A cauchy noise image restoration method based on ratio type sparse constraint

Abstract

The application discloses a Cauchy noise image restoration method based on a ratio type sparse constraint, and belongs to the technical field of digital image processing. The method uses the non-local similarity of an image, takes a structure group formed by similar blocks as a training object, learns an orthogonal dictionary, performs sparse representation on the structure group, and applies a non-convex constraint on representation coefficients. Firstly, a Cauchy noise image after preprocessing is blocked, a most similar group of image block vectors is extracted in a search window with a reference block as a center after being structured, then the orthogonal dictionary is trained by using the structure group, the correlation in the structure group is enhanced by using joint coding, and a norm is used as a regularization term to perform sparse constraint on a coefficient matrix, and finally, the Cauchy noise is removed and local texture details are restored. The proposed model is solved by using an alternating direction multiplier method, most of the noise can be effectively removed and image texture information can be reserved, and therefore, the method can be used for Cauchy noise image restoration.

Description

Technical Field

[0001] This invention belongs to the field of digital image processing technology, and relates to the joint encoding and ratio-based encoding of image structure groups. A sparse representation method with regular constraints is used to recover images containing Cauchy noise while preserving image texture details. Background Technology

[0002] Most existing noisy image restoration methods are designed for common noise models such as Gaussian noise, salt-and-pepper noise, or Poisson noise. However, in complex imaging environments or under non-ideal transmission conditions, image noise often exhibits strong peaks and heavy-tailed distribution characteristics. Traditional noise models based on the Gaussian assumption are difficult to accurately characterize the statistical laws of such noise. Cauchy noise, as a typical heavy-tailed noise, has obvious long-tailed distribution characteristics, easily generates anomalous perturbations with large amplitudes, and its variance does not have a finite value. This makes it difficult for traditional restoration methods relying on quadratic error criteria or Gaussian statistical assumptions to achieve ideal results in Cauchy noise environments, leading to problems such as incomplete noise suppression, severe loss of image details, and insufficient algorithm robustness.

[0003] Currently, research methods for noisy image restoration mainly include spatial filtering methods, transform domain sparse representation methods, nonlocal self-similarity methods, and deep learning methods. Spatial filtering methods, such as mean filtering, median filtering, and bilateral filtering, have advantages such as simple implementation and low computational cost, but they often struggle to simultaneously suppress noise and preserve edges under strong noise conditions. Transform domain sparse representation methods utilize the sparse prior of the image in a specific transform domain or dictionary to achieve image restoration through sparse constraints, offering advantages in local texture and edge preservation. Nonlocal self-similarity methods search for similar blocks in the image and utilize the correlation and low-rank characteristics between block groups to achieve restoration, effectively preserving repetitive structural information in the image. While deep learning methods have shown good performance in noisy image restoration tasks, they typically rely on a large number of training samples matched to the noise model; their generalization performance remains somewhat insufficient when the noise distribution changes or the real-world scene is complex.

[0004] Therefore, in view of the shortcomings of existing Cauchy noise image restoration methods in terms of heavy-tailed anomalous noise suppression, image detail preservation, and restoration robustness, it is necessary to study an image restoration method that integrates non-local similarity and sparse representation priors. This method can fully explore the correlation structure between image blocks under a unified model framework, and further enhance the model's representation ability and restoration performance through orthogonal dictionary learning and non-convex sparse constraints, thereby improving the overall restoration effect of Cauchy noise images. Summary of the Invention

[0005] The purpose of this invention is to address the shortcomings of existing methods for restoring Cauchy noise images by proposing a method based on nonlocal similarity and sparse representation. This method constructs a unified model combining orthogonal dictionary learning and nonconvex sparse constraints. To fully utilize the nonlocal similarity within structural groups, a joint encoding strategy is employed to collaboratively represent image patches; simultaneously, a ratio-based constraint is applied to the sparse representation coefficients. Sparsity constraints are applied to further enhance the sparsity of the coefficients. An orthogonal dictionary is learned for each structure group, which not only improves the dictionary's representational power but also enhances the quality of the reconstructed image. Specifically, this includes the following steps:

[0006] (1) Input an image containing Cauchy noise. The method extracts overlapping image patches from the image, searches for the set of image patches with the highest similarity to the reference patch within a set search window, and vectorizes and stacks them into a similar patch matrix to obtain a structure group, so as to mine the non-local self-similarity prior information inside the image.

[0007] (2) Taking the selected structure groups as the objects, the singular value decomposition method is used to learn the corresponding orthogonal dictionary for each structure group; then, similar image blocks within the group are jointly encoded under this dictionary, and the representation coefficients of each image block are arranged into a coefficient matrix by column; in order to characterize the joint sparsity of image blocks within the structure group in the representation domain, the coefficient matrix is ​​first calculated row by row. Norms are used to characterize the overall response of each dictionary atom across the entire structure set, and then the resulting row norm vector is applied. Non-convex sparse constraints are applied; finally, a Cauchy noise image restoration model based on structure group ratio sparse constraints is established:

[0008]

[0009] in The image is to be recovered. It is an observed Cauchy noise image. It is a data fidelity model for discrete Cauchy noise derived from maximum a posteriori estimation and Bayes' theorem, used to measure the image to be restored. With observation images Consistency under the Cauchy noise assumption, where This represents the natural logarithm function, which acts on the vector within the parentheses element-wise. Represents the element-wise residual vector. This indicates that the residual vector is squared element by element. This is the scaling parameter for Cauchy noise, used to control the sensitivity of the data fidelity term to the residuals. A larger value leads to a wider residual distribution, a heavier tail, and a greater likelihood of extreme values. It is a Euclidean inner product, specifically representing a vector. Each element and the total vector The elements are multiplied element by element and then summed. L is the size of the image, i.e., the number of pixels. It is the extraction operator used to extract the structure group of the i-th reference block. This represents a structural group composed of M vectorized similar blocks arranged together. These represent the vectors in the first and second columns of the structural group, i.e., the first and second similar image patches. It is the reconstruction error term of the sparse representation, represented as a matrix. The square of the Frobenius norm; It is a regularization constraint term in the sparse representation, representing the first regularization of the coefficient matrix. Calculated by row Norm, then apply a norm to the resulting row norm vector. Non-convex sparse constraints, It is an orthogonal dictionary. Represents the identity matrix. This represents the regularization parameter, which is used to balance the relative weights between data fidelity terms and regularization terms.

[0010] The innovation of this invention is to propose a Cauchy noise image restoration model based on ratio-based sparse constraints. This method first divides the image into blocks and searches for several most similar image blocks to the target image block, forming a structure group. Then, an adaptive orthogonal dictionary is learned for each structure group, and the image blocks within the group are jointly encoded under the corresponding dictionary to fully utilize non-local similarity. Simultaneously, it employs... The norm imposes sparsity constraints on the representation coefficients, thereby enhancing the joint sparse representation capability; finally, the specific solution steps of the model are given.

[0011] The beneficial effects of this invention are: Orthogonal dictionary learning and continuous updating are performed on each structure group, improving the dictionary's adaptive representation ability of image structural features and simplifying sparse coding. Norm constraints enhance the sparsity constraints on image coefficients, resulting in more accurate sparse coefficients and improved final recovery performance.

[0012] This invention is mainly verified by simulation experiments, and all steps and conclusions have been verified to be correct on MATLAB 2022a. Attached Figure Description

[0013] Figure 1 This is a flowchart of the present invention;

[0014] Figure 2 is the original image of the 256×256 sized house used in the simulation of this invention;

[0015] Figure 3 is the original image of the 256×256 size clock used in the simulation of this invention;

[0016] Figure 4 shows the noise scale parameters for different methods. The results of the restoration of the house image;

[0017] Figure 5 shows the noise scale parameters obtained by different methods. The results of the restoration of the house image;

[0018] Figure 6 shows the noise scale parameters for different methods. The result of restoring the clock image;

[0019] Figure 7 shows the noise scale parameters obtained by different methods. The result of restoring the clock image. Detailed Implementation

[0020] Reference Figure 1 A method for restoring Cauchy noise images based on ratio-type sparse constraints is described below, with the following specific steps:

[0021] Step 1, Obtaining the structure group:

[0022] (1a) Input an observation image with Cauchy noise, which is 256×256 in size. The observed image is then subjected to median filtering to obtain a preprocessed image. ;

[0023] (1b) Set the sliding window size to 8×8, and in the preprocessed image In this process, 20 row positions are selected at equal intervals starting from the first row, and 20 column positions are selected at equal intervals starting from the first column, thereby determining 20×20 reference positions, and 400 reference image blocks of size 8×8 are extracted based on the reference positions;

[0024] (1c) Take the i-th reference image block as the target image block, and construct a square search region with a side length of 30 centered on the target image block; when the search region exceeds the image boundary, the excess part is truncated. Within the search region, a sliding window of size 8×8 with a step size of 1 is used to extract candidate image blocks in order from left to right and from top to bottom, and select the 80 image blocks most similar to the target image block. These 80 similar image blocks are converted into column vectors of length 64 to form a structure group matrix of size 64×80. ;

[0025] (1d) Repeat step (1c), change the index i of the target image block, and traverse all the image blocks divided in step (1b) until all image blocks are processed, thereby obtaining the set of all structure group matrices corresponding to the whole image.

[0026] Step 2: Taking the selected structure groups as the object, apply singular value decomposition to each structure group to obtain an orthogonal dictionary, and then use joint encoding and... The norm pairs are used for sparse encoding of the sparse matrix under the corresponding dictionary. Finally, a Cauchy noise image restoration model based on nonlocal self-similarity and sparse representation is established:

[0027] Equation (1)

[0028] in The image is to be recovered. It is an observed Cauchy noise image. It is a data fidelity model for discrete Cauchy noise derived from maximum a posteriori estimation and Bayes' theorem, used to measure the image to be restored. With observation images Consistency under the Cauchy noise assumption, where This represents the natural logarithm function, which acts on the vector within the parentheses element-wise. Represents the element-wise residual vector. This indicates that the residual vector is squared element by element. This is the scaling parameter for Cauchy noise, used to control the sensitivity of the data fidelity term to the residuals. A larger value leads to a wider residual distribution, a heavier tail, and a greater likelihood of extreme values. It is a Euclidean inner product, specifically representing a vector. Each element and the total vector The elements are multiplied element by element and then summed. L is the size of the image, i.e., the number of pixels. It is the extraction operator used to extract the structure group of the i-th reference block. This represents a structural group composed of M vectorized similar blocks arranged together. These represent the vectors in the first and second columns of the structural group, i.e., the first and second similar image patches. It is the reconstruction error term of the sparse representation, represented as a matrix. The square of the Frobenius norm, It is a regularization constraint term in the sparse representation, representing the first regularization of the coefficient matrix. Calculated by row Norm, and then apply a norm to the resulting row norm vector. Non-convex sparse constraints, It is an orthogonal dictionary. Represents the identity matrix. This represents the regularization parameter, which is used to balance the relative weights between data fidelity terms and regularization terms.

[0029] Step 3: Solve the above Cauchy noise image restoration model using the alternating direction multiplier method:

[0030] Introducing auxiliary variables , as well as ,because Then the Lagrange augmented form of the model is obtained as follows:

[0031] Equation (2)

[0032] in For penalty parameters, It is a Lagrange multiplier.

[0033] (3a) Fixed Then, in equation (2) regarding The subproblems are:

[0034] Equation (3)

[0035] This subproblem can be solved by... Solving for the problem using singular value decomposition, i.e. The optimal orthogonal dictionary is obtained as follows:

[0036] Equation (4)

[0037] (3b) Fixed Then, in equation (2) regarding The subproblems are:

[0038] Equation (5)

[0039] Combining and simplifying the last two terms of equation (5), we get:

[0040] Equation (6)

[0041] Since the objective function in equation (6) is a least squares problem, its optimal solution is obtained by solving the following normal equation:

[0042] Equation (7)

[0043] The optimal solution is:

[0044] Equation (8)

[0045] (3c) Fixed Then, in equation (2) regarding The subproblems are:

[0046] Equation (9)

[0047] set up as well as Then the minimization subproblem simplifies to:

[0048] Equation (10)

[0049] if Then any satisfying The matrices are all solutions to minimizing subproblems; if ,but It means minimizing the solution to the subproblem; if and Taking the derivative of the objective function, we get:

[0050] Equation (11)

[0051] Clearly, there exists a positive number. Make If given and set ,for Solving the minimization subproblem becomes a problem about The problem of finding the roots of a univariate equation, in other words, if let Substituting into equation (11) and simplifying, we obtain a equation about The cubic equation of one variable:

[0052] Equation (12)

[0053] The cubic equation can be solved using Cardan's formula, and one of the real roots is expressed as:

[0054] Equation (13)

[0055] Equation (14)

[0056] therefore, The solution can be written in the following form:

[0057] Equation (15)

[0058] Among them when At any time, satisfying matrix All of these are solutions to the subproblem.

[0059] (3d) fixed Then, in equation (2) regarding The subproblems are:

[0060] Equation (16)

[0061] If let The second term in equation (16) has the following equivalence relation:

[0062] Equation (17)

[0063] To each and Decomposition by row normalization can be represented as:

[0064] Equation (18)

[0065] Equation (19)

[0066] in and These represent the diagonal elements as and Each row vector in The diagonal matrix of the norm, and Indicates by and The matrix formed by the unit vectors corresponding to each row vector in equation (17) is such that, according to the Cauchy-Schwarz inequality, the terms on the right-hand side of equation (17) satisfy the lower bound:

[0067] Equation (20)

[0068] in and Representing matrices respectively and The k-th row, where n represents the row number, and the condition for equality in equation (20) to hold is: Additionally, the first term in equation (16) It can be represented as of Norm, which is the following equivalence relation:

[0069] Equation (21)

[0070] in This has already been obtained through (3c), therefore It can be considered as a constant, and further, a lower bound of the objective function of equation (16) can be obtained:

[0071] Equation (22)

[0072] Clearly, this lower bound can be separated into terms related to... The scalar optimization problem for each diagonal element can be solved using the soft thresholding method:

[0073] Equation (23)

[0074] in This represents the diagonal element-wise soft threshold contraction operator, combined with the equality condition in equation (20). Therefore, in the (3d) subproblem regarding The optimal solution to the sparse coding problem is:

[0075] Equation (24)

[0076] (3e) Fixed Then, in equation (2) regarding The subproblems are:

[0077] Equation (25)

[0078] According to the matrix and By definition, the summation term of the first term of the objective function in equation (24) can be:

[0079] Equation (26)

[0080] Therefore, the objective function is multiple functions related to the image. of The sum of squares of norms is essentially a representation of the graph. The least squares problem can be solved by finding the optimal solution by solving the corresponding normal equations below:

[0081] Equation (27)

[0082] because Extracting operators for image patches This can be simplified to diagonal form, therefore we get:

[0083] Equation (28)

[0084] (3f) Fixed Then regarding The subproblems are:

[0085] Equation (29)

[0086] For seeking in equation (28) The optimization subproblem, let:

[0087] Equation (30)

[0088] Therefore, the optimization subproblem becomes a problem about... The problem of solving a quadratic function in one variable can be solved quickly using Newton's method:

[0089] Equation (31)

[0090] in The first derivative represents, Representing its second derivative, the variables are modified using the gradient and curvature information of the current point through the above update method, so that they gradually approach the optimal solution of equation (29).

[0091] (3g) Update Lagrange multipliers:

[0092] Equation (32)

[0093] Equation (33)

[0094] Equation (34)

[0095] (3h) Repeat steps (3a) to (3g) until the estimated image meets the conditions or the number of iterations reaches the preset upper limit.

[0096] The effects of this invention can be further illustrated by the following simulation experiments:

[0097] I. Experimental Conditions and Content

[0098] Experimental conditions: The experiment used grayscale images of the house (Figure 2(a)) and the clock (Figure 3(a)) with a size of 256×256. The peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM) were used to objectively evaluate the restoration results. PSNR was defined as:

[0099] Equation (35)

[0100] in and These are the original image and the restored image, respectively. L represents the total number of pixels in the image. A higher PSNR value indicates a higher quality restored image. SSIM is defined as:

[0101] Equation (36)

[0102] in and yes and The mean, and yes and standard deviation yes and covariance, and These are two constants to avoid instability. SSIM ranges from 0 to 1, with higher values ​​indicating higher quality recovered images.

[0103] Experimental content: Scale parameters of Cauchy noise This value is used to characterize the dispersion of noise distribution; the larger the value, the stronger the noise perturbation and the more difficult the image restoration. (The remaining text appears to be incomplete and requires further context.) and Under these conditions, the method described in this invention is compared with the classic TV method in the field of Cauchy noise image restoration, as well as the currently leading GBLR and OGSWHL methods. The TV method utilizes the gradient sparsity of the image and... Norm-constrained regularization terms, in the Cauchy noise scale parameter The recovery results under the given conditions are shown in Figures 4(a) and 5(a), with respect to the scale parameter. The restoration results under the given conditions are shown in Figures 6(a) and 7(a). The GBLR method utilizes the low-rank property of the structure group matrix obtained from the image to transform the Cauchy noise image restoration process into a low-rank matrix restoration problem. It employs the kernel norm to impose low-rank constraints on the structure group and uses the alternating direction multiplier method to solve the problem. The Cauchy noise scale parameter... The recovery results under the given conditions are shown in Figures 4(b) and 5(b), with respect to the scale parameter. The recovery results under the given conditions are shown in Figure 6(b) and Figure 7(b). The OGSWHL method uses overlapping image sparsity instead of single gradient sparsity, avoiding the gradient artifacts generated by the TV method. It also incorporates the hyper-Laplacian prior, which addresses the issue of Cauchy noise scale parameters. The recovery results under the given conditions are shown in Figures 4(c) and 5(c), with respect to the scale parameter. Recovery results under the conditions are as follows Figure 6(c) , 7(c) The method of this invention sets the image patch size. The search window size is The search involves finding 80 similar image patches and 400 structure groups, with a regularization parameter of [value missing]. , At Cauchy noise scale parameters The recovery results under the given conditions are shown in Figures 4(d) and 5(d), with respect to the scale parameter. The recovery results under the given conditions are shown in Figure 6(d) and Figure 7(d).

[0104] To more clearly compare the image restoration performance of different methods under Cauchy noise conditions, typical local areas were marked and magnified with white rectangles in the experimental results, as shown in Figures 2(b) and 3(b). In Figures 4, 5, 6, and 7, (e)–(h) are magnified local images, corresponding to the TV method, GBLR method, OGSWHL method, and the method of this invention, respectively. Observation reveals that although the TV method can preserve important details, it produces step artifacts in flat areas and still has residual noise in the restored image; the GBLR method's result image has good visual effects and preserves a large amount of texture details, but still has a small amount of residual noise; the OGSWHL method reduces gradient artifacts to some extent, but the texture in the restored image is over-smoothed, resulting in the loss of many texture details and poor image clarity; the method of this invention improves upon the shortcomings of the above methods, preserving a large amount of texture details while effectively removing noise, achieving the best visual effect. In comparison, the restoration result of the method of this invention is the best.

[0105] Table 1. PSNR index for different recovery methods

[0106]

[0107] Table 1 shows the PSNR index of the restoration results of each method, where a higher PSNR value indicates a better restored image. As can be seen from the table, the method of the present invention has a significant improvement over other methods, and this result is consistent with the restored image.

[0108] Table 2 SSIM index for different recovery methods

[0109]

[0110] Table 2 shows the SSIM values ​​of the restoration results of each method. A higher SSIM value indicates that the restoration result is closer to the real image. It can be seen that the method of the present invention has the highest SSIM value, and the restoration result is closer to the original image. This result is consistent with the restored image.

[0111] The above experiments show that the restored images obtained by the present invention are not only rich in detail, but also have good visual effects and objective evaluation indicators. Therefore, the present invention is effective for image restoration of Cauchy noise.

Claims

1. A method for restoring Cauchy noise images based on ratio-type sparse constraints, comprising the following steps: (1) Input an image containing Cauchy noise. The method extracts overlapping image patches from the image, searches for the set of image patches with the highest similarity to the reference patch within a set search window, and vectorizes and stacks them into a similar patch matrix to obtain a structure group, so as to mine the non-local self-similarity prior information inside the image. (2) Taking the selected structure groups as the objects, the singular value decomposition method is used to learn the corresponding orthogonal dictionary for each structure group; then, similar image blocks within the group are jointly encoded under this dictionary, and the representation coefficients of each image block are arranged into a coefficient matrix by column; in order to characterize the joint sparsity of image blocks within the structure group in the representation domain, the coefficient matrix is ​​first calculated row by row. Norms are used to characterize the overall response of each dictionary atom across the entire structure set, and then the resulting row norm vector is applied. Ratio-based non-convex sparse constraints; Finally, a Cauchy noise image restoration model that integrates nonlocal similarity and sparse representation is established. ; in The image is to be recovered. It is an observed Cauchy noise image. It is a data fidelity model for discrete Cauchy noise derived from maximum a posteriori estimation and Bayes' theorem, used to measure the image to be restored. With observation images Consistency under the Cauchy noise assumption, where This represents the natural logarithm function, which acts on the vector within the parentheses element-wise. Represents the element-wise residual vector. This indicates that the residual vector is squared element by element. This is the scaling parameter for Cauchy noise, used to control the sensitivity of the data fidelity term to the residuals. A larger value leads to a wider residual distribution, a heavier tail, and a greater likelihood of extreme values. It is a Euclidean inner product, specifically representing a vector. Each element and the total vector The elements are multiplied element by element and then summed. L is the size of the image, i.e., the number of pixels. It is the extraction operator used to extract the structure group of the i-th reference block. This represents a structural group composed of M vectorized similar blocks arranged together. These represent the vectors in the first and second columns of the structural group, i.e., the first and second similar image patches. It is the reconstruction error term of the sparse representation, represented as a matrix. The square of the Frobenius norm, It is a regularization constraint term in the sparse representation, representing the first regularization of the coefficient matrix. Calculated by row Norm, then apply a norm to the resulting row norm vector. Non-convex sparse constraints, It is an orthogonal dictionary. Represents the identity matrix. This represents the regularization parameter, which is used to balance the relative weights between data fidelity terms and regularization terms.

2. The Cauchy noise image restoration method based on ratio-type sparse constraints according to claim 1 is characterized in that when solving the model in step (2) using the alternating direction multiplier method, an auxiliary variable is introduced. , as well as ,because Then the Lagrange augmented form of the model is obtained as follows: ; in For penalty parameters, For Lagrange multipliers; (2a) Fixed Then the denoising model regarding The subproblems are: ; This subproblem can be solved using singular value decomposition; (2b) Fixed Then the denoising model regarding The subproblems are: ; This subproblem can be solved using the least squares method; (2c) Fixed Then, in the denoising model, regarding The subproblem is ; This subproblem can be solved using Cardan's formula for cubic equations; (2d) Fixed Then the denoising model regarding The subproblem is ; This problem can be solved using the soft thresholding method; (2e) Fixed Then the denoising model regarding The subproblem is ; This problem can be solved using Newton's iteration method; (2f) Fixed Then the denoising model regarding The subproblem is ; This problem can be solved using the least squares method; (2g) Repeat steps (2a) to (2f) until the number of iterations reaches the preset upper limit or the stopping condition is met.

3. The Cauchy noise image restoration method based on ratio-type sparse constraints according to claim 2 is characterized in that when solving the sub-problem in step (2a) using singular value decomposition, the following steps can be followed: (2a1) According to the dictionary The m-th atom, which is the m-th column vector of the dictionary. and its corresponding sparse matrix The m-th row vector The reconstruction error was obtained. ; (2a2) Update the residual matrix Perform singular value decomposition on it. Using left singular value matrix The m-th column vector Update the m-th atom of the dictionary Repeat this process until the dictionary is completely updated. All the atoms in it.

4. The Cauchy noise image restoration method based on ratio-type sparse constraints according to claim 2 is characterized in that when solving the subproblem in step (2b) using the least squares method, the following steps can be followed. (2b1) Combining and simplifying the last two terms in subproblem (2b), we get: ; ; (2b2) ​​Its optimal solution is obtained by solving the following normal equation: ; therefore, The optimal solution is: ; Therefore, the optimal solution to this subproblem can be obtained.

5. The Cauchy noise image restoration method based on ratio-type sparse constraints according to claim 2 is characterized in that when solving the subproblem in step (2c) using the cubic root formula, the following steps can be followed: (2c1) Let as well as Then the minimization subproblem simplifies to ; (2c2) If Then any satisfying The matrices are all solutions to minimizing subproblems; if ,but To minimize the solution to the subproblem; if and Taking the derivative of the objective function, we can obtain ; (2c3) Clearly, there exists a positive number. Make If given and set ,for Solving the minimization subproblem becomes a problem about The problem of finding the roots of a univariate equation, in other words, if let Substituting into equation (11) and simplifying, we obtain a equation about The cubic equation of one variable: ; The cubic equation can be solved using Cardan's formula, and one of the real roots is expressed as: ; ; (2c4) Combining the above steps, The solution can be written in the following form: ; Among them when At any time, arbitrarily satisfy matrix All of these are solutions to the subproblem.

6. The Cauchy noise image restoration method based on ratio-type sparse constraints according to claim 2 is characterized in that when solving the sub-problem in step (2d) using the soft thresholding method, the following steps can be followed: (2d1) Let Then the second term in (2d) has the following equivalence relation: ; To each and Decomposition by row normalization can be represented as: ; ; in and These represent the diagonal elements as and Each row vector in The diagonal matrix of the norm, and Indicates by and The matrix formed by the unit vectors corresponding to each row vector in the matrix; (2d2) According to the Cauchy-Schwarz inequality, the right-hand side of the equivalence relation has a lower bound, which can be expressed as: ; in and Representing matrices respectively and In the k-th row of the equation, where n represents the row number, the condition for the inequality to hold true is: ;in addition, It can be represented as of Norm, which is the following equivalence relation: ; in Already obtained, therefore This can be considered a constant. Combining the above two steps, we can obtain a lower bound for the objective function: ; (2d3) Clearly, this lower bound can be separated into terms about The scalar optimization problem for each diagonal element is solved using a soft thresholding operator: ; After obtaining the scalar value of each element, combine it with the equality conditions. Sub-problems about The optimal solution to the sparse coding problem is... ; Therefore, the optimal solution to this subproblem can be obtained.

7. The Cauchy noise image restoration method based on ratio-type sparse constraints according to claim 2 is characterized in that when solving the sub-problem in step (2e) using the Newton iteration method, the following steps can be followed: (2e1) Let the objective function corresponding to subproblem (2e) be expressed as: ; (2e2) Then the target vector It can be obtained through the following iterations ; in The first derivative represents, Representing its second derivative, the variables are corrected using the gradient and curvature information of the current point through the above update method, so that they gradually approach the optimal solution of the subproblem in step (2e).

8. The Cauchy noise image restoration method based on ratio-type sparse constraints according to claim 2 is characterized in that when solving the subproblem in step (2f) using the least squares method, the following steps can be followed: (2f1) Based on the matrix and By definition, the first term of the summation in the objective function (2f) can be: ; Therefore, the objective function is multiple functions related to the image. of The sum of squares of the norm is essentially a representation of the graph. The least squares problem can be solved by finding the optimal solution by solving the corresponding normal equations below: ; (2f2) Due to Extracting operators for image patches This can be simplified to diagonal form, therefore we get: ; Therefore, the optimal solution to this subproblem can be obtained, which is the image to be recovered.

Images