CN115131226B - Image restoration method based on wavelet tensor low-rank regularization - Google Patents

Abstract

The invention discloses an image restoration method based on wavelet tensor low-rank regularization. Belonging to the technical field of digital image processing. The method is an image restoration method for constructing tensors for subband coefficients after image wavelet transformation and performing low-rank constraint. The method comprises the steps of firstly carrying out wavelet decomposition on an image to obtain sub-band coefficients, then stacking two-dimensional coefficients corresponding to different sub-bands into third-order tensors, utilizing tensor kernel norms to constrain low-rank characteristics of the tensor kernel norms, and finally solving an image restoration model under the wavelet tensor low-rank constraint through an alternate direction iterative algorithm. The invention expresses wavelet sub-band coefficients in tensor form, fully excavates the relativity among the wavelet coefficients of the image, utilizes tensor kernel norms to carry out low-rank constraint on constructed third-order tensors, and uses an alternate direction iterative algorithm to efficiently solve the sub-problems about the wavelet coefficient tensors and the restored image. The invention can improve the estimation precision of the wavelet coefficient of the image, so that the restored image is clearer and the details are richer, thereby being applicable to restoration of the degraded image.

Description

Image restoration method based on wavelet tensor low-rank regularization

Technical Field

The invention belongs to the technical field of digital image processing, and particularly relates to an image restoration method for constructing a third-order wavelet coefficient tensor and performing low-rank constraint by using a tensor kernel norm, which is used for high-quality restoration of a degraded image.

Background

The image restoration technology is used for solving the quality degradation problems of damage, blurring and the like of the image in the process of acquisition and storage, so that the restored image is as clear as possible, and the technology is widely used in the fields of aerospace, medical treatment, digital communication and the like. The general process of image restoration is to carry out reasonable mathematical modeling on image degradation, and then to carry out inverse processing on the degraded image by combining with the prior information of the degraded image, so as to restore high-quality restored images. Since the image restoration problem belongs to the category of solving the inverse problem, the problem is usually pathological, and a unique solution is difficult to obtain, the improvement restoration method by effectively utilizing the prior information of the image is a research hot spot of the technology.

The early image restoration model based on total variation can keep good edge characteristics of the image and effectively inhibit noise, and is widely researched and applied. In recent years, non-local characteristics of images have been paid attention to, and are also applied to fields such as image denoising, and the quality of restored images has been greatly improved. Considering that the two-dimensional wavelet decomposition of an image can capture various characteristics such as smoothness, edges, details and the like of the image through different sub-band coefficients, designing a proper regularization term to accurately and effectively utilize the correlation between the sub-band coefficients and the inner sub-band coefficients of each wavelet of the image is a key for improving the restoration quality of the image.

Disclosure of Invention

The invention aims to provide an image restoration method based on wavelet tensor low-rank regularization, aiming at the difficulties of an image restoration model based on a regularization method. The method fully considers the internal structural property and mutual correlation of the wavelet coefficients, stacks the two-dimensional wavelet sub-band coefficients into a third-order tensor, and utilizes the tensor kernel norm to carry out low-rank constraint on the wavelet coefficient tensor, thereby improving the estimation precision of the wavelet coefficients.

The method specifically comprises the following steps:

(1) Inputting a web Performing one-layer two-dimensional redundant wavelet decomposition on the image x to be restored to obtain four two-dimensional wavelet subband coefficients;

(2) Sequentially stacking the four two-dimensional wavelet sub-band coefficients obtained in step (1) in a matrix form to form a third-order tensor

(3) The strong correlation among different wavelet sub-band coefficients is utilized, the tensor kernel norms are adopted to carry out low-rank constraint on the third-order tensor of the wavelet coefficients, and the third-order tensor is combinedCan be expressed as:

Wherein the method comprises the steps of Denote the core tensor, U, V and W denote three orthogonal singular vector matrices, x ₁、×₂ and x ₃ denote the Tucker mode-1 product, tucker mode-2 product and Tucker mode-3 product, respectively; based on this, third-order tensor/>Nuclear norm/>Defined as its core tensor/>Is one norm/>Can be expressed as:

(4) On the basis of constructing wavelet coefficient tensors and defining tensor kernel norm constraint, an image restoration model under the wavelet tensor low-rank constraint is established:

Where y represents a degraded image, H represents a degradation matrix, Representing the square of the two norms of the vector, x representing the image to be restored, λ being the regularization parameter; to solve the restoration model, first, the auxiliary variable/>, is introducedThe restoration model is rewritten into a multivariate minimization problem:

Where beta is the penalty parameter and, Is normalized Lagrangian multiplier,/>The square of the Frobenius norm representing the tensor; solving it by using an alternating direction iterative algorithm, the multivariate minimization problem can be decomposed into tensors/>, with respect to the image x and wavelet coefficients to be restoredAnd performing alternate iterative solution:

(4a) For the variable x in the model, give The restoration model becomes solving a sub-problem with respect to the image x to be restored:

Wherein the method comprises the steps of Representing the ith two-dimensional wavelet subband coefficient slice,/>Representation/>Corresponding auxiliary variable,/>Representation/>The corresponding normalized Lagrangian multiplier, which is a least squares problem, can directly obtain its closed solution by matrix inversion;

(4b) After obtaining the image x to be restored, the method is as follows The sub-problem of (2) can be expressed as:

According to Higher order singular value decomposition/>The sub-problem can be rewritten as to/>U, V and W polynary minimization problem:

Where (-) ^T denotes the transpose of the matrix, Representation/>A unit matrix of a size, I ₄ denotes a unit matrix of a size of 4×4; the multiple minimization problem can be decomposed into sub-problems for each variable and solved alternately;

(5) And (3) repeating the step (4) until the variation between the two adjacent reconstruction results is smaller than the iteration termination threshold or the maximum iteration times are met.

The innovation point of the invention is to construct a third-order tensor of the two-dimensional wavelet sub-band coefficient; performing low-rank constraint on wavelet coefficient tensors by using tensor kernel norms, and improving estimation accuracy of the wavelet coefficients of the image; and solving a wavelet tensor low-rank reconstruction model by using an alternate direction iterative algorithm, and applying the method to restoration of the degraded image.

The invention has the beneficial effects that: utilizing a layer of redundant wavelet decomposition to fully mine smooth characteristics and detail information of the image; the three-order tensor is adopted to represent the two-dimensional wavelet sub-band coefficient, so that the structural property in the scale and the correlation among the scales are well reserved, the tensor kernel norm is adopted to apply low-rank constraint on the three-order tensor, the estimation precision of the image wavelet coefficient is improved, and therefore, the final restored result graph not only has a good overall visual effect, but also reserves a large amount of detail information such as outlines, edges and textures in the image, and the whole estimation result is closer to a true value.

The invention is mainly verified by adopting a simulation experiment method, and all steps and conclusions are verified to be correct on MATLAB 9.5.

Drawings

FIG. 1 is a block diagram of the workflow of the present invention;

FIG. 2 is a Lena artwork used in the simulation of the present invention in an image inpainting scenario;

FIG. 3 is a test chart after overlaying text on Lena artwork in an image patch scene;

FIG. 4 is a reconstruction result of FIG. 3 using the SALSA method in an image inpainting scene;

FIG. 5 is a reconstruction result of FIG. 3 using the SKR method in an image inpainting scene;

FIG. 6 is a reconstruction result of FIG. 3 using the BPFA method in an image inpainting scenario;

FIG. 7 is a reconstruction result of FIG. 3 using IRCNN method in an image inpainting scene;

FIG. 8 is a reconstruction result of FIG. 3 using the IDBP method in an image inpainting scenario;

FIG. 9 is a reconstruction result of FIG. 3 using the method of the present invention in an image inpainting scenario;

FIG. 10 is a House artwork used in the simulation of the present invention in an image deblurring scenario;

FIG. 11 is a test chart of House artwork corrupted by Gaussian blur kernel in an image deblurring scene;

FIG. 12 is a reconstruction of FIG. 11 using the SALSA method in an image deblurring scene;

FIG. 13 is a reconstruction of FIG. 11 using the SA-DCT method in an image deblurring scene;

FIG. 14 is a reconstruction result of FIG. 11 using the IDD-BM3D method in an image deblurring scene;

FIG. 15 is a reconstruction of FIG. 11 using the IRCNN method in an image deblurring scene;

FIG. 16 is a reconstruction of FIG. 11 using the IDBP method in an image deblurring scene;

FIG. 17 is a reconstruction of FIG. 11 using the method of the present invention in an image deblurring scene;

FIG. 18 is a C.man artwork used in the simulation of the present invention in an image denoising scenario;

FIG. 19 is a test chart after applying noise to the C.man artwork in an image denoising scene;

FIG. 20 is a reconstruction result of FIG. 19 using NLM method in an image denoising scene;

FIG. 21 is a reconstruction result of FIG. 19 using the SA-DCT method in an image denoising scene;

FIG. 22 is a reconstruction of FIG. 19 using the OWT-SURE-LET method in an image denoising scenario;

FIG. 23 is a reconstruction of FIG. 19 using IRCNN method in an image denoising scenario;

fig. 24 is a reconstruction of fig. 19 using the method of the present invention in an image denoising scenario.

Detailed Description

Referring to fig. 1, the invention is an image restoration method based on wavelet tensor low-rank regularization, comprising the following specific steps:

and step 1, performing one-layer two-dimensional redundant wavelet decomposition on the image to obtain wavelet sub-band coefficients.

(1A) Respectively passing the input original airspace image x through an up-sampling low-pass filter and an up-sampling high-pass filter to obtain two images with the same size as the original image;

(1b) And (2) respectively passing the two images obtained in the step (1 a) through up-sampling low-pass and high-pass filters to finally obtain four subband images with the same size as the original image, wherein the subband images can be expressed as follows:

α_i(x)＝Φ_ix(i＝1，2，3，4)

Wherein Φ _i denotes a filter matrix, and when i=1, 2,3,4, α _i (x) respectively denotes a low-frequency two-dimensional wavelet subband coefficient α _L,L and high-frequency two-dimensional wavelet subband coefficients α _L,H、α_H,L and α _H,H, wherein the low-frequency two-dimensional wavelet subband coefficients reflect the smooth characteristic of the whole image and are most similar to the original image, and are called as approximation coefficients or approximation coefficients, and the high-frequency two-dimensional wavelet subband coefficients retain the local characteristics of the contour, edge, texture, and the like of the image and are called as detail coefficients;

and 2, constructing a wavelet sub-band coefficient tensor and performing low-rank constraint on the tensor coefficient tensor by using a tensor kernel norm.

(2A) After the wavelet sub-band coefficients are obtained, in order to preserve the internal structural property and mutual correlation of the wavelet sub-band coefficients, the advantage of the multi-dimensional data potential structure can be preserved by means of tensors, and four two-dimensional wavelet sub-band coefficients are sequentially stacked in a matrix form to form a third-order tensorThe binding α _i(x)＝Φ_i x (i=1, 2,3, 4) can be expressed as:

Where vec (·) represents the operator of matrix vectorization, Representing an ith two-dimensional wavelet sub-band coefficient slice;

(2b) From the visual effect, although the wavelet subband coefficient pixel gray values are different, the textures and the outline structures of the wavelet subband coefficient pixel gray values are highly similar, so that the wavelet coefficient tensor is considered to have obvious low-rank characteristics; similar to the low rank characteristic of the matrix kernel norms used for constraining the matrix, the tensor kernel norms are the sum of tensor singular values, are convex approximations of tensor rank functions, and can be used for applying low rank constraint to third-order wavelet coefficient tensors; in addition, the higher order singular value decomposition is taken as a special case of the Tucker decomposition, and each component matrix obtained after the decomposition is further subjected to orthogonal constraint and is also considered to be the multi-linear popularization of matrix singular value decomposition, so that the method is based on the following steps of It can be seen that the third order tensor/>Nuclear norm/>And can be defined as/>

And 3, establishing an image restoration model under the wavelet tensor low-rank constraint and solving.

(3A) Image restoration model under wavelet tensor low rank constraint:

Where y represents a degraded image, H represents a degradation matrix, Representing the square of the two norms of the vector, x representing the image to be restored, λ being the regularization parameter; to solve the restoration model in (3 a), an auxiliary variable/>, is introducedThe restoration model is rewritten into a multivariate minimization problem:

Where beta is the penalty parameter and, Is normalized Lagrangian multiplier,/>The square of the Frobenius norm representing the tensor; solving it by using an alternating direction iterative algorithm, the multivariate minimization problem can be decomposed into tensors/>, with respect to the image x and wavelet coefficients to be restoredAnd performing alternate iterative solution:

(3b) For the variable x in (3 a), given The restoration model becomes solving a sub-problem with respect to the image x to be restored:

Wherein the method comprises the steps of Representing the ith two-dimensional wavelet subband coefficient slice,/>Representation/>Corresponding auxiliary variable,/>Representation/>The corresponding normalized Lagrangian multiplier, which is a least squares problem, can be directly inverted by a matrix to obtain its closed solution:

(3c) After obtaining the image x to be restored, the method is as follows The sub-problem of (2) can be expressed as:

According to Higher order singular value decomposition/>The sub-problem can be rewritten as to/>U, V and W polynary minimization problem:

Where (-) ^T denotes the transpose of the matrix, Representation/>A unit matrix of a size, I ₄ denotes a unit matrix of a size of 4×4; the multivariate minimization problem can be decomposed into four sub-problems for each variable and solved alternately:

3c1) With respect to The sub-problem of (2) can be solved by a soft threshold algorithm:

wherein soft (·) is a soft threshold function point by point:

3c2) The sub-problem with U can be expressed as:

Since matrices V and W are both orthogonal matrices, the sub-problem with U can be rewritten as:

wherein unfold ₁ (·) represents the expansion of the tensor in the Tucker mode-1, meaning that all column fibers in the tensor are used as column vectors to form a matrix, pair Singular value decomposition is performed:

wherein P ₁、Q₁ is Sigma ₁ is a singular value matrix, and finally, the closed solution of U can be obtained:

3c3) The sub-problem with V can be expressed as:

Wherein unfold ₂ (. Cndot.) represents the expansion of the tensor in the Tucker mode-2, meaning that all the row fibers in the tensor are used as column vectors to form a matrix, and the same applies to Singular value decomposition is performed:

Wherein P ₂、Q₂ is Sigma ₂ is a singular value matrix, and finally, the closed solution of V can be obtained:

3c4) The sub-problem with W can be expressed as:

wherein unfold ₃ (-) represents the expansion of the tensor in the Tucker mode-3, meaning that all the tube fibers in the tensor are matrix-formed as column vectors, and similarly, for Singular value decomposition is performed:

Wherein P ₃、Q₃ is Sigma ₃ is a singular value matrix, and finally a closed solution of W can be obtained:

3c5) Repeating steps 3c 1) to 3c 4) to alternately solve the solutions of the wavelet coefficient tensors

(4) And (3) repeating the step until the variation between the two adjacent reconstruction results is smaller than the iteration termination threshold or the maximum iteration times are met.

The effect of the invention can be further illustrated by the following simulation experiments:

1. experimental conditions and content

Experimental conditions: the experimental scene comprises an image inpainting experiment, an image deblurring experiment and an image denoising experiment, and the experimental image adopts standard digital image test pictures, which are respectively shown in figures 2, 10 and 18; the experimental result evaluation index adopts peak signal-to-noise ratio PSNR and structural similarity SSIM to objectively evaluate the restoration result, and the higher the PSNR value and the SSIM value, the better the restoration result, and the closer to the real image.

The experimental contents are as follows: in the image inpainting scene, the restoration result is compared with the method of the present invention using SALSA method, SKR method, BPFA method, IRCNN method and IDBP method, which are currently representative in the image inpainting field. In the image deblurring scene, the restoration results are compared with the method of the invention by using SALSA method, SKR method, BPFA method, IRCNN method and IDBP method which are representative in the image deblurring field at present. In the image denoising scene, the restoration result is compared with the method of the invention by using the SALSA method, the SKR method, the BPFA method, the IRCNN method and the IDBP method which are representative in the image denoising field at present.

Experiment 1: in the image inpainting scene, the method and the SALSA method, the SKR method, the BPFA method, the IRCNN method and the IDBP method are used for respectively restoring the image (figure 3) of which the text is covered in figure 2. The SALSA method is a method for quickly solving the unconstrained optimization problem by using the total variation, and the restoration result is shown in fig. 4; the SKR method utilizes a self-adaptive kernel regression function method, and the method is characterized by the pixel position and intensity of a local structure in an image, so that the restoration result is shown in fig. 5; the BPFA method is a method for mining spatial correlation of an image by dictionary learning under a non-parametric bayesian framework, and the restoration result is shown in fig. 6; the IRCNN method and the IDBP method are two reconstruction methods based on deep learning, and the restoration results are respectively shown in fig. 7 and fig. 8. In the experiment, the regularization parameter beta=0.06, lambda=0.85, the maximum iteration time T=100 and the iteration termination threshold eta=1×10 ^-6 are set; the final restoration result is fig. 9.

As can be seen from the restoration results and the partial area enlarged diagrams of the methods of fig. 4, 5, 6, 7, 8 and 9, comparing the SALSA method, the SKR method, the BPFA method, the IRCNN method and the IDBP method with the method of the present invention, the restoration effect of the method of the present invention on the detail portion is superior to other comparison methods.

TABLE 1 PSNR/SSIM indicators for different restoration methods

Image processing apparatus

SALSA method

SKR method

BPFA method

IRCNN method

IDBP method

The method of the invention

Lene

34.10/0.9619

31.89/0.9150

33.14/0.7897

35.32/0.9638

36.48/0.9768

37.02/0.9801

Table 1 shows the PSNR and SSIM index conditions for each recovery method, wherein higher PSNR and SSIM values indicate better recovery; the method has the advantages that under the image patching scene, compared with other methods, PSNR value and SSIM value are both greatly improved, which indicates that the restoration result of the method is closer to the real image, and the result is consistent with the restoration effect graph.

Experiment 2: in the image deblurring scene, the image (fig. 11) of fig. 10 damaged by the gaussian blur kernel is restored by the method of the present invention and the SALSA method, the SA-DCT method, the IDD-BM3D method, the IRCNN method and the IDBP method, respectively. The SALSA method is a method for quickly solving the unconstrained optimization problem by using the total variation, and the restoration result is shown in fig. 12; the SA-DCT method is an image processing method that implements a hard threshold algorithm and wiener filtering in the discrete cosine transform domain of shape adaptation, and the restoration result is fig. 13; the IDD-BM3D method is a classical denoising algorithm commonly used for image restoration, and is known as a modified version of BM3D, and the restoration result is shown in FIG. 14; the IRCNN method and the IDBP method are two reconstruction methods based on deep learning, and the restoration results thereof are respectively fig. 15 and 16. In the experiment, the regularization parameter beta=0.045, lambda=0.75, the maximum iteration time T=100 and the iteration termination threshold eta=1×10 ^-6 are set in the method; the final restoration result is fig. 17.

As can be seen from the restoration results and the enlarged partial areas of the methods of fig. 12, 13, 14, 15, 16 and 17, the restoration effect of the method of the present invention on the detail part is superior to other comparison methods as can be seen by comparing the SALSA method, the SA-DCT method, the IDD-BM3D method, the IRCNN method and the IDBP method with the method of the present invention.

TABLE 2 PSNR/SSIM indicators for different recovery methods

Image processing apparatus

SALSA method

SA-DCT method

IDD-BM3D method

IRCNN method

IDBP method

The method of the invention

House

34.51/0.8804

35.92/0.9053

32.75/0.8005

35.98/0.8853

34.88/0.8712

37.02/0.9801

Table 2 shows the PSNR and SSIM index conditions for each recovery method, wherein higher PSNR and SSIM values indicate better recovery; the method has the advantages that under the image deblurring scene, compared with other methods, the PSNR value and the SSIM value are both greatly improved, and the restoration result of the method is closer to a real image, and is consistent with a restoration effect graph.

Experiment 3: in the image denoising scene, the image (figure 19) after the noise addition of figure 18 is restored by the method and NLM method, SA-DCT method, OWT-SURE-LET method and IRCNN method respectively. The NLM method is a classical denoising method for weighting and averaging non-local information of an image, and the restoration result is shown in FIG. 20; the SA-DCT method is an image processing method that implements a hard threshold algorithm and wiener filtering in the discrete cosine transform domain of shape adaptation, and the restoration result is fig. 21; the OWT-SURE-LET method is a threshold denoising method using orthogonal wavelets, and the restoration result is fig. 22; the IRCNN method is a reconstruction method based on deep learning, and the restoration results are respectively shown in fig. 23. In the experiment, the regularization parameter beta=0.055, lambda=0.8, the maximum iteration time T=100 and the iteration termination threshold eta=1×10 ^-6 are set in the method; the final restoration result is fig. 24.

As can be seen from the restoration results and the enlarged partial areas of the methods of fig. 20, 21, 22, 23 and 24, the restoration effect of the method of the present invention on the detail part is better than that of other comparative methods by comparing the NLM method, the SA-DCT method, the OWT-SURE-LET method and IRCNN method with the method of the present invention.

TABLE 3 PSNR/SSIM indicators for different recovery methods

Image processing apparatus	NLM method	SA-DCT method	OWT-SURE-LET method	IRCNN method	The method of the invention
						C.man	28.17/0.7845	29.11/0.8519	27.38/0.7601	30.08/0.8727	30.26/0.8732

Table 3 shows the PSNR and SSIM index conditions for each recovery method, wherein higher PSNR and SSIM values indicate better recovery; the method has the advantages that under the image denoising scene, compared with other methods, PSNR value and SSIM value are both greatly improved, which indicates that the restoration result of the method is closer to the real image, and the result is consistent with the restoration effect graph.

The three experiments show that the method has obvious restoration effect, rich restored image content and higher objective evaluation index under the experimental scenes of image inpainting, image deblurring and image denoising, so that the method is effective for restoring the degraded image.

Claims (3)

1. An image restoration method based on wavelet tensor low-rank regularization comprises the following steps:

(1) Inputting a web Performing one-layer two-dimensional redundant wavelet decomposition on the image x to be restored to obtain four two-dimensional wavelet subband coefficients;

(2) Sequentially stacking the four two-dimensional wavelet sub-band coefficients obtained in step (1) in a matrix form to form a third-order tensor

(3) The strong correlation among different wavelet sub-band coefficients is utilized, the tensor kernel norms are adopted to carry out low-rank constraint on the third-order tensor of the wavelet coefficients, and the third-order tensor is combinedCan be expressed as:

Wherein the method comprises the steps of Denote the core tensor, U, V and W denote three orthogonal singular vector matrices, x ₁、×₂ and x ₃ denote the Tucker mode-1 product, tucker mode-2 product and Tucker mode-3 product, respectively; based on this, third-order tensor/>Nuclear norm/>Defined as its core tensor/>Is one norm/>Can be expressed as:

(4) On the basis of constructing wavelet coefficient tensors and defining tensor kernel norm constraint, an image restoration model under the wavelet tensor low-rank constraint is established:

Where y represents a degraded image, H represents a degradation matrix, Representing the square of the two norms of the vector, x representing the image to be restored, λ being the regularization parameter; to solve the restoration model, first, the auxiliary variable/>, is introducedThe restoration model is rewritten into a multivariate minimization problem:

Where beta is the penalty parameter and, Is normalized Lagrangian multiplier,/>The square of the Frobenius norm representing the tensor; solving it by using an alternating direction iterative algorithm, the multivariate minimization problem can be decomposed into tensors/>, with respect to the image x and wavelet coefficients to be restoredAnd performing alternate iterative solution:

(4a) For the variable x in the model, give The restoration model becomes solving a sub-problem with respect to the image x to be restored:

Wherein the method comprises the steps of Representing the ith two-dimensional wavelet subband coefficient slice,/>Representation/>Corresponding auxiliary variable,/>Representation/>The corresponding normalized Lagrangian multiplier, which is a least squares problem, can directly obtain its closed solution by matrix inversion;

(4b) After obtaining the image x to be restored, the method is as follows The sub-problem of (2) can be expressed as:

According to Higher order singular value decomposition/>The sub-problem can be rewritten as to/>U, V and W polynary minimization problem:

Where (-) ^T denotes the transpose of the matrix, Representation/>A unit matrix of a size, I ₄ denotes a unit matrix of a size of 4×4; the multiple minimization problem can be decomposed into sub-problems for each variable and solved alternately;

(5) And (3) repeating the step (4) until the variation between the two adjacent reconstruction results is smaller than the iteration termination threshold or the maximum iteration times are met.

2. The method for restoring an image based on wavelet tensor low-rank regularization according to claim 1, wherein in the step (1), a layer of two-dimensional redundant wavelet decomposition is implemented on the image x to be restored to obtain a low-frequency two-dimensional wavelet subband coefficient and three high-frequency two-dimensional wavelet subband coefficients, wherein the low-frequency two-dimensional wavelet subband coefficients reflect the overall smooth characteristic of the image, and the high-frequency two-dimensional wavelet subband coefficients retain the local characteristics of the image such as contour, edge, texture and the like.

3. The image restoration method based on wavelet tensor low-rank regularization according to claim 1, wherein the model solving problem in the step (4 b) is obtained by:

4b1) With respect to The sub-problem of (2) can be solved by a soft threshold algorithm:

wherein soft (·) is a soft threshold function point by point:

4b2) The sub-problem with U can be expressed as:

Since matrices V and W are both orthogonal matrices, the sub-problem with U can be rewritten as:

wherein unfold ₁ (·) represents the expansion of the tensor in the Tucker mode-1, meaning that all column fibers in the tensor are used as column vectors to form a matrix, pair Singular value decomposition is performed:

wherein P ₁、Q₁ is Sigma ₁ is a singular value matrix, and finally, the closed solution of U can be obtained:

4b3) The sub-problem with V can be expressed as:

Wherein unfold ₂ (. Cndot.) represents the expansion of the tensor in the Tucker mode-2, meaning that all the row fibers in the tensor are used as column vectors to form a matrix, and the same applies to Singular value decomposition is performed:

Wherein P ₂、Q₂ is Sigma ₂ is a singular value matrix, and finally, the closed solution of V can be obtained:

4b4) The sub-problem with W can be expressed as:

wherein unfold ₃ (-) represents the expansion of the tensor in the Tucker mode-3, meaning that all the tube fibers in the tensor are matrix-formed as column vectors, and similarly, for Singular value decomposition is performed:

Wherein P ₃、Q₃ is Sigma ₃ is a singular value matrix, and finally a closed solution of W can be obtained:

4b5) Repeating steps 4b 1) to 4b 4) to alternately solve the solution of the wavelet coefficient tensor 。