Image denoising algorithm based on weighted lp norm combined image group sparsity and low rank

Abstract

The invention discloses an image denoising algorithm based on weighted lp norm combined image group sparsity and low-rank performance, and belongs to the field of image denoising. According to the algorithm, image denoising is realized by combining two directions of image low-rank performance and group sparsity, and non-local group sparsity and low-rank performance of the image can be considered at the same time, so that image denoising is more flexible and efficient. When the local structure of the image is expressed, the adaptive dictionary learning method has better adaptability and robustness than the traditional over-complete dictionary learning method, and the structure information of the image can be better expressed. A weighted lp norm is used for replacing an l1 norm in a previous algorithm, so that an optimal solution can be better obtained, and algorithm convergence is accelerated. And meanwhile, an image denoising task is well completed by using adaptive parameter setting and an alternating direction multiplier method. Experimental results show that the algorithm of the invention has good convergence, shows good performance in the aspects of PSNR and SSIM, and is superior to many classic and currently popular de-noising algorithms.

Description

Technical Field

[0001] The present invention belongs to the field of image denoising, and particularly relates to a method based on weighted p A denoising algorithm that combines the sparsity and low rank properties of image groups. Background Art

[0002] Image processing is one of the most important branches in the field of signal processing, and image denoising is an important task in image processing. Image denoising plays an important role in aerospace, medicine, geology and other fields. It lays the foundation for further image processing such as image recognition and image classification. Usually, the image noise model is defined as y=x+n, where x represents a clean and noise-free image, and n represents the noise model. Generally, it is assumed that n has a mean of 0 and a variance of σ. 2 The image denoising algorithm is a Gaussian white noise (AWGN), and y represents the noisy image, i.e., the input image of the denoising algorithm. The purpose of image denoising is to recover x from y. In the past few decades, the field of image denoising has flourished, and image denoising algorithms have emerged in an endless stream. Most of the current algorithms for solving the image denoising problem are based on image prior information. By finding appropriate image prior information to model the image denoising problem, image sparsity prior plays a vital role in the current image processing field. The traditional group sparsity (GSC) algorithm only imposes simple sparsity on the group coefficients, which has certain limitations on image modeling, making the image denoising effect not obvious enough. Summary of the invention

[0003] The purpose of the present invention is to overcome the shortcomings of the above background and provide a method based on weighted l p The image denoising algorithm based on the norm and the image group sparsity and low rank overcomes the defects of low precision, poor robustness and complicated manual parameter adjustment of traditional image denoising algorithms. In addition, compared with other traditional denoising algorithms, the denoising index results of the present invention are better. In order to achieve the above technical objectives, the technical solution of the present invention is as follows:

[0004] A weighted l p The image denoising algorithm that combines the image group sparsity and low rank mainly achieves the purpose of image denoising by modeling the group sparsity and low rank of image blocks. If the low rank and sparse constraints are directly imposed on the matrix representation A of the image block through the norm, the flexibility of the model will be limited, which will make the problem difficult to solve. Therefore, the variable B is introduced to relax the constraints. i is the sparse matrix A that combines each group i The obtained low-rank approximation.

[0005] The traditional l0 norm minimization is an NP-hard optimization problem, so it is usually relaxed to a convex l1 norm minimization problem. However, in some image restoration problems, the l1 norm minimization problem can only be regarded as an estimate of the l0 norm minimization, which cannot obtain an ideal sparse solution. p The minimization of the norm constrains the image sparse solution. For the matrix A, the weighted l p The norm is expressed as:

[0006]

[0007] Among them, 0 <p≤1,σ i is the i-th singular value of A, w i ≥0 is applied to σ i A method based on weighted l p The denoising model of the image denoising algorithm that combines the sparsity and low rank of the image group is:

[0008]

[0009] The denoising model is not easy to solve. If the denoising model is solved directly, it is difficult and cumbersome. Therefore, this study divides the model into two parts: i and A i The sub-problems of are optimized respectively to realize the solution of the algorithm. Regarding the sub-problem of x, the present invention uses the Alternating Direction Method of Multipliers (ADMM) to realize its secondary optimization.

[0010] In the parameter setting part, in traditional algorithms, parameter setting is often performed based on past experience. This parameter setting method is too dependent on experience and has great limitations, which is not conducive to coping with parameter changes in various models. The parameters used for different models may be different. If the parameters are set only based on past experience, the optimal solution may not be obtained, or the local optimal solution may be mistaken for the global optimal solution. The present invention uses an adaptive parameter adjustment method to update parameters, which can improve the reliability of the algorithm more than the traditional model parameter setting based on past experience. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] Attached Figure 1 It is a basic flow chart of the present invention;

[0012] Attached Figure 2 It is an overall block diagram of the present invention. DETAILED DESCRIPTION

[0013] The content of the present invention is further described below in conjunction with the accompanying drawings.

[0014] Attached Figure 2 The one shown is based on weighted l p The image denoising algorithm combining the sparsity and low rank of the image group with the norm includes the following steps:

[0015] S1: First obtain the non-local similar blocks of the image through block matching;

[0016] S2: Combined with adaptive dictionary learning, low-rank approximation of non-local approximation blocks is obtained;

[0017] S3: Combined weighted l p The norm combines the sparsity and low rank of the image group to build a denoising model;

[0018] S4: The denoising model is not easy to solve. If the denoising model is solved directly, it is difficult and cumbersome. Therefore, the present invention divides the model into multiple sub-problems for optimization to achieve the solution of the algorithm: by fixing x and B i , then A i The solution will be simplified to:

[0019]

[0020] To solve A i The optimization problem is to firstly calculate each image block group X in the image i , using the adaptive dictionary D i Solve the above equation:

[0021]

[0022] Among them, R i x=D i G i , And v = ηρλ. For the closed solution of the above equation, the present invention proposes to use generalized soft threshold (GST) to solve it. GST can more efficiently obtain the converged solution A i :

[0023]

[0024] The edge and texture information in the image is usually represented by a larger sparse coefficient. Therefore, the weight should change with the size of the sparse coefficient, and a smaller weight should be applied to a larger sparse coefficient; on the contrary, a larger weight should be applied to a smaller sparse coefficient. Therefore, the weight is defined as:

[0025]

[0026] Where W = (w1, w2, ..., w i ), P i =(p1, p2, ... pi ), in order to satisfy the above equation, ε is taken as a constant close to 0.

[0027] In the model, by fixing A i , construct solution B i Sub-questions:

[0028]

[0029] where ||·|| * is the nuclear norm, ||B i || * =∑ j ψ i,j , j = (1, 2, ..., d), d = min (b, m). In addition, ψ i，j Represents the matrix B i The j-th singular value of . There exists a closed-form solution:

[0030]

[0031] Where Soft(·) represents the soft threshold algorithm. i Performing singular value decomposition (SVD) yields

[0032] In the model, by fixing A i and B i , to construct the subproblem of solving x:

[0033]

[0034] S5: Use the alternating direction multiplier method to solve the quadratic optimization problem about x. Introduce an auxiliary variable f with the constraint x=f, and the above formula can be rewritten as follows:

[0035]

[0036] ADMM can be used to The solution process is decomposed into the solution of the following sub-problems:

[0037]

[0038] Among them, k is the kth iteration, g is the Lagrange multiplier, and the update of x and f is completed. For the convenience of description, the following description omits the superscript k.

[0039] The quadratic optimization problem of x is described above. By fixing g and f, The x update problem can be solved by gradient descent method and can be rewritten as follows:

[0040]

[0041] Among them, β represents the step size, and q represents the gradient direction of x. For this problem, the closed solution of x can be obtained by solving the following formula:

[0042]

[0043] The The quadratic optimization problem is to fix x, A i , and g, then The f-update problem in has the following closed form solution:

[0044]

[0045] Where E is the identity matrix,

[0046] Regarding parameter setting, in traditional algorithms, parameter setting is often performed based on past experience. This parameter setting method is too dependent on experience and has great limitations, which is not conducive to coping with parameter changes in various models. The present invention uses adaptive parameter adjustment to update parameters, which can improve the reliability of the algorithm more than setting parameters based on past experience in traditional models.

[0047] The present invention uses the following method to update the parameters ρ, η and the sparsity and low-rank constraint coefficients λ and τ involved in the algorithm:

[0048]

[0049] Where θ and φ are proportional factors. To satisfy the above parameter update formula, ε is taken as a constant close to 0. i Pick The estimated standard deviation of i Take i The present invention adopts an iterative regularization method to estimate the noise variance To update:

[0050]

[0051] in is the scale factor.

[0052] Table 1 is a comparison of the experimental results of the present invention and other image denoising algorithms, and the data set is the public data set Set12:

[0053] Table 1 Comparison of experimental results of the present invention and other image denoising algorithms

[0054]

[0055] As shown in the table, the image denoising algorithm proposed in the present invention is reliable in the field of image denoising and can effectively improve the denoising performance of the image. The denoising performance is better than that of traditional algorithms and currently popular image denoising methods, and it also has excellent performance compared with the image denoising method based on deep learning. The algorithm plays a good role in promoting the subsequent image processing process.

Claims

1. An image denoising algorithm based on weighted lp norm combining sparsity and low rank of image group, characterized by: Image denoising is performed by integrating image group sparsity and low rank; The group sparsity is to expand the traditional independent image block sparsity into group sparsity, which is used for the non-local similarity of images and the local sparsity of images; The group low rank property is to apply the inter-group low rank property on each group sparse coefficient to further utilize the inter-group similarity to improve the denoising effect and retain the image detail information.

2. According to claim 1, an image denoising algorithm based on weighted lp norm combining sparsity and low rank of image group, characterized in that: The denoising algorithm uses weighted p The norm and nuclear norm regularize the sparsity and low rank of the image; The weighted p The norm is used as a penalty for sparsity between groups; The nuclear norm acts as a low rank penalty.

3. According to claim 2, an image denoising algorithm based on weighted lp norm combining sparsity and low rank of image group, characterized in that: The denoising model that regularizes the sparsity and low rank of the image by using the weighted lp norm and nuclear norm solves the problem that a single group of sparse models is prone to over-smoothing in various image restoration tasks. The invention proposes to regularize the sparsity coefficient A of each group by i The inter-group low rank is applied at the same time to further utilize the inter-group similarity to improve the denoising effect and the retention of image detail information. Both sparsity and low rank penalties are used as dual regularization. The denoising model of the present invention is as follows:

4. According to claim 3, an image denoising algorithm based on weighted lp norm combining sparsity and low rank of image group, characterized in that: The complex optimization problem of the denoising model is divided into multiple sub-problems and optimized separately. i and generalized soft threshold (GST) to more efficiently obtain the converged solution A i : Among them, the weight W = (w1, w2, ..., w i ), v=ηρλ. To ensure that the solution always holds, ε is taken as a constant approaching 0.

5. According to claim 3, an image denoising algorithm based on weighted lp norm combining sparsity and low rank of image group, characterized in that: The optimization problem can be divided into multiple sub-problems and optimized separately, and the converged solution B can be obtained by using soft threshold. i :

6. According to claim 3, an image denoising algorithm based on weighted lp norm combining sparsity and low rank of image group, characterized in that: The optimization problem can be divided into multiple sub-problems and optimized separately. An auxiliary variable f with the constraint x=f is introduced, and the alternating direction method of multipliers (ADMM) is used for secondary optimization to obtain the converged solution x: Where k is the kth iteration and g is the Lagrange multiplier.

7. According to claim 3, an image denoising algorithm based on weighted lp norm combined with image group sparsity and low rank, characterized in that: The parameters involved in the denoising model are adjusted using the adaptive parameter adjustment method: where θ, φ, is the scale factor. i Pick The estimated standard deviation of .

8. According to claim 7, an image denoising algorithm based on weighted lp norm combined with image group sparsity and low rank, characterized in that: The parameters involved in the denoising model are adjusted using the adaptive parameter adjustment method: Where ε is a constant approaching 0, γ i Take i The estimated standard deviation of .

9. According to claim 1, an image denoising algorithm based on weighted lp norm combining sparsity and low rank of image group, characterized in that: The overall algorithm flow includes the following steps: S1: Initialization: Input the degraded image y=x+n, where x represents the clean image and n represents the noise model. Generally, it is assumed that n has a mean of 0 and a variance of σ. 2 Gaussian white noise, initialized Parameters b, m, W, ρ, η, and μ; S2: Start iteration, from 1 to the maximum number of iterations Max_Iter; S3: Image processing: Calculation of parameter σ e , x (k) Divide into a group of size Overlapping blocks; S4: Non-local similarity block calculation: For each block x in the overlapping block group i Processing, from group X i Find non-local similar blocks in; S5: Sub-question A i Optimization: Use principal component analysis to calculate X i The adaptive dictionary D i and through Perform A i Updates; S6: Through [U i ,∏ i ,V i ]=SVD(A i )Get U i ,∏ i , V i ; S7: Other sub-problems and related parameter updates: Update η, τ, ρ, λ, And end the loop; S8: Secondary Optimization: Update f using the alternating direction multiplier method (k+1) , g (k+1) , x (k+1) ; S9: Output denoised image

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