CN109671029A

CN109671029A - Image denoising algorithm based on gamma norm minimum

Info

Publication number: CN109671029A
Application number: CN201811492110.9A
Authority: CN
Inventors: 王洪雁; 王拓; 张莉彬
Original assignee: Dalian University
Current assignee: Dalian University
Priority date: 2018-12-07
Filing date: 2018-12-07
Publication date: 2019-04-23
Anticipated expiration: 2038-12-07
Also published as: CN109671029B

Abstract

The invention belongs to digital image processing fields, are related to a kind of Image denoising algorithm based on gamma norm minimum.The algorithm is first to noise image overlap partition, structural similarity index adaptable search and the most like several non local image blocks of current image block are then based on to form similar image block matrix, and then using the unbiased approximate matrix rank function of non-convex gamma norm to construct low-rank denoising model, optimization problem solving is finally denoised to gained low-rank based on singular value decomposition, and denoising image block is reassembled as denoising image.Simulation result show with existing PID, NLM, BM3D with NNM algorithm compares, and the mentioned algorithm of the present invention can effectively eliminate Gaussian noise, and can preferably restore original image details.

Description

Image denoising algorithm based on gamma norm minimization

Technical Field

The invention belongs to the field of digital image processing, and particularly relates to an image denoising algorithm based on gamma norm minimization.

Background

Digital images are inevitably contaminated by noise during acquisition and transmission, resulting in imagesLoss of detail and degradation of quality, which in turn affects subsequent image processing. The purpose of image denoising is to recover the original image x from the noise image y as accurately as possible and to retain important detail features such as edges and textures. The degradation model of the denoising problem can be expressed as: y x + v, where v is generally assumed to be a mean of 0 and a variance of σ_n ²White gaussian noise. Due to the unsuitability of the image denoising problem, denoising by using the priori knowledge representing the statistical characteristics of the image is very important.

In recent years, numerous image noise reduction algorithms have been proposed in succession, which can be roughly classified into the following two categories: based on a local prior method and based on a non-local Self-Similarity (NSS) prior method. The local prior-based method mainly comprises a Wavelet Shrinkage method (WS), a Total Variation method (TV) and an improved Image Denoising method (PID) based on an annealing algorithm. However, these methods only consider local a priori knowledge, neglect similarity a priori information between non-local image blocks, and thus result in poor denoising performance of the algorithm. The image non-local similarity prior refers to: a certain local image block in a given image is searched for a large number of image blocks similar to the given image block by utilizing the linear correlation characteristic among non-local similar blocks in the image, and practice shows that NSS becomes very effective prior information in an image recovery task. Based on this, Buades a et al propose a non-local mean denoising (NLM) algorithm, which can obtain better denoising performance by weighted averaging of similar image blocks, but when the noise level is higher, the similar image blocks contain a large amount of noise, so that the denoised image has a large-area fuzzy phenomenon. Aiming at the problem, Dabov K proposes a three-dimensional Block Matching (BM 3D Filtering, 3D) algorithm, which constructs 2D image blocks with similar structures as three-dimensional data and then performs joint denoising to improve denoising performance, but the algorithm application is limited by high time complexity. Based on this, Dong W et al propose a Non-local centralized sparse Representation (NCSR) algorithm, construct similar image blocks as a matrix, and improve denoising performance by using a group sparse method. However, the denoised image obtained by the algorithm has the problems of strong artifact phenomenon, fuzzy key details and the like. In response to the above problems, researchers have noted that: the non-local similar image blocks are constructed into a matrix in a vector form, and the matrix has low-rank characteristics and sparse singular values, so that the denoising performance can be improved by using low-rank prior information. Based on this, Ji H et al propose a Nuclear Norm Minimization (NNM) algorithm, which utilizes a Nuclear Norm approximation matrix rank function to construct a low-rank denoising model, thereby obtaining better denoising performance. However, the singular values are processed in a balanced mode, the prior knowledge that the singular values reflect image textures, details and the like to different degrees is ignored, and therefore the improvement of the denoising performance is limited. To address this problem, Gu S et al propose a Weighted Nuclear Norm Minimization (WNNM) algorithm that utilizes a Weighted Nuclear Norm approximation matrix rank function to improve denoising performance by assigning different weights to singular values. However, the use of the kernel norm to approximate the matrix rank function excessively penalizes large singular values, so that the denoising problem can only obtain a suboptimal solution.

Disclosure of Invention

In order to improve the image denoising performance under the Gaussian noise condition, the invention provides an image denoising algorithm based on gamma norm minimization based on a low-rank theory.

The method comprises the following steps of firstly, overlapping and blocking a noise image, constructing a similar image block matrix by adopting a block matching algorithm based on SSIM, then constructing a low-rank denoising model by utilizing a non-convex gamma norm of an approximate unbiased approximate rank function, finally solving the obtained non-convex problem based on singular value decomposition, and recombining the denoised image blocks to obtain the denoised image. The technical scheme of the invention is as follows:

1. establishing a low-rank denoising model

The low-rank denoising method is one of the research hotspots in the current image denoising field, and the principle thereof can be described as follows: of size M x NThe noisy image y is overlappingly divided into n sizes ofImage block y_i1, 2.., n. Then searching the window with the size of L multiplied by L and the current image block y_iM most similar image blocks are constructed into a similar image block matrix Y in the form of column vectors_i∈R^d×mI.e. Y_i＝(y_i,1,y_i,2,...,y_i,m)，y_i,mRepresenting the current image block y_iIs similar to the image block. Based on this, the low-rank denoising problem can be expressed as the following optimization problem:

wherein ,Y_iFor a matrix of noise-like image blocks, X_iTo denoise the similar image block matrix, | ·| non-woven_FIs Frobenius norm, rank (X)_i) Is a matrix X_iλ is a regularization parameter to balance the regularization term and the fidelity term.

It should be noted that the above-mentioned rank minimization problem is an NP-hard problem, and it is very difficult to directly solve the problem. To address this problem, Gu S et al propose a kernel norm minimization algorithm, which converts the non-convex optimization problem into a convex optimization problem using a kernel norm approximation matrix rank function, thereby making the optimization problem (1) easy to solve, which can be expressed as follows:

wherein ,||X_i||_*Is a matrix X_iThe nuclear norm of (d). The solution to problem (2) can be expressed as:

wherein ,Y_iIs decomposed into Y_i＝PΛQ^T，S_λ(Λ) is a soft threshold operator on the diagonal matrix Λ and the parameter λ, defined as:

S_λ(Λ)_jj＝max(Λ_jj-λ,0) (4)

and executing the above operations on each overlapped image block, and reconstructing the original image by recombining all the image blocks. However, the kernel norm is a biased estimator, and the approximate matrix rank function may excessively penalize a larger singular value of the matrix, so that the kernel norm minimization problem cannot obtain an optimal solution, and the denoising performance is reduced. Aiming at the problem, the non-convex gamma norm is adopted to replace the nuclear norm so as to obtain the approximate unbiased estimation of the rank function, and further improve the performance of the low-rank denoising model.

2. Adaptive similar image block search

The traditional searching method for the non-local similar image blocks based on the Euclidean distance does not consider the structural similarity of the image blocks, so that the non-local similar image blocks are not searched accurately. In order to improve the searching accuracy of similar image blocks, the invention provides a self-adaptive similar image block searching method based on a structural similarity index. The Structural Similarity Index (SSIM) is a comprehensive image similarity evaluation Index, which considers three different characteristics of brightness, contrast and structure between images and can better evaluate the similarity degree of the two images. Given two images x and y, the SSIM is defined as shown in equation (5):

wherein ,μ_x，μ_y，σ_x ²，σ_y ²Mean and variance, σ, of the images x, y, respectively_xyIs the covariance of the images x, y. Furthermore, C₁＝(k₁L)²，C₂＝(k₂L)²To ensure that the denominator is notConstant of zero, L255 is the maximum value of the pixel, k₁＝0.01，k₂0.03 is the default constant.

The main idea of the self-adaptive similar image block searching method is as follows: given a current image block y_iAnd the target data set is used for calculating the structural similarity indexes of the current image block and each image block of the target data set, wherein the larger the value of the structural similarity indexes is, the more similar the two image blocks are, and then m image blocks which are most similar to the current image block are searched. Wherein, the number m of similar image blocks needs to be determined adaptively according to the noise level. Then, each similar image block is converted into a column vector and is sequentially arranged from left to right according to the descending order of similarity to form a similar image block matrix Y_iThe construction process is shown in fig. 2.

In summary, the provided similar image block searching method can fully utilize the non-local self-similar prior information of the image to improve the searching accuracy of the similar image block. Based on the obtained similar image block matrix Y_iThe image denoising model based on gamma norm minimization is as follows.

3. Establishing a gamma norm minimization model

The gamma norm is a matrix extension of the non-convex MCP function, which can approximate the rank function nearly unbiased compared to the kernel norm of the biased estimate. Let singular value decomposition of matrix X be X ═ U Σ V^TWherein U is [ U ═ U₁,u₂,...,u_n]，V＝[v₁,v₂,...,v_n]，Σ＝diag(λ₁,λ₂,...,λ_n) And λ₁≥λ₂≥...≥λ_n≧ 0, then the gamma norm may be defined as follows:

wherein ,[x]₊＝max(x,0)。

Based on the characteristic that the gamma norm can approximate a rank function almost unbiased, the invention uses the low-rank term in the alternative formula (1) to construct a non-convex gamma norm minimum denoising model, which can be expressed as follows:

wherein ,Y_iFor the matrix of noise-like image blocks,to denoise the similar image block matrix, | X_i||_γIs a matrix X_iGamma norm of (d).

4. Proposed non-convex model solution

For convenience of description, let X be X_i，Y＝Y_iThen the solution to the optimization problem (8) can be derived as follows:

through simple matrix operations, equation (9) can be re-expressed as:

fixed u_i and v_iFor y (X) with respect to λ_iDifferentiating to obtain:

order toEquation (11) can be expressed as:

finishing to obtain:

wherein ,S_λ,γ(. cndot.) is a non-convex soft shrink operator. Will be provided withCan be substituted by the formula (10):

order toSince the first term at the right end of the above equation is independent of the variable X to be optimized, then y (X) the minimization problem with respect to X is equivalent toAbout ξ_iThe minimization problem of (2).Can be re-expressed as follows:

wherein ,when lambda is less than ξ_iWhen the gamma value is less than or equal to gamma,can be expressed as:

when ξ_iWhen the alpha value is larger than gamma value,can be expressed as:

to pairAbout ξ_iDifferentiating to obtain:

due to the fact thatTo make it possible toAbout ξ_iTake the minimum value u_i and v_iThe left and right singular vectors corresponding to the minimum singular value of the matrix Y are respectively taken, and ξ is obtained_i＝σ_i. Thus, the optimal solution of the gamma norm minimization problem (8) can be expressed as follows:

wherein ,Y＝U₁Σ₁V₁ ^TIs a singular value decomposition of the matrix Y, sigma₁＝diag(σ₁,σ₂,...,σ_n)，Σ_λ,γ＝diag(S_λ,γ(σ₁),S_λ,γ(σ₂),...,S_λ,γ(σ_n) ) is a diagonal matrix whose diagonal elements are:

the optimization problem is solved for each overlapped image block in sequence, and then all the denoised similar image block matrixes { X can be obtained_i1, 2.., n, and reconstructing each denoised image block into a denoised image x. In practical application, the steps can be repeatedly iterated to obtain better denoising performance.

The process of the image denoising algorithm based on the gamma norm minimization comprises the following steps:

inputting: noisy image y

1: initialization:y⁽⁰⁾＝y

2：for k＝1:K

3: iterative regularization:

4：for y^(k)of each overlapping image block y_i

5: similar image block matrix Y constructed based on SSIM image block searching method_i

6: for Y_iSingular value decomposition: [ U, Σ, V)]＝SVD(Y_i)

7: evaluating an estimation value based on the model of formula (8):

8：end for

9: polymerization of X_iTo reconstruct a denoised image

10：end for

And (3) outputting: de-noised image

And 3, the iterative regularization process is used for improving the denoising performance of the algorithm.

The invention has the beneficial effects that: the method disclosed by the invention has the advantages that the similar image blocks are searched by utilizing the structural similarity index to avoid the defect of inaccurate searching of the similar image blocks in the traditional method, the gamma norm is approximately unbiased to approximate the matrix rank function to construct the low-rank denoising model, the problem of poor estimation precision of the traditional matrix rank function approximation method is solved, and the non-convex problem is solved based on singular value decomposition. Simulation results show that compared with the existing PID, NLM, BM3D and NNM algorithms, the algorithm can obviously eliminate Gaussian noise, better avoid artifacts and detail blurring phenomena, and can restore original image details with higher precision.

Drawings

FIG. 1 is a flow chart of an implementation of the present invention;

FIG. 2 is a similar image block matrix construction process;

FIG. 3 is six test images;

FIG. 4 shows the denoising results (σ) of the House image by different algorithms_n50). (a) Noise image (b) PID (c) NLM (d) BM3D (e) NNM (f) algorithm;

FIG. 5 shows the denoising results (σ) of Boats images by different algorithms_n100). (a) Noise image (b) PID (c) NLM (d) BM3D (e) NNM (f).

Detailed Description

The invention is further described with reference to the following drawings and detailed description.

The specific steps of the image denoising algorithm based on the gamma norm minimization are as follows:

1. establishing a low-rank denoising model

The principle of the low-rank denoising method can be described as follows: dividing a noise image y of size M × N into N noise images of size M × NImage block y_i1, 2.., n. Then searching the window with the size of L multiplied by L and the current image block y_iM most similar image blocks are constructed into a similar image block matrix Y in the form of column vectors_i∈R^d×mI.e. Y_i＝(y_i,1,y_i,2,...,y_i,m)，y_i,mRepresenting the current image block y_iIs similar to the image block. Based on this, the low-rank denoising problem can be expressed as the following optimization problem:

2. Adaptive similar image block search

The Structural Similarity Index (SSIM) is a comprehensive image Similarity evaluation Index, which can evaluate the Similarity of two images by considering three different characteristics of brightness, contrast and structure between the images, and given two images x and y, the SSIM is defined as formula (5):

wherein ,μ_x，μ_y，σ_x ²，σ_y ²Mean and variance, σ, of the images x, y, respectively_xyIs the covariance of the images x, y. Furthermore, C₁＝(k₁L)²，C₂＝(k₂L)²To ensure a constant with a denominator that is not zero, L255 is the maximum value of the pixel, k₁＝0.01，k₂0.03 is the default constant.

The main scheme of the self-adaptive similar image block search is as follows: given a current image block y_iAnd the target data set is used for calculating the structural similarity indexes of the current image block and each image block of the target data set, wherein the larger the value of the structural similarity indexes is, the more similar the two image blocks are, and then m image blocks which are most similar to the current image block are searched. Wherein, the number m of similar image blocks needs to be determined adaptively according to the noise level. Then, each similar image block is converted into a column vector and is sequentially arranged from left to right according to the descending order of similarity to form a similar image block matrix Y_iThe construction process is shown in fig. 2.

3. Establishing a gamma norm minimization model

wherein ,[x]₊＝max(x,0)。

4. Proposed non-convex model solution

through simple matrix operations, equation (9) can be re-expressed as:

fixed u_i and v_iFor y (X) with respect to λ_iDifferentiating to obtain:

order toEquation (11) can be expressed as:

finishing to obtain:

when ξ_iWhen the alpha value is larger than gamma value,can be expressed as:

to pairAbout ξ_iDifferentiating to obtain:

wherein ,is a singular value decomposition of the matrix Y, sigma₁＝diag(σ₁,σ₂,...,σ_n)，Σ_λ,γ＝diag(S_λ,γ(σ₁),S_λ,γ(σ₂),...,S_λ,γ(σ_n) ) is a diagonal matrix whose diagonal elements are:

inputting: noisy image_y

1: initialization:y⁽⁰⁾＝y

2：for k＝1:K

3: iterative regularization:

4：fory^(k)of each overlapping image block y_i

6: for Y_iSingular value decomposition: [ U, Σ, V)]＝SVD(Y_i)

7: evaluating an estimation value based on the model of formula (8):

8：end for

9: polymerization of X_iTo reconstruct a denoised image

10：end for

And (3) outputting: de-noised image

The beneficial effects of the present invention can be further illustrated by the following experiments:

the experimental conditions are as follows:

to verify the denoising performance of the proposed algorithm, six images in a standard image library were selected for testing, which are 256 × 256Monarch, 720 × 576Boats, 512 × 512Lena, 512 × 512 peper, 256 × 256House and 720 × 576Barbara images, respectively, as shown in fig. 3. PID, NLM, BM3D and NNM algorithms are adopted as comparison algorithms, and the comparison algorithms are compared with the proposed algorithms respectively from two aspects of denoising performance and running time. The experimental environment is as follows: the processor Intel Core i7-7700, the main frequency 3.60GHz, the memory 8GB, the operating system 64 bits Windows 10, the simulation software Matlab R2014 a.

In order to quantitatively evaluate the denoising performance of the proposed algorithm, the invention adopts Peak Signal to Noise Ratio (PSNR) defined by Ghanbari M and the like as an evaluation index. Given two images x and y, the PSNR is defined as shown in equation (38):

where MSE represents the Mean Square Error (Mean Square Error) and mxn is the size of the two images.

According to the experiment, the search window size L × L is set to 30 × 30, and the image block sizeThe number m of similar image blocks and the iteration number K of the algorithm need to be determined in a self-adaptive mode according to the noise level. For the noise variance σ_n≤20，20＜σ_n≤40，40＜σ_n≤60 and σ_nNoise image > 60, the image block sizes are set to 6 × 6, 7 × 7, 8 × 8 and 9 × 9, respectively, and the number of similar image blocks m is set to 70, 90, 120 and 140, respectively. Accordingly, the number of iterations K is set to 8, 12, 14, and 14, respectively. The parameters δ, λ and γ are set to 0.1, 0.001 and 100, respectively.

The experimental contents are as follows:

experiment 1: zero-mean white gaussian noise is added to the six test images to generate a noise image. Due to space limitation, the invention only displays the denoising results of the four noise level algorithms. Wherein for the noise variance σ_n＝10，σ_n＝30，σ_n＝50 and σ_nThe different algorithm denoise results are shown in table 1, with the experimental peak PSNR shown in bold, 100 gaussian white noise. As shown in Table 1, the denoising performance of the NLM algorithm is not obvious, and the proposed algorithm can obtain higher PSNR in almost every case. Compared with PID, NLM, BM3D and NNM algorithms, the PSNR of the algorithm is respectively and averagely improved by 0.32dB, 2.76dB, 0.35dB and 1.99dB, and the PSNR is changed along with the improvement of the noise levelIs better and more remarkable. The above phenomenon is because the proposed algorithm improves similar image block search accuracy by using SSIM for block matching, and improves matrix rank function approximation accuracy based on non-convex gamma norm. Therefore, compared with the other four algorithms, the algorithm has better denoising performance under different noise levels.

TABLE 1 PSNR comparison of different denoising algorithms

Experiment 2: for the noise variance σ_n50 House image and noise variance σ_nThe denoising results of different algorithms are shown in fig. 4 and 5, respectively, for a Boats image of 100. As can be seen from the figure, the NLM denoised image has a large-area fuzzy phenomenon and a poor denoising effect, which is caused by the fact that a large amount of noise exists in the similar image blocks searched by the NLM algorithm and only the weighted average similar image blocks are used for denoising. The NNM algorithm can effectively avoid the phenomenon of large-area blurring of images by using matrix low-rank prior, but the NNM algorithm can balance and process each singular value, so that the denoising images have the problems of blurring and losing key details and the like, for example, part of masts of a ship in FIG. 5(e) are obviously lost. The PID algorithm utilizes a spatial domain and frequency domain combined processing and annealing algorithm, so that the problem of loss of key details of a denoised image is effectively solved, and the detail blurring phenomenon still exists. The BM3D algorithm obtains a good denoising effect by using a joint filtering method, but the denoised image has a strong artifact phenomenon. The provided algorithm adopts a similar image block searching method based on a structural similarity index and utilizes a non-convex gamma norm approximate to an unbiased approximate matrix rank function to overcome the defects, so that the denoising performance of the algorithm is improved. As can be seen from fig. 4(f) and fig. 5(f), the artifact phenomenon of the de-noised image of the proposed algorithm is weak, and key detail features such as edges and textures can be better recovered. In conclusion, under the condition of different levels of Gaussian noise, the denoising performance of the algorithm is remarkably improved compared with other four algorithms, the phenomena of blurring and artifacts of a denoised image can be effectively avoided, and the key detail information of the original image can be well recovered.

Experiment 3: the time complexity is an important index for evaluating the running speed of the algorithm. From theoretical analysis, the time complexity of the PID algorithm is wherein ,σ_sStandard deviation of space domain Gaussian function; the time complexity of NLM algorithm is O (MNL)²d) (ii) a The time complexity of the BM3D algorithm isThe time complexity of NNM and proposed algorithm is O (K (MNL)²d+MN²)). From the above analysis, it can be seen that NNM and the proposed algorithm are time consuming compared to the other three algorithms due to singular value decomposition and multiple iterative operations. Six test images in the standard image library were selected for the evaluation of the average run times of the different algorithms on the above-mentioned computing platform, and the test results are shown in table 2. As can be seen from the table, the BM3D algorithm runs much faster than the other four algorithms, and the average time consumption is short. The main reasons are that the frequency domain discrete Fourier transform operation adopted by the PID algorithm, the image block extraction and similar image block search operation adopted by the NLM, the NNM and the proposed algorithm are time-consuming, and the NNM and the proposed algorithm have long operation time because of singular value decomposition and multiple iterative denoising operations on a similar image block matrix. However, although the BM3D algorithm includes image block extraction and similar image block search operations, the algorithm is significantly accelerated due to the optimized code. In addition, the running time of the algorithm is similar to that of the NNM algorithm, but the average time consumption is longer than that of the NNM algorithm due to the adoption of the similar image block searching method based on the structural similarity index. However, the benefit obtained by sacrificing the operation speed is that the denoising performance of the proposed algorithm is outstanding under different noise levels, not only gaussian noise can be better eliminated, but also artifacts and detail blurring phenomena can be better avoided.

Table 2 average run time comparison of different algorithms

Algorithm	PID	NLM	BM3D	NNM	Proposed algorithm
						Mean time(s)	307	132	3	462	473

In summary, the invention provides an image denoising algorithm based on gamma norm minimization, aiming at the problems of artifacts and detail blurring of denoised images under the Gaussian noise condition, based on a non-convex gamma norm which can approximate to an unbiased approximate rank function and a low-rank denoising model. Firstly, the noise image is overlapped and partitioned, and a similar image block matrix is constructed by adopting a block matching algorithm based on SSIM, so that the defect that similar image block searching is inaccurate in the traditional method is avoided. And then, constructing a low-rank denoising model by using a non-convex gamma norm which is approximate to an unbiased approximate rank function, thereby solving the problem of poor estimation accuracy of the traditional rank function approximation method. And finally, solving the obtained non-convex problem based on singular value decomposition and recombining the de-noised image blocks to obtain a de-noised image. Simulation results show that compared with the existing mainstream algorithm, the algorithm can better eliminate Gaussian noise and can better avoid artifacts and detail blurring phenomena.

Claims

1. The image denoising algorithm based on the gamma norm minimization is characterized by comprising the following steps of:

firstly, establishing a low-rank denoising model

The principle of the low-rank denoising method can be described as follows: dividing a noise image y of size M × N into N noise images of size M × NImage block y_iI 1, 2.. times.n, and then searching for the current image block y in a window of size lxl_iM most similar image blocks are constructed into a similar image block matrix Y in the form of column vectors_i∈R^d×mI.e. Y_i＝(y_i,1,y_i,2,…,y_i,m)，y_i,mRepresenting the current image block y_iThe m-th similar image block of (b),

based on this, the low-rank denoising problem can be expressed as the following optimization problem:

wherein ,Y_iFor a matrix of noise-like image blocks, X_iTo denoise the similar image block matrix, | ·| non-woven_FIs Frobenius norm, rank (X)_i) Is a matrix X_iλ is a regularization parameter to balance the regularization term and the fidelity term;

adaptive similar image block search

wherein ,μ_x，μ_y，σ_x ²，σ_y ²Mean and variance, σ, of the images x, y, respectively_xyIs the covariance of the image x, y, and, in addition, C₁＝(k₁L)²，C₂＝(k₂L)²To ensure a constant with a denominator that is not zero, L255 is the maximum value of the pixel, k₁＝0.01，k₂0.03 is the default constant;

the main scheme of the self-adaptive similar image block search is as follows: given a current image block y_iAnd a target data set, calculating the structural similarity index of the current image block and each image block of the target data set, wherein the larger the value of the structural similarity index is, the larger the structural similarity index is, the structural similarity index isThe two image blocks are more similar to each other, and then m image blocks most similar to the current image block are searched, wherein the number m of the similar image blocks needs to be determined in a self-adaptive manner according to the noise level, then all the similar image blocks are converted into column vectors, and the column vectors are sequentially arranged from left to right according to the descending order of the similarity to form a similar image block matrix Y_i；

Thirdly, establishing a gamma norm minimization model

The gamma norm is a matrix extension of the non-convex MCP function, which can approximate the rank function almost unbiased compared to the kernel norm with biased estimation, assuming that the singular value of matrix X is decomposed into X ═ U Σ V^TWherein U is [ U ═ U₁,u₂,…,u_n]，V＝[v₁,v₂,...,v_n]，Σ＝diag(λ₁,λ₂,...,λ_n) And λ₁≥λ₂≥...≥λ_n≧ 0, then the gamma norm may be defined as follows:

wherein ,[x]₊＝max(x,0)；

Based on the characteristic that the gamma norm can approximate to a rank function almost unbiased, a non-convex gamma norm minimization denoising model is constructed by using a low-rank term in the alternative expression (1), and the non-convex gamma norm minimization denoising model can be expressed as follows:

wherein ,Y_iFor the matrix of noise-like image blocks,to denoise the similar image block matrix, | X_i||_γIs a matrix X_iThe gamma norm of (d);

solving the proposed non-convex model

For convenience of description, let X be X_i，Y＝Y_iThen the solution to the optimization problem (5) can be derived as follows:

through simple matrix operations, equation (6) can be re-expressed as:

fixed u_i and v_iFor y (X) with respect to λ_iDifferentiating to obtain:

order toEquation (8) can be expressed as:

finishing to obtain:

wherein ,S_λ,γ(. is a non-convex soft shrink operator, willCan be substituted by the formula (7):

order toSince the first term at the right end of the above equation is independent of the variable X to be optimized, then y (X) the minimization problem with respect to X is equivalent toAbout ξ_iThe problem of minimization of (a) is,can be re-expressed as follows:

when ξ_iWhen the alpha value is larger than gamma value,can be expressed as:

to pairAbout ξ_iDifferentiating to obtain:

due to the fact thatTo make it possible toAbout ξ_iTake the minimum value u_i and v_iThe left and right singular vectors corresponding to the minimum singular value of the matrix Y are respectively taken, and ξ is obtained_i＝σ_i，

Thus, the optimal solution of the gamma norm minimization problem (5) can be expressed as follows:

the optimization problem is solved for each overlapped image block in sequence, and then all the denoised similar image block matrixes { X can be obtained_iAnd (4) reconstructing each denoised image block into a denoised image x, wherein the steps can be repeatedly iterated in practical application.

2. The image denoising algorithm based on gamma norm minimization of claim 1, wherein the algorithm flow is as follows:

inputting: noisy image y

1: initialization:

2：for k＝1:K

3: iterative regularization:

4：for y^(k)of each overlapping image block y_i

6: for Y_iSingular value decomposition: [ U, Σ, V)]＝SVD(Y_i)

7: evaluating an estimation value based on the model of the formula (5):

8：end for

9: polymerization of X_iTo reconstruct a denoised image

10：end for

And (3) outputting: de-noised image