Image mixed noise removal method based on low-rank function and multi-directional total variation

Abstract

The present invention discloses a method for removing mixed image noise based on low-rank function and multi-directional total variation. In combination with the low-rank structural characteristics of the image, an improved low-rank function is used to approximate the rank function, and in combination with a regularization method based on multi-directional total variation, a mixed image denoising method based on low-rank function and multi-directional total variation is designed. This method adopts an improved low-rank function to better utilize the inherent low-rank structural characteristics of the image; on the other hand, in order to fully utilize the correlation existing in all possible directions in the image, a multi-directional total variation regularization term is introduced. That is, using multi-directional adjacent domain pixel points as input data, this will fully consider the texture continuity between the current pixel and all its adjacent pixels, and maximize the use of the redundancy of this image in the spatial domain.

Description

Technical Field

[0001] The present invention relates to the technical field of image denoising, and in particular to a method for removing mixed image noise based on a low-rank function and multi-directional total variation. Background Art

[0002] During the process of image acquisition and transmission, images are often contaminated by different types of noise, such as Gaussian noise, salt and pepper noise, random noise, and stripe noise. These mixed noises seriously reduce the visual quality of the image and greatly affect the subsequent image segmentation, classification, target detection and other tasks.

[0003] At present, the methods for removing mixed noise in images mainly include: the traditional two-stage method, the mixed noise removal method based on joint modeling, the method based on low-rank matrix recovery, and the mixed noise removal method based on deep learning. The traditional two-stage method usually adopts a method combining different denoising models with median filtering. Generally, the noise-contaminated area of ​​the image is detected first, and then the image denoising problem is converted into an image restoration problem. Therefore, this type of method relies on the accuracy of noise detection. The mixed noise removal method based on joint modeling does not require prior detection, but jointly models the mixed noise of Gaussian and salt and pepper, Gaussian and random. However, due to the large difference in the distribution of Gaussian noise and salt and pepper noise or random noise, the computational complexity of their joint modeling is high. The method based on low-rank matrix recovery mainly combines the high correlation between image blocks, so that the image data composed of image blocks usually presents low-rank structural characteristics. Therefore, the method of low-rank matrix recovery can be used to remove noise in the image. The advantage of the method based on low-rank matrix recovery is that it can simultaneously process multiple different types of mixed noise without prior information about noise, and has good tolerance for different noises. However, most of the methods based on low-rank matrix recovery treat each singular value equally in the low-rank function, while in fact the contributions of all non-zero singular values ​​of the nuclear norm are different. Therefore, this method may not be appropriate. A mixed noise removal method based on the weighted Schatten-p norm model. Although this method performs weighted processing on different singular values, the P value and weight coefficient of the weighted Schatten-p norm need to be set in advance, which easily leads to the instability of the performance of this method. Although the mixed noise removal method based on deep learning has achieved good denoising effect, the mixed noise removal method based on deep learning requires a large amount of training data as support and is prone to overfitting.

[0004] The hybrid denoising method based on low-rank matrix recovery takes advantage of the intrinsic low-rank structure of the image, uses the nuclear norm of the matrix to design the rank function, and uses the 1 norm instead of the zero norm to describe the sparsity of salt and pepper noise. Finally, the original non-convex optimization problem is transformed into a convex optimization problem, and it is solved iteratively, achieving a good effect of removing mixed noise. However, this method is based on the nuclear norm strategy of the matrix, ignoring the fact that all non-zero singular values ​​in the nuclear norm of the matrix contribute differently to image restoration, and treating each singular value equally, which easily leads to the denoised image being too smooth. On the other hand, the hybrid denoising method based on low-rank matrix recovery uses the L1 norm as the sparse regularization term, ignoring the texture correlation and redundancy in multiple directions in the image, which easily causes the restored image to have a step effect. Summary of the invention

[0005] In view of the above-mentioned deficiencies in the prior art, the present invention provides an image mixed noise removal method based on low-rank function and multi-directional total variation, which solves the problem that the hybrid denoising method based on low-rank matrix recovery ignores the fact that all non-zero singular values ​​in the matrix nuclear norm contribute differently to image restoration and treats each singular value equally, resulting in the denoised image being too smooth, and ignores the texture correlation and redundancy in multiple directions in the image, resulting in a step effect in the restored image.

[0006] In order to achieve the above-mentioned object of the invention, the technical solution adopted by the present invention is: a method for removing mixed noise of an image based on a low-rank function and multi-directional total variation, comprising the following steps:

[0007] S1, dividing the degraded image polluted by mixed noise into multiple image blocks;

[0008] S2, using a block matching algorithm to search for image blocks with high similarity to the image block among the remaining image blocks except the image block, i.e. similar image blocks, for each image block;

[0009] S3, arranging similar image blocks of the same image block into a matrix, that is, obtaining a structure group;

[0010] S4, for each structure group, a rank function and a multi-directional total variation method are used to jointly model the image mixed noise to obtain a mixed noise removal model;

[0011] S5, repeating step S4 until each structure group is modeled, using the alternating direction multiplier method to solve each mixed noise removal model K times to obtain K original noise-free structure groups;

[0012] S6. Merge the K original noise-free structure groups to obtain a denoised image.

[0013] Furthermore, the formula for removing the mixed noise model in step S4 is:

[0014]

[0015] Z i =L i +S i +N i

[0016]

[0017]

[0018] Among them, ‖L i ‖ GN is a low-rank function based on Gaman-norm, ‖L i ‖ MOTV is the i-th original noise-free structure group L i The multi-directional total variation norm of , χ is a constant factor greater than 0, ‖S i ‖1 is S i The 1-norm of N i is the i-th structure group Z i Gaussian noise in S i is the i-th structure group Z i Salt and pepper noise in is Gaussian noise N i The square of the F norm, K is the number of structure groups obtained in step S3, is the i-th original noise-free structure group L i The jth singular value after singular value decomposition, r is the number of singular values, is the pixel point (L i ) x,y The four adjacent pixels in the horizontal and vertical directions, k1, k2 = 1, 2, 3, 4, is the pixel point (L i ) x,y The four adjacent pixels in the diagonal direction of , x, y are the original noise-free structure group L i The position of the pixel point, m is the original noise-free structure group L i The number of rows, n is the original noise-free structure group L i The number of columns, w1, w2 are weight factors, α, β, γ are regularization parameters.

[0019] The beneficial effects of the above further scheme are as follows: the constructed mixed noise removal model, on the one hand, mines the overall information in the image, that is, the low-rank function based on Gaman-norm is used to approximate the low-rank structure of the image, which can obtain a tighter approximation than the traditional method, thereby improving the performance of image denoising; on the other hand, it also considers the local structural direction information of the image (based on multi-directional total variation). This is conducive to avoiding local over-smoothing of the denoised image. :

[0020] Furthermore, in step S5, solving the mixed noise removal model K times by the alternating direction multiplier method includes the following sub-steps:

[0021] A1. According to the mixed noise removal model, the Lagrangian function of the mixed noise removal model is obtained by using the augmented Lagrangian method;

[0022] A2. Divide the Lagrangian function into sub-problems and obtain 4 sub-problems;

[0023] A3. Solve the four sub-problems alternately to obtain the original noise-free structure group.

[0024] Furthermore, the formula of the Lagrangian function in step A1 is:

[0025]

[0026] Among them, l(L i ,X i ,S i ,N i ,θ1,θ2) is the Lagrangian function, L i is the i-th original noise-free structure group, S i is the i-th structure group Z i Salt and pepper noise in X i For L i The equivalent auxiliary matrix, N i is the i-th structure group Z i Gaussian noise in, θ1, θ2 are Lagrange multipliers, ‖L i ‖ GN is a low-rank function based on Gaman-norm, ‖L i ‖ MOTV is the i-th original noise-free structure group L i The multi-directional total variation norm of , ‖S i ‖1 is S i The 1-norm of is Gaussian noise N i The square of the F-norm, α, β, γ are regularization parameters, <,> is the inner product, and μ is the penalty parameter.

[0027] The beneficial effect of the above further scheme is that the constructed Lagrangian function transforms the constrained original problem into an unconstrained optimization problem, which ultimately helps to use the alternating direction multiplier method to alternately solve the hybrid denoising model and try to make the obtained solution converge to the optimal solution of the original problem.

[0028] Furthermore, the solution method for the first sub-problem in step A3 is: using the convex difference method and the neighboring point method to solve the first sub-problem model, and obtaining the original noise-free structure group solved for the k+1th time, wherein the first sub-problem model is:

[0029]

[0030] The original noise-free structure group solved for the k+1th time is:

[0031] in, is the original noise-free structure group solved for the k+1th time, is the minimum point L when the function is minimized i Assign the value of is the i-th original noise-free structure group L i The jth singular value after singular value decomposition, χ is a constant factor greater than 0, r is the number of singular values, μ is the penalty parameter, L i is the i-th original noise-free structure group, Z i is the i-th structure group, is the equivalent auxiliary matrix of the k-th solution, is the salt and pepper noise solved for the kth time, is the Gaussian noise solved for the kth time, is the Lagrange multiplier solved for the kth time, is the square of the norm of F, U and V are matrices A respectively i The two orthogonal matrices after singular value decomposition are: Right now is the matrix A i The diagonal matrix after singular value decomposition, is the matrix A i The singular value vector of To make the function The minimum point at which the minimum value is obtained The diagonal matrix composed of is the singular value vector, σ i is the i-th original noise-free structure group L i The singular value vector after singular value decomposition.

[0032] Furthermore, the solution method for the second sub-problem in step A3 is: solve the second sub-problem model to obtain the equivalent auxiliary matrix of the k-th solution Among them, the second sub-problem model is:

[0033]

[0034] ‖X i ‖ MOTV =‖Dx i ‖ 2,1

[0035]

[0036] The equivalent auxiliary matrix of the k-th solution is:

[0037] in, is the minimum point X when the function is minimized i Assign the value of X i For L i The equivalent auxiliary matrix of i ‖ μOTV For X i is the multi-directional total variation norm, ‖L i ‖ MOTV is the i-th original noise-free structure group L i The multi-directional total variation norm of , ‖‖ 2,1 is the mixed l after the difference matrix transformation 2,1 Norm, l 2,1 The norm is the result of finding the 2 norm of each row and adding the 2 norms of all rows. D is the difference operator, x i For X i The vector representation of X i For L i The equivalent auxiliary matrix of is the original noise-free structure group solved for the kth time, is the Lagrange multiplier solved for the k-1th time, μ is the penalty parameter, is the square of the F norm, is the pixel point (L i ) x,y The four adjacent pixels in the horizontal and vertical directions, k1, k2 = 1, 2, 3, 4, is the pixel point (L i ) x,y The four adjacent pixels in the diagonal direction of , x, y are the original noise-free structure group L i The position of the pixel point, m is the original noise-free structure group L iThe number of rows, n is the original noise-free structure group L i The number of columns, w1, w2 are weight factors.

[0038] Furthermore, the solution method for the third sub-problem in step A3 is: the ADMM method is used to solve the third sub-problem model to obtain the salt and pepper noise solved for the kth time Among them, the model of the third sub-problem is:

[0039]

[0040]

[0041] T β / μ (Q (s,t) )=max(|Q (s,t) |-β / μ,0)sign(Q (s,t) )

[0042] Salt and pepper noise solved for the kth time for:

[0043] in, is the minimum point S when finding the minimum value of the function i Assign the value of β is the regularization parameter, μ is the penalty parameter, ‖S i ‖1 is S i The 1-norm of S i is the i-th structure group Z i Salt and pepper noise in is the original noise-free structure group solved for the kth time, is the Gaussian noise solved for the k-1th time, is the Lagrange multiplier solved for the k-1th time, is the square of the F norm, Q (s,t) is the (s,t)th element of the matrix Q, T β / μ (·) is a soft threshold operator, s = 1, 2, ... M, t = 1, 2, ... N, M is the number of rows of the matrix Q, N is the number of columns of the matrix Q, |Q (s,t) |To obtain Q (s,t) The absolute value of , sign(·) is the sign function.

[0044] Furthermore, the method for solving the fourth sub-problem in step A3 is: solving the fourth sub-problem model to obtain the Gaussian noise of the k-th solution Among them, the fourth sub-problem model is:

[0045]

[0046] Gaussian noise of the kth solution for:

[0047] in, is the minimum point N when finding the minimum value of the function i Assign the value of γ is the regularization parameter, is the square of the F norm, is the Gaussian noise solved for the kth time, N i is the i-th structure group Z i Gaussian noise in, μ is the penalty parameter, is the original noise-free structure group solved for the kth time, is the salt and pepper noise solved for the kth time, is the Lagrange multiplier solved for the k-1th time.

[0048] The beneficial effect of the above further scheme is that the main idea of ​​the four problem models and solution methods is based on the variable separation method, that is, fixing other variables and alternately finding the optimal solution of the corresponding variables. This helps to make the method simple and efficient.

[0049] In summary, the beneficial effects of the present invention are as follows: the present invention combines the low-rank structural characteristics of the image, adopts an improved low-rank function to approximate the rank function, and combines the regularization method based on multi-directional total variation to design an image hybrid denoising method based on low-rank function and multi-directional total variation. This method adopts an improved low-rank function to better utilize the inherent low-rank structural characteristics of the image; on the other hand, in order to make full use of the correlation in all possible directions in the image, a multi-directional total variation regularization term is introduced. That is, using multi-directional adjacent domain pixel points as input data, this will fully consider the texture continuity between the current pixel and all its adjacent pixels, and maximize the use of the redundancy of this image in the spatial domain. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 This is a flowchart of a method for removing mixed noise from an image based on low-rank function and multi-directional total variation;

[0051] Figure 2 The position relationship diagram of 8 adjacent pixels is used to calculate the multi-directional total variation norm;

[0052] Figure 3 This is a comparison chart of the hybrid denoising effect of the test image Peppers. DETAILED DESCRIPTION

[0053] The specific implementation modes of the present invention are described below so that those skilled in the art can understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation modes. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the attached claims, these changes are obvious, and all inventions and creations utilizing the concept of the present invention are protected.

[0054] like Figure 1 As shown, a method for removing mixed noise from an image based on a low-rank function and multi-directional total variation includes the following steps:

[0055] In this embodiment, the embodiment of the present invention: the degraded image specifically takes a Peppers image as an example, and its size is 512x512. In the present invention, the variance of the Gaussian noise is 20, and the density of the salt and pepper noise is 30%.

[0056] S1, dividing the degraded image polluted by mixed noise into multiple B×B image blocks;

[0057] S2, using a block matching algorithm to search for image blocks with high similarity to the image block among the remaining image blocks except the image block, i.e. similar image blocks, for each image block;

[0058] Initialize related parameters, including: row value B of the image block matrix, number of similar image blocks, regularization factors α, β, γ. In the present invention, the row value B of the image block matrix is ​​8, the number of similar image blocks is 32, and the values ​​of α, β, γ are 0.05, 0.10, 0.05 respectively.

[0059] S3, similar image blocks of the same image block are arranged into a matrix, that is, a structure group Z is obtained i ;

[0060] S4, for each structure group, a rank function and a multi-directional total variation method are used to jointly model the image mixed noise to obtain a mixed noise removal model;

[0061] The formula for removing the mixed noise model in step S4 is:

[0062]

[0063] Z i =L i +S i +N i

[0064]

[0065]

[0066] Among them, ‖Li ‖ GN is a low-rank function based on Gaman-norm, ‖L i ‖ MOTV is the i-th original noise-free structure group L i The multi-directional total variation norm of , χ is a constant factor greater than 0, the value of χ is 0.2, ‖S i ‖1 is S i The 1 norm is mainly used to describe the structure group Z i The sparsity of salt and pepper noise, N i is the i-th structure group Z i Gaussian noise in S i is the i-th structure group Z i Salt and pepper noise in is Gaussian noise N i The square of the F norm, K is the number of structure groups obtained in step S3, is the i-th original noise-free structure group L i The jth singular value after singular value decomposition, r is the number of singular values, is the pixel point (L i ) x,y The four adjacent pixels in the horizontal and vertical directions, k1, k2 = 1, 2, 3, 4, is the pixel point (L i ) x,y The four adjacent pixels in the diagonal direction of , x, y are the original noise-free structure group L i The position of the pixel point, m is the original noise-free structure group L i The number of rows, n is the original noise-free structure group L i The number of columns, w1, w2 are weight factors, α, β, γ are regularization parameters.

[0067] like Figure 2 As shown in FIG. 1 , a schematic diagram of a pixel point * and its neighboring points in different directions, where · corresponds to the two horizontal neighboring pixels and the two vertical neighboring pixels of the pixel point *, and × is the four diagonal neighboring pixels of the pixel point *. The present invention assumes that the correlation between two pixels is inversely proportional to the geometric distance between them. Therefore, in the multi-directional total variation norm, the weight factor w1 is set to 1, and the weight factor w2 is set to

[0068] S5, repeating step S4 until each structure group is modeled, using the alternating direction multiplier method to solve each mixed noise removal model K times to obtain K original noise-free structure groups;

[0069] Initialization: k = 0 means the 0th iteration, The superscript 0 means L i ,X i ,S i The initial value is taken.

[0070] The alternating direction multiplier method in step S5 solves the mixed noise removal model K times, including the following sub-steps:

[0071] A1. According to the mixed noise removal model, the Lagrangian function of the mixed noise removal model is obtained by using the augmented Lagrangian method;

[0072] The formula of the Lagrangian function in step A1 is:

[0073]

[0074] Among them, l(L i ,X i ,S i ,N i ,θ1,θ2) is the Lagrangian function, L i is the i-th original noise-free structure group, S i is the i-th structure group Z i Salt and pepper noise in X i For L i The equivalent auxiliary matrix, N i is the i-th structure group Z i Gaussian noise in, θ1, θ2 are Lagrange multipliers, ‖L i ‖ GN is a low-rank function based on Gaman-norm, ‖L i ‖ MOTV is the i-th original noise-free structure group L i The multi-directional total variation norm of , ‖S i ‖1 is S i The 1-norm of is Gaussian noise N i The square of the F-norm, α, β, γ are regularization parameters, <,> is the inner product, and μ is the penalty parameter.

[0075] A2. Divide the Lagrangian function into sub-problems and obtain 4 sub-problems;

[0076] A3. Solve the four sub-problems alternately, that is, find the value of one of the variables and keep the other variables fixed, to obtain the original noise-free structure group.

[0077] The solution method for the first sub-problem in step A3 is: use the convex difference method and the neighboring point method to solve the first sub-problem model, and obtain the original noise-free structure group solved for the k+1th time, where the first sub-problem model is:

[0078]

[0079] The original noise-free structure group solved for the k+1th time is:

[0080] in, is the original noise-free structure group solved for the k+1th time, is the minimum point L when the function is minimized i Assign the value of is the i-th original noise-free structure group L i The jth singular value after singular value decomposition, χ is a constant factor greater than 0, r is the number of singular values, μ is the penalty parameter, L i is the i-th original noise-free structure group, Z i is the i-th structure group, is the equivalent auxiliary matrix of the k-th solution, is the salt and pepper noise solved for the kth time, is the Gaussian noise solved for the kth time, is the Lagrange multiplier solved for the kth time, is the square of the norm of F, U and V are matrices A respectively i The two orthogonal matrices after singular value decomposition are: Right now is the matrix A i The diagonal matrix after singular value decomposition, is the matrix A i The singular value vector of To make the function The minimum point at which the minimum value is obtained The diagonal matrix composed of is the singular value vector, σ i is the i-th original noise-free structure group L i The singular value vector after singular value decomposition.

[0081] The solution method for the second sub-problem in step A3 is: solve the second sub-problem model and obtain the equivalent auxiliary matrix of the k-th solution Among them, the second sub-problem model is:

[0082]

[0083] ‖X i ‖ MOTV =‖Dx i ‖ 2,1

[0084]

[0085] The equivalent auxiliary matrix of the k-th solution is:

[0086] in, is the minimum point X when the function is minimized i Assign the value of X i For L i The equivalent auxiliary matrix of i ‖ MOTV For X i is the multi-directional total variation norm, ‖L i ‖ MOTV is the i-th original noise-free structure group L i The multi-directional total variation norm of , ‖‖ 2,1 is the mixed l after the difference matrix transformation 2,1 Norm, l 2,1 The norm is the result of finding the 2 norm of each row and adding the 2 norms of all rows. D is the difference operator, x i For X i The vector representation of X i For L i The equivalent auxiliary matrix of is the original noise-free structure group solved for the kth time, is the Lagrange multiplier solved for the k-1th time, μ is the penalty parameter, is the square of the F norm, is the pixel point (L i ) x,y The four adjacent pixels in the horizontal and vertical directions, k1, k2 = 1, 2, 3, 4, is the pixel point (L i ) x,y The four adjacent pixels in the diagonal direction of , x, y are the original noise-free structure group L i The position of the pixel point, m is the original noise-free structure group L i The number of rows, n is the original noise-free structure group L i The number of columns, w1, w2 are weight factors.

[0087] The solution to the third sub-problem in step A3 is: use the ADMM method to solve the third sub-problem model and obtain the salt and pepper noise solved for the kth time Among them, the model of the third sub-problem is:

[0088]

[0089]

[0090] T β / μ (Q (s,t) )=max(|Q (s,t) |-β / μ,0)sign(Q (s,t) )

[0091] Salt and pepper noise solved for the kth time for:

[0092] in, is the minimum point S when finding the minimum value of the function i Assign the value of β is the regularization parameter, μ is the penalty parameter, ‖S i ‖1 is S i The 1-norm of S i is the i-th structure group Z i Salt and pepper noise in is the original noise-free structure group solved for the kth time, is the Gaussian noise solved for the k-1th time, is the Lagrange multiplier solved for the k-1th time, is the square of the F norm, Q (s,t) is the (s,t)th element of the matrix Q, T β / μ (·) is a soft threshold operator, s = 1, 2, ... M, t = 1, 2, ... N, M is the number of rows of the matrix Q, N is the number of columns of the matrix Q, |Q (s,t) |To obtain Q (s,t) The absolute value of , sign(·) is the sign function.

[0093] The solution method for the fourth sub-problem in step A3 is: solve the fourth sub-problem model to obtain the k-th solved Gaussian noise Among them, the fourth sub-problem model is:

[0094]

[0095] Gaussian noise of the kth solution for:

[0096] in, is the minimum point N when finding the minimum value of the function i Assign the value of γ is the regularization parameter, is the square of the F norm, is the Gaussian noise solved for the kth time, N i is the i-th structure group Z i Gaussian noise in, μ is the penalty parameter, is the original noise-free structure group solved for the kth time, is the salt and pepper noise solved for the kth time, is the Lagrange multiplier solved for the k-1th time.

[0097] S6. Merge the K original noise-free structure groups to obtain a denoised image.

[0098] In order to verify the denoising performance of the present invention, the test image Peppers is selected as an example in the simulation, and its pixel size is 512×512. At the same time, the present invention is compared with the other two mixed noise filtering methods, including document 1: Y. Xiao, T. Zeng, J. Yu, M. Ng. Restoration of images corrupted by mixed Gaussian-impulse noise via L1-L0 minimization. Pattern Recognition, 2011, 44(8): 1708-1720. and document 2: Y. Xie, SH Gu, Y. Liu, WM Zuo, WS Zhang, L. Zhang. Weighted Schatten p-Norm Minimization for Image Denoising and Background Subtraction. IEEE Transactions on Image Processing, 2016, 25(10): 4842-4857. The parameters of these methods are set according to the values ​​recommended in the original text. The algorithm simulation environment of the present invention is a 64-bit Windows 7 operating system, a 2.6G CPU, a 16G memory, and Matlab 2018.

[0099] Figure 3 The comparison of the mixed denoising effect of the test image Peppers is given. The image (a) shows an image contaminated by mixed noise, with a Gaussian noise variance of 20 and a salt and pepper noise density of 30%; the image (b) is the result of denoising in reference 1, the image (c) is the result of denoising in reference 2, and the image (d) is the result of denoising in the present invention. Figure 2It can be seen from the hybrid denoising of the Peppers image that the denoising effect of the present invention has a good edge preservation effect, and the texture structure is relatively clear without excessive smoothing. The main reason is that the present invention utilizes the low-rank structure sparse characteristics of the image and information in multiple directions to achieve a good hybrid noise removal effect.

[0100] To further objectively evaluate the hybrid denoising algorithm, the present invention uses peak signal-to-noise ratio (PSNR) and structural similarity index measure (SSIM) to evaluate the image quality after hybrid denoising. Table 1 shows the performance comparison results of the three denoising methods for the test image Peppers when the Gaussian noise variance and salt and pepper noise density are σ=15, s=20% and σ=25, s=28% respectively. It can be seen from Table 1 that the method proposed in the present invention has been improved to varying degrees in terms of peak signal-to-noise ratio and structural similarity indicators, and has better denoising effect than references 1 and 2. The effectiveness of the method proposed in the present invention is further verified by examples.

[0101] Table 1 Results of Peppers image hybrid denoising

[0102]

[0103]

Claims

1. A method for removing mixed image noise based on low-rank function and multi-directional total variation, characterized in that: The following steps are involved: S1, dividing the degraded image polluted by mixed noise into multiple image blocks; S2, using a block matching algorithm to search for image blocks with high similarity to the image block among the remaining image blocks except the image block, i.e. similar image blocks; S3, arranging similar image blocks of the same image block into a matrix, that is, obtaining multiple structure groups; S4, for each structure group, a rank function and a multi-directional total variation method are used to jointly model the image mixed noise to obtain a mixed noise removal model; S5, repeating step S4 until each structure group is modeled, using the alternating direction multiplier method to solve each mixed noise removal model K times to obtain K original noise-free structure groups; S6, merging the K original noise-free structure groups to obtain a denoised image; The formula for removing the mixed noise model in step S4 is: Z i =L i +S i +N i Among them, ‖L i ‖ GN is a low-rank function based on Gaman-norm, ‖L i ‖ MOTV is the i-th original noise-free structure group L i The multi-directional total variation norm of , χ is a constant factor greater than 0, ‖S i ‖1 is S i The 1-norm of N i is the i-th structure group z i Gaussian noise in S i is the i-th structure group Z i Salt and pepper noise in is Gaussian noise N i The square of the F norm, K is the number of structure groups obtained in step S3, is the i-th original noise-free structure group L i The jth singular value after singular value decomposition, r is the number of singular values, is the pixel point (L i ) x,y 4 adjacent pixels in the horizontal and vertical directions, is the pixel point (L i ) x,y The four adjacent pixels in the diagonal direction of , x, y are the original noise-free structure group L i The position of the pixel point, m is the original noise-free structure group L i The number of rows, n is the original noise-free structure group L i The number of columns, w1, w2 are weight factors, α, β, γ are regularization parameters.

2. The image mixed noise removal method based on low-rank function and multi-directional total variation according to claim 1 is characterized in that: The alternating direction multiplier method in step S5 solves the mixed noise removal model K times, including the following sub-steps: A1. According to the mixed noise removal model, the Lagrangian function of the mixed noise removal model is obtained by using the augmented Lagrangian method; A2. Divide the Lagrangian function into sub-problems and obtain 4 sub-problems; A3. Solve the four sub-problems alternately to obtain the original noise-free structure group.

3. The image mixed noise removal method based on low-rank function and multi-directional total variation according to claim 2 is characterized in that: The formula of the Lagrangian function in step A1 is: in, is the Lagrangian function, L i is the i-th original noise-free structure group, S i is the i-th structure group Z i Salt and pepper noise in X i For L i The equivalent auxiliary matrix, N i is the i-th structure group Z i Gaussian noise in, θ1, θ2 are Lagrange multipliers, ‖L i ‖ GN is a low-rank function based on Gaman-norm, ‖X i ‖ MOTV For X i The multi-directional total variation norm of , ‖S i ‖1 is S i The 1-norm of is Gaussian noise N i The square of the F-norm, α, β, γ are regularization parameters, <,> is the inner product, and μ is the penalty parameter.

4. The image mixed noise removal method based on low-rank function and multi-directional total variation according to claim 2 is characterized in that: The solution method for the first sub-problem in step A3 is: use the convex difference method and the neighboring point method to solve the first sub-problem model, and obtain the original noise-free structure group solved for the k+1th time, where the first sub-problem model is: The original noise-free structure group solved for the k+1th time is: in, is the original noise-free structure group solved for the k+1th time, is the minimum point L when the function is minimized i Assign the value of is the i-th original noise-free structure group L i The jth singular value after singular value decomposition, χ is a constant factor greater than 0, r is the number of singular values, μ is the penalty parameter, L i is the i-th original noise-free structure group, Z i is the i-th structure group, is the equivalent auxiliary matrix of the k-th solution, is the salt and pepper noise solved for the kth time, is the Gaussian noise solved for the kth time, is the Lagrange multiplier solved for the kth time, is the square of the norm of F, U and V are matrices A respectively i The two orthogonal matrices after singular value decomposition are: Right now is the matrix A i The diagonal matrix after singular value decomposition, is the matrix A i The singular value vector of To make the function The minimum point at which the minimum value is obtained The diagonal matrix composed of is the singular value vector, σ i is the i-th original noise-free structure group L i The singular value vector after singular value decomposition.

5. The image mixed noise removal method based on low-rank function and multi-directional total variation according to claim 4 is characterized in that: The solution method for the second sub-problem in step A3 is: solve the second sub-problem model and obtain the equivalent auxiliary matrix of the k-th solution Among them, the second sub-problem model is: ‖X i ‖ MOTV =‖Dχ i ‖ 2,1 The equivalent auxiliary matrix of the k-th solution is: in, is the minimum point X when the function is minimized i Assign the value of X i For L i The equivalent auxiliary matrix of i ‖ MOTV For X i The multi-directional total variation norm of , ‖L i ‖ MOTV is the i-th original noise-free structure group L i The multi-directional total variation norm of , ‖ ‖ 2,1 is the mixture after difference matrix transformation norm, The norm is the result of finding the 2 norm of each row and adding the 2 norms of all rows. D is the difference operator. For X i The vector representation of X i For L i The equivalent auxiliary matrix of is the original noise-free structure group solved for the kth time, is the Lagrange multiplier solved for the k-1th time, μ is the penalty parameter, is the square of the F norm, is the pixel point (L i ) x,y The four adjacent pixels in the horizontal and vertical directions, k1, k2 = 1, 2, 3, 4, is the pixel point (L i ) x,y The four adjacent pixels in the diagonal direction of , x, y are the original noise-free structure group L i The position of the pixel point, m is the original noise-free structure group L i The number of rows, n is the original noise-free structure group L i The number of columns, w1, w2 are weight factors.

6. The image mixed noise removal method based on low-rank function and multi-directional total variation according to claim 4 is characterized in that: The solution to the third sub-problem in step A3 is: use the ADMM method to solve the third sub-problem model and obtain the salt and pepper noise solved for the kth time Among them, the model of the third sub-problem is: T β / μ (Q (s,t) )=max(|Q (s,t) |-β / μ,0)sign(Q (s,t) ) Salt and pepper noise solved for the kth time for: in, is the minimum point S when finding the minimum value of the function i Assign the value of β is the regularization parameter, μ is the penalty parameter, ‖S i ‖1 is S i The 1-norm of S i is the i-th structure group Z i Salt and pepper noise in is the original noise-free structure group solved for the kth time, is the Gaussian noise solved for the k-1th time, is the Lagrange multiplier solved for the k-1th time, is the square of the F norm, Q (s,t) is the (s,t)th element of the matrix Q, T β / μ (·) is a soft threshold operator, s = 1, 2, ... M, t = 1, 2, ... N, M is the number of rows of the matrix Q, N is the number of columns of the matrix Q, |Q (s,t) |To obtain Q (s,t) The absolute value of , sign(·) is the sign function.

7. The image mixed noise removal method based on low-rank function and multi-directional total variation according to claim 4 is characterized in that: The solution method for the fourth sub-problem in step A3 is: solve the fourth sub-problem model to obtain the k-th solved Gaussian noise Among them, the fourth sub-problem model is: Gaussian noise of the kth solution for: in, is the minimum point N when finding the minimum value of the function i Assign the value of γ is the regularization parameter, is the square of the F norm, is the Gaussian noise solved for the kth time, N i is the i-th structure group Z i Gaussian noise in, μ is the penalty parameter, is the original noise-free structure group solved for the kth time, is the salt and pepper noise solved for the kth time, is the Lagrange multiplier solved for the k-1th time.

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