, a line graph is shown in Figure4. Figure 4 reflects that the PSNR of the proposed algorithm and WPNM algorithm decreases gradually when the noise standard deviation increases. And the PSNR of the WNPSEC algorithm is higher than competing method under different noise standard deviations σ.Figure 4
The average PSNR (dB) results of nonconvex weightedlp norm minimization and WPNSEC algorithms on 16 gray images.


## 4.3. Results on Gray Noisy Images Denoising

The proposed WPNSEC algorithm is compared with several related classical algorithms, including BM3D [35], EPLL [36], NCSR [37], WNNM [20], and WSNM [24]. In order to evaluate the performance of the WPNSEC algorithm in gray image denoising, the peak signal-to-noise ratio (PSNR) and feature similarity index (FSIM) are used as the standard of the experimental results. The PSNR values of the proposed algorithm and all competing algorithms on 10 test images are listed in Table 1, and the FSIM values of the proposed algorithm and all competing algorithms on 10 test images are displayed in Table 2. The noise level is σ = 10, σ = 20, σ = 30, and σ = 50 from top to bottom. An overall impression can be observed from Table 1; when the noise level increases from 20 to 50, the improvements of WPNSEC increase 0.60dB, 0.44dB, and 0.52dB on average, respectively. And the average PSNR of proposed algorithm increases by 0.96 dB, 1.32 dB, 0.66 dB, 0.63 dB, and 0.56 dB over the BM3D, EPLL, NCSR, WNNM, and WSNM. In summary, although the PSNR value decreases with the increase of the standard deviation, most of the PSNR values of the proposed algorithm are higher than those in other related algorithms and the average PSNR is higher than that in all competing algorithms. In order to observe the comparison between our proposed algorithm intuitively and the competing methods on PSNR and FSIM, a line graph of the correlation algorithm is shown in Figure 5, which is drawn by average PSNR (dB) and FSIM of different denoising algorithms on 16 gray images with standard deviation σ=20,30,40,50,60,80. At the same time, in order to reflect the superiority of the WPNSEC algorithm, the average running time of all algorithms is shown in Table 3.Table 1
Denoising PSNR (dB) results of different denoising algorithms under different standard deviationsσ (the highest PSNR values are bold).
σ=10 Image BM3D EPLL NCSR WNNM WSNM WPNSEC Monarch 33.24 33.17 33.21 33.40 33.51 34.93 Couple 32.17 32.08 32.50 32.84 32.93 35.81 House 34.42 34.21 35.16 35.42 35.57 36.83 Lena 32.89 33.16 34.31 34.62 34.81 35.49 Parrot 32.59 32.21 33.65 34.44 34.49 35.75 C. man 32.52 32.55 33.17 33.48 33.62 34.24 Boat 32.57 32.64 32.91 32.89 32.97 33.70 Leaves 31.84 32.17 33.04 33.11 33.35 34.95 Peppers 33.24 33.29 33.27 33.34 33.47 34.67 Airplane 33.32 33.21 33.41 33.52 33.74 34.61 Average 32.88 32.87 33.46 33.71 33.85 35.10  σ=20  Image BM3D EPLL NCSR WNNM WSNM WPNSEC Monarch 30.35 30.48 30.69 31.10 31.13 32.37 Couple 30.76 30.54 30.56 30.82 30.83 32.63 House 33.77 32.98 33.97 34.01 34.05 34.08 Lena 31.60 32.61 32.92 33.12 33.13 32.86 Parrot 29.96 29.97 32.16 30.19 30.21 33.03 C. man 30.28 30.34 30.48 30.75 30.77 31.47 Boat 29.69 30.66 30.74 31.00 31.02 31.26 Leaves 29.08 30.76 30.59 30.74 30.96 31.12 Peppers 31.14 31.17 31.26 31.53 31.55 31.86 Airplane 30.44 32.41 32.58 32.82 32.82 33.06 Average 30.71 31.19 31.60 31.65 31.74 32.26  σ=30  Image BM3D EPLL NCSR WNNM WSNM WPNSEC Monarch 28.36 28.35 28.46 28.91 28.93 28.96 Couple 30.08 28.61 28.57 28.98 29.02 30.68 House 32.08 31.22 32.07 32.52 32.54 32.69 Lena 29.55 31.41 29.43 29.83 31.48 30.08 Parrot 29.13 28.07 30.38 28.33 28.33 30.61 C. man 28.63 28.36 28.58 28.80 28.83 29.31 Boat 29.11 28.89 28.94 29.24 29.26 29.52 Leaves 27.82 27.38 28.14 28.61 28.62 28.64 Peppers 29.28 28.35 31.11 29.48 29.54 30.56 Airplane 27.56 30.41 30.70 30.87 30.92 31.21 Average 29.16 29.11 29.64 29.56 29.75 30.23  σ=50  Image BM3D EPLL NCSR WNNM WSNM WPNSEC Monarch 25.51 25.77 25.78 26.18 26.30 26.32 Couple 26.46 26.23 26.19 27.45 26.71 28.30 House 29.69 28.76 29.62 30.23 30.31 30.29 Lena 27.07 26.42 27.12 27.16 27.28 27.40 Parrot 25.89 25.83 27.88 26.00 26.10 28.23 C. man 26.13 26.03 26.15 26.42 26.44 26.62 Boat 26.78 26.65 26.67 26.97 27.01 26.88 Leaves 24.75 24.36 24.96 25.49 25.54 25.69 Peppers 26.68 26.62 28.07 26.81 26.94 28.77 Airplane 25.10 27.88 28.18 28.44 28.49 28.46 Average 26.41 26.46 27.06 27.12 27.11 27.70Table 2
Denoising FSIM results of different denoising algorithms under different standard deviationsσ (the highest FSIM values are bold).
σ=10 Image BM3D EPLL NCSR WNNM WSNM WPNSEC Monarch 0.9578 0.9564 0.9624 0.9550 0.9567 0.9661 Couple 0.9794 0.9781 0.9818 0.9815 0.9843 0.9897 House 0.9511 0.9462 0.9601 0.9589 0.9590 0.9592 Lena 0.9653 0.9633 0.9819 0.9820 0.9831 0.9872 Parrot 0.9367 0.9364 0.9618 0.9549 0.9563 0.9621 C. man 0.9553 0.9561 0.9553 0.9530 0.9541 0.9590 Boat 0.9723 0.9726 0.9813 0.9814 0.9832 0.9879 Leaves 0.9492 0.9512 0.9730 0.9628 0.9642 0.9743 Peppers 0.9587 0.9573 0.9611 0.9530 0.9543 0.9612 Airplane 0.9396 0.9468 0.9563 0.9515 0.9532 0.9591 Average 0.9565 0.9564 0.9675 0.9634 0.9648 0.9706  σ=20  Image BM3D EPLL NCSR WNNM WSNM WPNSEC Monarch 0.9298 0.9294 0.9320 0.9176 0.9187 0.9389 Couple 0.9573 0.9461 0.9555 0.9532 0.9541 0.9583 House 0.9190 0.9173 0.9201 0.9177 0.9182 0.9202 Lena 0.9541 0.9551 0.9598 0.9601 0.9603 0.9357 Parrot 0.9345 0.9302 0.9348 0.9212 0.9237 0.9356 C. man 0.9097 0.9095 0.9086 0.9096 0.9125 0.9175 Boat 0.9551 0.9537 0.9553 0.9561 0.9574 0.9621 Leaves 0.9527 0.9493 0.9469 0.9296 0.9326 0.9503 Peppers 0.9322 0.9314 0.9327 0.9149 0.9161 0.9341 Airplane 0.8995 0.9172 0.9185 0.9116 0.9142 0.9218 Average 0.9344 0.9339 0.9367 0.9292 0.9308 0.9375  σ=30  Image BM3D EPLL NCSR WNNM WSNM WPNSEC Monarch 0.9013 0.8994 0.9082 0.8898 0.8965 0.9183 Couple 0.9279 0.9273 0.9285 0.9323 0.9344 0.9311 House 0.8991 0.8987 0.9000 0.8930 0.8952 0.9023 Lena 0.9563 0.9531 0.9446 0.9452 0.9449 0.9453 Parrot 0.9194 0.9175 0.9208 0.8999 0.9051 0.9279 C. man 0.8632 0.8651 0.8707 0.8737 0.8739 0.8746 Boat 0.9351 0.9293 0.9309 0.9346 0.9351 0.9390 Leaves 0.9266 0.9185 0.9279 0.9041 0.9184 0.9284 Peppers 0.8996 0.9042 0.9126 0.8897 0.8931 0.9115 Airplane 0.8791 0.8788 0.8897 0.8865 0.8895 0.8967 Average 0.9108 0.9092 0.9130 0.9049 0.9086 0.9175  σ=50  Image BM3D EPLL NCSR WNNM WSNM WPNSEC Monarch 0.8601 0.8552 0.8615 0.8517 0.8573 0.8821 Couple 0.8746 0.8765 0.8800 0.8913 0.8928 0.8922 House 0.8573 0.8493 0.8675 0.8706 0.8741 0.8834 Lena 0.8921 0.8896 0.9191 0.9177 0.9182 0.9210 Parrot 0.8931 0.8779 0.8926 0.8855 0.8860 0.8994 C. man 0.7922 0.7869 0.8091 0.8279 0.8292 0.8350 Boat 0.8876 0.8815 0.8847 0.8954 0.8953 0.8939 Leaves 0.8564 0.8475 0.8865 0.8754 0.8791 0.9011 Peppers 0.8693 0.8531 0.8685 0.8489 0.8504 0.8777 Airplane 0.8360 0.8256 0.8378 0.8420 0.8432 0.8483 Average 0.8619 0.8543 0.8707 0.8705 0.8726 0.8834Table 3
Average run time(s) of all methods on the BSD68 dataset.
Algorithms BM3D EPLL NCSR WNNM WSNM WPNSEC Times 2.67 103.78 374.01 169.32 174.21 89.64Figure 5
The average PSNR (dB) and FSIM results of different denoising algorithms on BSD68 dataset.
(a) (b)In order to display the visual quality of the proposed denoising algorithm, this part uses theleaves test image with σ = 20 to carry out a simulation experiment, and the experimental results are shown in Figure 6. It can be clearly seen from the highlighted red window in Figure 6 that the WPNSEC algorithm recovers the texture of the leaves well, but other competing denoising algorithms contain more noise and produce many artifacts. In Figure 6(c), since the EPLL algorithm ignores the nonlocal self-similarity, the details of the texture and structure edges are seriously lost, which affects the visual perception. The WPNSEC algorithm that is based on similar patches group preserves the image features better while effectively removing the image noise. According to the above analysis, the proposed algorithm has robust denoising performance, which can not only obtain higher PSNR and FSIM indices but also produce better visual quality.Figure 6
Denoising results on imageleaves by different methods (σ = 20). (a) Truth. (b) Noise. (c) BM3D, PSNR=29.08dB. (d) NCSR, PSNR=30.76dB. (e) WNNM, PSNR=30.59dB. (f) EPLL, PSNR=30.74dB. (g) WSNM, PSNR=30.96dB. (h) WPNSEC, PSNR=31.12dB.
(a) (b) (c) (d) (e) (f) (g) (h)In Algorithm1, the size of an image patch is l2×m, wherel represents the size of the image patch yi andm is the number of similar patches in a similar patches group. The constructing dictionary is the main computation in each iteration (Step 5 in Algorithm 1), and it costs O(min{l2m2,l4m}). The generalized soft threshold algorithm (Step 8 in Algorithm 1) only costs O(Km)), whereK represents the number of iterations in the generalized soft threshold algorithm. Therefore, the cost for the proposed WPNSEC image denoising algorithm is K1×S×O(min{l2m2,l4m}+Km), K1 represents the number of iterations of Algorithm 1, andS represents the number of image patches in a sliding window.


## 4.4. Results on Color Noisy Images Denoising

In this subsection, the multichannel WPNSEC method is compared with other competing denoising methods, including CBM3D [38], MLP [39], DnCNN [40], TNRD[41], and MC-WNNM [33], on the 24 color images displayed in Figure 7 (from left to right and from top to bottom: gate, red door, caps, bikes, sailing 1, flower buildings, sailing 4, couple, stream, rapids, girl, ocean, lighthouse 1, house, painted house, parrots, plane, woman hat, sailing 2, sailing 3, statue, woman, and lighthouse 2). In order to obtain the color noise images, additive white Gaussian noise is added to each channel of the clear image, respectively: σr=40, σg=20, and σb=30.Figure 7
The 24 color images from the Kodak PhotoCD Dataset (http://r0k.us/graphics/kodak/).The PSNR values of competing methods on the Kodak PhotoCD dataset are reported in Table4 and the PSNR average values of all methods on the CBSD68 dataset are reflected in Table 5. The highest PSNR values for each color image are bold. Table 4 shows that most of the PSNR values of our proposed algorithm are higher than the PSNR values of other competing methods and that the average value of the PSNR is the highest. The improvements of multichannel WPNSEC achieve 1.84dB, 0.43dB, 0.29dB, 8.39dB, and 0.27dB over CBM3D, MLP, TNDR, DnCNN, and MC-WNNM on average, respectively. At the same time, in order to reflect the high efficiency of the proposed algorithm, the average running time of each algorithm is shown in Table 6.Table 4
Denoising PSNR (dB) results of different denoising algorithms on color images (the highest PSNR values are bold).
σ r = 40 σ g = 20 σ b = 30 Image CBM3D MLP TNRD DnCNN MC-WNNM Ours Gate 25.24 25.70 25.74 20.47 25.83 26.13 Red door 28.27 30.12 30.21 20.47 29.75 30.19 Caps 28.81 31.19 31.49 20.53 31.54 31.64 Woman hat 27.95 29.88 29.86 20.47 29.94 30.37 Bikes 25.03 26.00 26.18 20.52 25.74 26.52 Sailing 1 26.24 26.84 26.90 20.66 27.00 27.42 Flower 27.88 30.28 30.40 20.52 30.53 30.47 Buildings 25.05 25.59 25.83 20.57 25.95 26.32 Sailing 2 28.44 30.75 30.81 20.50 30.95 31.04 Sailing 3 28.27 30.38 30.57 20.52 30.52 30.83 Sailing 4 26.95 28.00 28.14 20.52 28.23 28.43 Couple 28.76 30.87 31.05 20.60 31.12 31.41 Stream 23.76 23.95 23.99 20.52 24.01 24.63 Rapids 26.02 26.97 27.11 20.51 27.03 27.21 Girl 28.38 30.15 30.44 20.71 30.52 30.76 Ocean 27.75 28.82 28.87 20.52 29.00 29.27 Statue 27.90 29.57 29.80 20.56 29.60 29.76 Woman 25.77 26.40 26.41 20.53 26.52 26.73 Lighthouse 1 27.30 28.67 28.81 20.53 28.89 29.34 Plane 28.96 30.40 30.76 21.44 30.80 30.92 Lighthouse 2 26.54 27.53 27.60 20.51 27.59 27.91 House 27.05 28.17 28.27 20.51 28.00 28.27 Parrots 29.14 32.31 32.51 20.54 32.06 32.19 Painted house 25.75 26.41 26.53 20.59 26.64 27.42 Average 27.13 28.54 28.68 20.58 28.70 28.97Table 5
The average PSNR (dB) results of different denoising algorithms on the CBSD68 dataset.
Algorithms CBM3D MLP TNRD DnCNN MC-WNNM Ours Average 26.28 27.69 27.93 19.74 28.05 28.22Table 6
Average run time(s) of all methods on the CBSD68 dataset.
Algorithms CBM3D MLP TNRD DnCNN MC-WNNM Ours Times 3.69 3.92 - 190.73 1127.91 132.47The denoised images of our proposed algorithm and all competing methods on color imagebuildings are shown in Figure 8. In order to observe the details of the image more clearly, a red window is used to select a partial block of the image and zoom in it. Figure 8 shows that the denoising quality of the multichannel WPNSEC algorithm is better than other algorithms. DnCNN, MLP, and CBM3D can preserve as much edge information as possible when removing the noise, but these algorithms produce some distortion and artifacts. For example, in Figures 8(c), 8(d), and 8(f), there are a lot of noises on the windows of the house and a large amount of artifacts in the smooth area. TNRD method reduces noise and artifacts, so it has better denoising effect than DnCNN, MLP, and CBM3D. Due to the oversmoothing effect in the TNRD method, some boundaries and details are lost. For instance, in Figure 8(e), the red mark in the lower left corner of the red window is unclear. MC-WNNM is a state-of-the-art image denoising algorithm, but it is constructed based on convex optimization, so the sparse coefficients are a lack of robustness. Although the MC-WNNM algorithm performs well on denoising quality, it is not as good as the multichannel WPNSEC algorithm. As can be seen from Figure 8(g), the details of the letters and the red mark in the lower left window are slightly lost. In Figure 8(h), the edge and detail information can be well protected, the background is smoother, and it gets a better visual experience. Although our proposed algorithm does not achieve a clear representation of all the details (such as the red mark), it shows the best visual effect in the competing methods.Figure 8
Denoising results on imagebuildings under different methods. (a) Truth. (b) Noise. (c) CBM3D, PSNR=25.03dB. (d) MLP, PSNR=26.00dB. (e) TNRD, PSNR=26.18dB. (f) DnCNN, PSNR=20.52dB. (g) MC-WNNM, PSNR=26.95dB. (h) Ours, PSNR=26.52dB.
(a) (b) (c) (d) (e) (f) (g) (h)In Algorithm2, the size of a color image patch is 3l2m, wherel represents the size of the image patch yic andm is the number of similar patches in a similar patches group. UpdatingZ costs O(max{l4m,m3}) and updating α costs O(l4m+m3). The cost for updatingb and ρ can be ignored. Therefore, the cost of the ADMM algorithm is O((l4m+m3)K2), where K2 represents the number of iterations of Algorithm 2. In Algorithm 3, for multichannel color image denoising, the overall cost is O((l4m+m3)K2K3S), where K3 represents the number of iterations of Algorithm 3 andS represents the number of image patches in a sliding window.


## 5. Conclusion

Due to the influence of noise, there is a sparse error between the estimated sparse coefficient and the true sparse coefficient. The sparse error makes the image denoising more challenging. Aiming at this problem, a weightedlp norm sparse error constraint (WPNSEC) framework for image denoising is proposed. The image denoising is transformed into the problem reducing sparse error and finding the optimal sparse coefficient. Specifically, the suitability setting of powerp in the lp norm is analyzed in detail. In addition, the WPNSEC framework is extended to color image denoising, and a multichannel WPNSEC model is proposed. The proposed model is solved by the ADMM algorithm. Considering that the noise of the RGB channels has different statistics, a weight matrix is introduced. The experiment results of gray image denoising and color image denoising demonstrate that the WPNSEC algorithm and the multichannel WPNSEC algorithm outperform the current state-of-the-art denoising methods.



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*Source: 1262171-2019-05-09.xml*