A color image restoration method based on minimization of coefficient tensor nuclear norm
Through tensor representation and redundant dictionary learning, combined with tensor kernel norm constraints, the problem of existing matrix methods destroying data structures in image restoration is solved, and more accurate and detailed color image restoration is achieved.
Patent Information
- Application Number
- CN202210461053.8
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2022-04-28
- Publication Date
- 2025-05-23
- Estimated Expiration
- 2042-04-28
AI Technical Summary
Existing matrix-based image restoration methods will destroy the inherent structure and characteristics of the original data when processing high-dimensional data, resulting in inaccurate restoration results.
Tensors are used to represent images linearly, and redundant dictionaries are learned using the initial reconstruction image to enhance sparse representation ability, and tensor kernel norm constraints are used to make full use of the low-rank characteristics of the image.
The dimensionality reduction defect based on the matrix method is effectively avoided, more intrinsic spatial structure information of the image is retained, and the accuracy of linear representation of the image is improved, resulting in good visual effects of the reconstruction image and sufficient details are retained.
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Figure CN115239573B_ABST
Abstract
Description
Technical Field
[0001] The invention belongs to the technical field of digital image processing, and particularly relates to a method for realizing image restoration by minimizing the constraint of coefficient tensor nuclear norm. Background Art
[0002] Image restoration is a processing technology to improve image quality and is one of the main research directions in the field of image processing. Image restoration analyzes a known degraded image and estimates the result that is closest to the real image. It can be regarded as an inverse solution process of a signal. Directly solving this process will result in a non-unique solution space, that is, it has pathological characteristics. Therefore, it is often used to transform the ill-posed inverse problem into a well-posed problem from the perspective of regularization.
[0003] In recent years, image restoration methods based on sparse representation and low-rank matrix estimation have achieved great success. Since similar image blocks in natural images have similar structures, after vectorization, the matrix composed of similar image blocks has low-rank characteristics. Applying low-rank constraints to each matrix can restore the original image. Although traditional matrix-based low-rank representation methods can well represent low-dimensional data, for high-dimensional data such as videos and images, these methods need to convert data samples into vectors, which often destroys the inherent intrinsic structure, features and other information of the original data, resulting in inaccurate restoration results. Therefore, tensor-based data analysis methods have received increasing attention. Tensors can be regarded as high-dimensional extensions of matrices, which can not only effectively represent high-dimensional data, but also maintain the intrinsic structural characteristics of the original data. The tensor low-rank representation method is an extension of the matrix low-rank representation method. It can accurately restore low-rank tensors directly from damaged tensors. Image clustering and video / image denoising applications have verified the effectiveness of the algorithm. However, the solution complexity of such methods is high and they cannot be applied to image restoration fields such as deblurring.
[0004] Recent studies have shown that there are low-rank characteristics inside the image. If the original three-dimensional tensor is used as the unit of linear representation of the image, the defect of spatial structure information being destroyed after data dimensionality reduction can be overcome. In order to learn a highly redundant dictionary, an economical tensor singular value decomposition is implemented from the initial reconstructed image, so as to learn the dictionary to linearly represent the image, making the restored image more detailed. Summary of the invention
[0005] The purpose of the present invention is to address the shortcomings of existing matrix-based image restoration methods and propose a color image restoration method based on minimizing the coefficient tensor nuclear norm. The method first uses a tensor to linearly represent the image, and uses the tensor nuclear norm to constrain the low-rank characteristics of the image representation coefficients. At the same time, the redundant dictionary is learned using the initial reconstructed image to enhance the ability of linear representation of the image. Specifically, the following steps are included:
[0006] (1) Input a three-channel color degraded image and use the tensor robust principal component analysis method to perform initial reconstruction to obtain the initial reconstructed image right After performing economic tensor singular value decomposition, the redundant dictionary is obtained The specific steps are:
[0007] (1a) First, Perform a tensor singular value decomposition:
[0008]
[0009] in and is an orthogonal tensor satisfying for The conjugate transpose of Indicates that the first positive slice is the identity matrix, and the other positive slices are unit tensors of zero matrices. represents the diagonal tensor of the frontal slices, express and The tensor product between ;
[0010] (1b) Interception The non-zero part of and The lateral slicing of implements economical tensor singular value decomposition:
[0011]
[0012] in and Respectively represent the intercepted tensor and The tensor consisting of the first r lateral slices of , for The conjugate transpose of Represents a truncated tensor A tensor consisting of the first r rows and first r columns of each positive slice matrix, where r represents the tensor The maximum number of non-zero diagonal elements in each frontal slice is used to further calculate the redundant dictionary
[0013]
[0014] The redundant dictionary is obtained from
[0015] (2) Considering that each channel sub-image of a color image has highly similar structural features, the original image tensor has a low-rank characteristic. An image restoration model under low-rank constraints is established with the original image tensor as the processing unit:
[0016]
[0017] Where λ represents the regularization parameter, represents the degraded color image, represents the representation coefficient of the image to be restored, H represents the image degradation matrix, Representing a tensor The nuclear norm of in Representing a tensor The i-th frontal slice matrix after discrete Fourier transform on each tube fiber, Represents the frontal slice matrix The nuclear norm of the matrix The sum of the singular values, represents the square of the norm of the matrix F, Representing a tensor With tensor The tensor product between them, the operator unfold(·) represents the matrix of each positive slice of the vectorized tensor;
[0018] (3) The alternating direction multiplier method is used to iteratively solve the entire reconstruction model. First, the expression in (2) is converted into an augmented Lagrangian function:
[0019]
[0020] in is an auxiliary variable, defined as is the Lagrange multiplier, β is the penalty parameter, <·> represents the inner product of two tensors, and the third term Represents the square of the norm of the tensor F;
[0021] (3a) Given In this case, solve the variables in the augmented Lagrangian function in the t+1 iteration The subproblem can be transformed into solving a convex optimization problem on a three-dimensional tensor:
[0022]
[0023] The variables This problem is to transform all variables into the frequency domain, solve them using the proximal gradient method, and then inversely transform them into the spatial domain to obtain
[0024] (3b) Given In this case, solve the variables in the augmented Lagrangian function in the t+1 iteration The subproblem can be transformed into solving a least squares problem:
[0025]
[0026] The variables This problem is solved using the least squares method.
[0027] (3c) Update the Lagrange multiplier in (3):
[0028]
[0029] (4) Repeat steps (3a) to (3c) until the restored color image meets the convergence condition or the number of iterations reaches a preset upper limit.
[0030] The innovation of the present invention is to use tensors to linearly represent images, and use the initial reconstructed image to learn a redundant dictionary, thereby enhancing the sparse representation ability of the image; further, the representation coefficients of the image are constrained by the tensor nuclear norm to fully utilize the inherent low-rank characteristics of the image; in addition, the alternating direction multiplier method is used to solve the representation coefficients and the reconstructed image respectively for the reconstruction model, wherein the proximal gradient method is applied to solve the representation coefficient optimization problem containing the tensor nuclear norm constraint term, and the reconstructed image is obtained by calculating the approximate least squares solution, thereby realizing the restoration of the color image.
[0031] The beneficial effects of the present invention are as follows: the use of tensors to linearly represent images effectively avoids the defect of image dimensionality reduction required by matrix-based methods, and retains more intrinsic spatial structure information of the image; the tensor nuclear norm is used to constrain the representation coefficients of the image, making full use of the intrinsic low-rank characteristics of the image; the initial reconstructed image is used to learn a redundant dictionary, which improves the accuracy of the linear representation of the image, so that the final reconstructed image not only has a good overall visual effect, but also retains a large number of image details, and the estimation result is closer to the real image.
[0032] The present invention is mainly verified by simulation experiment method, and all steps and conclusions are verified to be correct on MATLAB8.0. BRIEF DESCRIPTION OF THE DRAWINGS
[0033] Figure 1 It is a work flow chart of the present invention;
[0034] Figure 2 is the original 3-channel color image used in the simulation of the present invention;
[0035] Figure 3 The restoration results of the image Bike with a pixel retention rate of 50% using various methods (SNN method, SKR method, JPG-SR method and the method of the present invention in order);
[0036] Figure 4The restoration results of the image Lena after the text information is applied using various methods (SNN method, SKR method, JPG-SR method and the method of the present invention in order);
[0037] Figure 5 It is the restoration result of the image Light after the motion blur kernel is applied using various methods (IDD-BM3D method, TRPCA method, JSM method, JPG-SR method and the method of the present invention in order);
[0038] Figure 6 The restoration results of the Castle image with 25% noise ratio added using various methods (HOSVD method, RTA-LSM method, TRPCA method, STROLLR method and the method of the present invention in order); DETAILED DESCRIPTION
[0039] Reference Figure 1 The present invention is a color image restoration method based on minimizing the coefficient tensor nuclear norm, and the specific steps include the following:
[0040] Step 1: Initially reconstruct the degraded color target image and obtain the dictionary.
[0041] (1a) Input a three-channel color degraded image and use the tensor robust principal component analysis method to perform initial reconstruction to obtain the initial reconstructed image right After performing economic tensor singular value decomposition, the redundant dictionary is obtained The specific steps are:
[0042] (1a1) First, the initial reconstructed image Perform a tensor singular value decomposition:
[0043]
[0044] in and is an orthogonal tensor satisfying for The conjugate transpose of Indicates that the first positive slice is the identity matrix, and the other positive slices are unit tensors of zero matrices. represents a diagonal tensor whose frontal slices are all diagonal matrices, express and The tensor product between ;
[0045] (1a2) Truncate the diagonal tensor The non-zero part of the front slice matrix is intercepted accordingly and The lateral slicing matrix implements economical tensor singular value decomposition:
[0046]
[0047] in and Respectively represent the intercepted tensor and The tensor consisting of the first r lateral slices of , for The conjugate transpose of Represents a truncated tensor A tensor consisting of the first r rows and first r columns of each positive slice matrix, where r represents the tensor The maximum number of non-zero diagonal elements in each frontal slice is used to further calculate the redundant dictionary
[0048]
[0049] The redundant dictionary is obtained from
[0050] Step 2: Establish an image restoration model with the coefficient tensor nuclear norm minimization constraint and estimate the coefficients.
[0051] (2a) The dictionary obtained using formula (3) Degraded color image of the target To make a linear representation:
[0052]
[0053] in Representing a tensor With tensor Considering that the sub-images of each channel of the color image have highly similar structural features, the original image tensor has a low-rank characteristic. The image restoration model under low-rank constraint is established with the original image tensor as the processing unit:
[0054]
[0055] Where λ represents the regularization parameter, represents the representation coefficient of the image to be restored, H represents the image degradation matrix, Representing a tensor The nuclear norm of in Representing a tensor The i-th frontal slice matrix after discrete Fourier transform on each tube fiber, Represents the frontal slice matrix The nuclear norm of the matrix The sum of the singular values, represents the square of the norm of the matrix F, and the operator unfold(·) represents the matrix of each positive slice of the vectorized tensor;
[0056] (2b) The alternating direction multiplier method is used to iteratively solve the entire reconstruction model and introduce auxiliary variables Convert the expression in equation (5) into the augmented Lagrangian function:
[0057]
[0058] in is the Lagrange multiplier, β is the penalty parameter, <·> represents the inner product of two tensors, and the third term Represents the square of the norm of the tensor F, alternately solving each optimization variable and updating the Lagrange multiplier;
[0059] (2c) Given In this case, solve the variables in the augmented Lagrangian function in the t+1 iteration The subproblem can be transformed into solving a convex optimization problem on a three-dimensional tensor:
[0060]
[0061] The variables This problem is to transform all variables into the frequency domain, solve them using the proximal gradient method, and then inversely transform them into the spatial domain to obtain To simplify the expression, the iterative superscript in equation (7) is omitted in the following solution steps:
[0062] (2c1) The three-dimensional tensor in the convex optimization problem of formula (7) and After performing discrete Fourier transform on each tube fiber, we get and The tensor product of tensors in the spatial domain is equivalent to their product in the frequency domain, so equation (7) is equivalent to being converted to the frequency domain as follows:
[0063]
[0064] (2c2) For the target tensor in equation (8) Solve each front slice matrix separately and get the solution:
[0065]
[0066] in and They are tensors and The corresponding i-th frontal slice matrix, Representation Matrix The nuclear norm of the matrix The sum of the singular values, represents the square of the norm of the matrix F;
[0067] (2c3) The proximal gradient method is used to iteratively solve the convex optimization problem under the nuclear norm constraint corresponding to equation (9). First, define the function visible is a differentiable convex function, and is a non-differentiable convex function, and its proximal mapping function is defined as:
[0068]
[0069] in express The proximal mapping function, T represents the introduced auxiliary variable, ρ represents the step size factor, due to the calculation of the proximal mapping function Equivalent to To perform singular value threshold operation, we can first Singular value decomposition yields:
[0070]
[0071] Where P is the left singular vector matrix, Q is the right singular vector matrix, Q T represents the transpose of Q, Σ is the singular value matrix, the threshold is set to 1 / ρ, and the singular value threshold operator is defined based on equation (11):
[0072]
[0073] Where max(·) represents the maximum value function, which is defined as:
[0074]
[0075] So the proximal mapping function Equivalent to the singular value threshold operator
[0076]
[0077] because The gradient of According to the proximal gradient method and the proximal mapping function, the iterative expression of the convex optimization problem in equation (9) is:
[0078]
[0079] where ρ is set as a matrix The maximum eigenvalue of the operator S 1 / ρThe definition of (·) is shown in formula (12), and finally the inverse discrete Fourier transform is used to obtain The optimal solution is:
[0080]
[0081] The operator IDFT(·) represents the inverse discrete Fourier transform of each tube fiber of the three-dimensional tensor, which can be solved to
[0082] (2d) Given In this case, solve the variables in the augmented Lagrangian function in the t+1 iteration The subproblem can be transformed into solving a least squares problem:
[0083]
[0084] The variables This problem is solved using the least squares method. To simplify the expression, the iterative superscript in equation (17) is omitted in the following solution steps:
[0085] (2d1) The three-dimensional tensor in the second term of equation (17) and The frontal slices are vectorized and the least squares problem of formula (17) is transformed into:
[0086]
[0087] in, represents the square of the norm of matrix F, H represents the image degradation matrix, represents the degraded color image, unfold(·) represents the frontal slice matrix of the vectorized tensor;
[0088] (2d2) For the matrix in equation (18) Solve using the least squares method:
[0089]
[0090] in(·) -1 represents matrix inversion, H T represents the transpose of the matrix H, I represents the identity matrix, and finally the matrix The column vectors of the matrix are used to obtain the variables in the original solution. The optimal solution of Solution of the sub-problem;
[0091] (2e) Update the Lagrange multiplier in equation (6) and use equation (20) to obtain:
[0092]
[0093] Step 3, repeat steps (2c) to (2e) until the restored color image meets the convergence condition or the number of iterations reaches a preset upper limit.
[0094] The effect of the present invention can be further illustrated by the following simulation experiment:
[0095] 1. Experimental conditions and contents
[0096] Experimental conditions: The experiment uses an artificially set random degradation process; the experimental images are all 3-channel color images of size w×h×3, and the size of each channel is w×h, such as Figure 2 As shown in the figure, the peak signal-to-noise ratio (PSNR) is used as the evaluation index of the experimental results to objectively evaluate the reconstruction effect, where PSNR is defined as:
[0097]
[0098] in is the original image, size is n 1 ×n 2 ×n 3 , The higher the PSNR value, the better the reconstruction result is, which is closer to the real image.
[0099] Experimental content: Under the above experimental conditions, the reconstruction results are compared with the method of the present invention using the SNN method, SKR method, JPG-SR method, IDD-BM3D method, TRPCA method, JSM method, HOSVD method, RTA-LSM method and STROLLR method, which are currently representative in the field of image restoration.
[0100] Experiment 1: The method of the present invention, the SNN method, the SKR method and the JPG-SR method were used to analyze Figure 2 The images in (a)-(d) are reconstructed with pixel retention rates of 30%, 50%, 60% and 80% and Gaussian noise with a standard deviation of 10 added. The SNN method defines the tensor trace norm based on the matrix trace norm study, extends the low-rank matrix completion problem to low-rank tensor completion, and then proposes three different solution algorithms for different solution scenarios. The reconstruction results at a pixel retention rate of 50% are as follows: Figure 3 (a) The SKR method uses an adaptive kernel regression function to characterize the local information of the image. The restoration effect often depends on the pixel position and the inherent local structure. The reconstruction result at a 50% pixel retention rate is: Figure 3(b) JPG-SR uses an effective mechanism to integrate the local sparsity and non-local self-similarity of the image, solving the problem of undesirable visual artifacts produced by the patch sparse representation model and over-smoothing effect produced by the group sparse representation model. The reconstruction result at 50% pixel retention rate is: Figure 3 (c) In the experiment, the method of the present invention sets the maximum number of iterations T = 50, and the reconstruction result at a pixel retention rate of 50% is: Figure 3 (d).
[0101] In addition, the above method and the test image were compared for text removal. The reconstruction result of the SNN method is Figure 4 (a); The reconstruction result of the SKR method is Figure 4 (b); The reconstruction result of the JPG-SR method is Figure 4 (c); The reconstruction result of the method of the present invention is Figure 4 (d).
[0102] from Figure 3 and Figure 4 It can be seen from the reconstruction results of each method and the magnified images of the local area that, compared with the SNN method, the SKR method and the JPG-SR method, the method of the present invention is superior to other compared methods in the details of the reconstruction results, and has better visual effects in both the restoration of edge and texture areas and the removal of noise.
[0103] Table 1 PSNR (dB) comparison of image inpainting experiments using different reconstruction methods
[0104]
[0105] Table 1 shows the PSNR indicators of the reconstruction results of each method at different pixel retention rates, where the higher the PSNR value, the better the reconstruction effect, and the highest PSNR value is marked in bold. It can be seen that the method of the present invention has a significant improvement compared with other methods, and this result is consistent with the reconstruction effect diagram.
[0106] Table 2 PSNR (dB) comparison of different reconstruction methods for image text removal experiments
[0107]
[0108] Table 2 shows the PSNR indicators of the reconstruction results of each method after the text is applied. Obviously, the method of the present invention achieves the highest PSNR in all cases, which further verifies the effective performance of the method of the present invention.
[0109] Experiment 2: The method of the present invention and the IDD-BM3D method, TRPCA method, JSM method and JPG-SR method were used to analyze Figure 2The images in (e)-(h) are restored after being affected by three different blur kernels, including 9×9 uniform blur kernel, Gaussian blur kernel and motion blur kernel. The blurring process is also accompanied by a standard deviation of Additive Gaussian white noise. IDD-BM3D is an improved version of the classic algorithm BM3D in the image deblurring scene. Its restoration result under the action of motion blur kernel is: Figure 5 (a); TRPCA is an extension of the robust principal component analysis method RPCA on high-order data. Under the dual constraints of low-rank tensor and sparsity, it can recover a relatively clean original low-rank tensor from the noise tensor. The restoration result under the action of the motion blur kernel is Figure 5 (b) JSM obtains the local smoothness characteristics and non-local self-similarity of the image through a joint statistical model. The restoration result under the action of the motion blur kernel is: Figure 5 (c); The restoration result of the JPG-SR method under the action of the motion blur kernel is: Figure 5 (d) In the experiment, the maximum number of iterations of the method of the present invention is set to T = 50, and the restoration result under the action of the motion blur kernel is: Figure 5 (e).
[0110] from Figure 5 It can be seen from the restoration results of each method and the magnified images of the local area that, compared with the IDD-BM3D method, the TRPCA method, the JSM method and the JPG-SR method, the method of the present invention is superior to other comparison methods in the details of the restoration results. The restored image has clearer and sharper image edges and texture details, and the deblurring effect is the best.
[0111] Table 3 PSNR (dB) comparison of different reconstruction methods for image deblurring experiments
[0112]
[0113]
[0114] Table 3 shows the PSNR indicators of the restoration results of each method under different blur kernels. It can be seen that the method of the present invention has achieved the highest PSNR value under different blur kernel scenarios, which objectively verifies the effectiveness of the method of the present invention.
[0115] Experiment 3: The method of the present invention and the HOSVD method, RTA-LSM method, TRPCA method and STROLLR method were used to analyze Figure 2In (i)-(l), the noise images with Gaussian noise ratios of 10%, 15%, 20%, 25% and 30% are reconstructed. HOSVD performs high-order singular value decomposition in the three-dimensional stack and filters these coefficients through hard thresholds. The restoration result after adding 25% noise ratio is Figure 6 (a); RTA-LSM constructs the approximate tensor as a Laplace mixture model and observes the noise with maximum a posteriori estimation. The restoration result after adding 25% noise ratio is Figure 6 (b); The restoration result of the TRPCA method after adding 25% noise ratio is Figure 6 (c) STROLLR combines the adaptive transformation sparsity of image blocks and the low rank of the data matrix formed by block matching, thereby making full use of the local sparsity and non-local self-similarity of natural images. The restoration result after adding 25% noise ratio is Figure 6 (d) In the experiment, the maximum number of iterations of the method of the present invention is set to T = 50. After adding 25% noise ratio, the restoration result is Figure 6 (e) as shown.
[0116] from Figure 6 It can be seen from the restoration results of each method and the magnified image of the local area that, compared with the HOSVD method, the RTA-LSM method, the TRPCA method and the STROLLR method, the image restored by the method of the present invention can not only effectively suppress noise, but also effectively retain structural information, and the denoising effect is the best.
[0117] Table 4 PSNR (dB) comparison of different methods for image denoising experiments
[0118]
[0119]
[0120] Table 4 shows the PSNR indicators of the restoration results of each method under different noise ratios; it can be seen that the method of the present invention achieves the highest PSNR value in almost all cases, and this result is consistent with the repetition effect diagram.
[0121] The above experiments show that the image restored by the method of the present invention is rich in content and has not only good visual effects but also good performance in objective evaluation indicators, which shows the effectiveness of the present invention in restoring color images.
Claims
1. A color image restoration method based on minimizing the coefficient tensor nuclear norm. The following steps are involved: (1) Input a three-channel color degraded image and use the tensor robust principal component analysis method to perform initial reconstruction to obtain the initial reconstructed image right After performing economic tensor singular value decomposition, the redundant dictionary is obtained The specific steps are: (1a) First, Perform a tensor singular value decomposition: in and is an orthogonal tensor satisfying for The conjugate transpose of Indicates that the first positive slice is the identity matrix, and the other positive slices are unit tensors of zero matrices. represents a diagonal tensor whose frontal slices are all diagonal matrices, express and The tensor product between ; (1b) Interception The non-zero part of and The lateral slicing of implements economical tensor singular value decomposition: in and Respectively represent the intercepted tensor and The tensor consisting of the first r lateral slices of , for The conjugate transpose of Represents a truncated tensor A tensor consisting of the first r rows and first r columns of each positive slice matrix, where r represents the tensor The maximum number of non-zero diagonal elements in each frontal slice is used to further calculate the redundant dictionary The redundant dictionary is obtained from (2) Considering that each channel sub-image of a color image has highly similar structural features, the original image tensor has a low-rank characteristic. An image restoration model under low-rank constraints is established with the original image tensor as the processing unit: Where λ represents the regularization parameter, represents the degraded color image, represents the representation coefficient of the image to be restored, H represents the image degradation matrix, Representing a tensor The nuclear norm of in Representing a tensor The i-th frontal slice matrix after discrete Fourier transform on each tube fiber, Represents the frontal slice matrix The nuclear norm of the matrix The sum of the singular values, represents the square of the norm of the matrix F, Representing a tensor With tensor The tensor product between them, the operator unfold(·) represents the matrix of each positive slice of the vectorized tensor; (3) The alternating direction multiplier method is used to iteratively solve the entire reconstruction model. First, the expression in (2) is converted into an augmented Lagrangian function: in is an auxiliary variable, defined as is the Lagrange multiplier, β is the penalty parameter, <·> represents the inner product of two tensors, and the third term Represents the square of the norm of the tensor F; (3a) Given In this case, solve the variables in the augmented Lagrangian function in the t+1 iteration The subproblem can be transformed into solving a convex optimization problem on a three-dimensional tensor: The variables This problem is to transform all variables into the frequency domain, solve them using the proximal gradient method, and then inversely transform them into the spatial domain to obtain (3b) Given In this case, solve the variables in the augmented Lagrangian function in the t+1 iteration The subproblem can be transformed into solving a least squares problem: The variables This problem is solved using the least squares method. (3c) Update the Lagrange multiplier in (3): (4) Repeat steps (3a) to (3c) until the restored color image meets the convergence condition or the number of iterations reaches a preset upper limit.
2. A color image restoration method based on minimization of coefficient tensor nuclear norm according to claim 1, It is characterized in that The model solution problem in step (3a) can be obtained by following the following steps. To simplify the expression, the iteration superscript in formula (3a) is omitted in the following solution steps: (3a1) The three-dimensional tensor of the convex optimization problem in (3a) and After performing discrete Fourier transform on each tube fiber, we get and And transform the convex optimization problem into: in and They are tensors and The corresponding i-th frontal slice matrix, Representation Matrix The nuclear norm of the matrix The sum of the singular values, represents the square of the norm of the matrix F; (3a2) The proximal gradient method is used to iteratively solve the convex optimization problem in (3a1). First, define the function visible is a differentiable convex function, and is a non-differentiable convex function, and its proximal mapping function is defined as: in express The proximal mapping function, T represents the introduced auxiliary variable, ρ represents the step size factor, due to the calculation of the proximal mapping function Equivalent to To perform singular value threshold operation, we can first Singular value decomposition gives Where P is the left singular vector matrix, Q is the right singular vector matrix, Q T represents the transpose of Q, Σ is the singular value matrix, on which the singular value threshold operator is defined Where max(·) represents the maximum value function, so the proximal mapping function Equivalent to the singular value threshold operator because The gradient of According to the proximal gradient method and the proximal mapping function, the iterative expression of the convex optimization problem in (3a1) is: where ρ is set as a matrix The maximum eigenvalue of is obtained by inverse discrete Fourier transform. The optimal solution is: The operator IDFT(·) represents the inverse discrete Fourier transform of each tube fiber of the three-dimensional tensor, which can be solved to 3. The color image restoration method based on minimization of coefficient tensor nuclear norm according to claim 1, It is characterized in that The model solution problem in step (3b) can be obtained by following the following steps. To simplify the expression, the iteration superscript in formula (3b) is omitted in the following solution steps: (3b1) Substitute the three-dimensional tensor in the second term of the least squares problem (3b) into and The frontal slice of is vectorized and the least squares problem is transformed into: in, represents the square of the norm of the matrix F, λ is the regularization parameter, β is the penalty parameter, H represents the image degradation matrix, represents the degraded color image, unfold(·) represents the frontal slice matrix of the vectorized tensor; (3b2) For each frontal slice matrix in (3b1) Solve using the least squares method: in(·) -1 represents matrix inversion, H T represents the transpose of the matrix H, I represents the identity matrix, and finally The column vectors of The optimal solution of The solution of the sub-problem.