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

An image restoration method based on wavelet tensor low-rank regularization

Technical Field

The present invention belongs to the technical field of digital image processing, and particularly relates to an image restoration method that constructs a third-order wavelet coefficient tensor and uses the tensor nuclear norm to perform low-rank constraint, for high-quality restoration of degraded images.

Background technique

Image restoration technology is used to solve the quality degradation problems such as damage and blur that occur during the acquisition and storage of images, so that the restored image is as clear as possible. This technology has been widely used in aerospace, medical, digital communications and other fields. The general process of image restoration is to make a reasonable mathematical model of image degradation, and then perform inverse processing on the degraded image itself in combination with its own prior information, so as to restore a high-quality restored image. Since the image restoration problem belongs to the category of solving inverse problems, the problem is usually ill-posed and it is difficult to obtain a unique solution. Therefore, the effective use of image prior information to improve the restoration method is a research hotspot of this technology.

Early image restoration models based on total variation can maintain good edge characteristics of images while effectively suppressing noise, and have been widely studied and applied. In recent years, the non-local characteristics of images have gradually received attention and have been applied to fields such as image denoising, greatly improving the quality of restored images. Considering that the two-dimensional wavelet decomposition of an image can capture the smoothness, edges, details and other characteristics of an image through different sub-band coefficients, designing appropriate regularization terms to accurately and effectively utilize the correlation between the coefficients of each wavelet sub-band and between the sub-band coefficients of the image is the key to improving the quality of image restoration.

Summary of the invention

The purpose of the present invention is to propose an image restoration method based on low-rank regularization of wavelet tensors to address the difficulties of image restoration models based on regularization methods. The method fully considers the internal structure of wavelet coefficients and the correlation between each other, stacks the two-dimensional wavelet subband coefficients into a third-order tensor, and uses the tensor nuclear norm to perform low-rank constraints on the wavelet coefficient tensor, thereby improving the estimation accuracy of the wavelet coefficients.

The specific steps include:

(1) Input a The image x to be restored is subjected to a one-layer two-dimensional redundant wavelet decomposition to obtain four two-dimensional wavelet subband coefficients;

(2) The four two-dimensional wavelet subband coefficients obtained in (1) are stacked in matrix form to form a third-order tensor

(3) Taking advantage of the strong correlation between different wavelet subband coefficients, the tensor nuclear norm is used to constrain the third-order tensor of wavelet coefficients to a low rank. The high-order singular value decomposition of can be expressed as:

in represents the core tensor, U, V and W represent three orthogonal singular vector matrices, × ₁ , × ₂ and × ₃ represent Tucker mode-1 product, Tucker mode-2 product and Tucker mode-3 product respectively; based on this, the third-order tensor/> The nuclear norm of Defined as its core tensor/> The one-norm of It can be expressed as:

(4) Based on constructing the wavelet coefficient tensor and defining the tensor nuclear norm constraint, an image restoration model under the low-rank constraint of the wavelet tensor is established:

Where y represents the degraded image, H represents the degradation matrix, represents the square of the vector's bi-norm, x represents the image to be restored, and λ is the regularization parameter. To solve the restoration model, we first introduce auxiliary variables. Rewrite the restoration model as a multivariate minimization problem:

Where β is the penalty parameter, is the normalized Lagrange multiplier,/> represents the square of the Frobenius norm of the tensor; using the alternating direction iterative algorithm to solve it, the multivariate minimization problem can be decomposed into the image x to be restored and the wavelet coefficient tensor/> Two sub-problems of , and solve them alternately and iteratively:

(4a) For the variable x in the model, given Then the restoration model becomes a sub-problem of solving the image x to be restored:

in represents the i-th two-dimensional wavelet subband coefficient slice,/> Indicates/> The corresponding auxiliary variables, /> Indicates/> The corresponding normalized Lagrange multiplier, this subproblem is a least squares problem, and its closed solution can be obtained directly by matrix inversion;

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

according to Higher-order singular value decomposition of This sub-problem can be rewritten as about/> Multivariate minimization problem of U, V and W:

where (·) ^T represents the transpose of the matrix, Indicates/> The size of the identity matrix, I ₄ represents the size of the identity matrix of 4×4; the multivariate minimization problem can be decomposed into sub-problems about each variable and solved alternately;

(5) Repeat step (4) until the variation between two adjacent reconstruction results is less than the iteration termination threshold or the maximum number of iterations is met.

The innovation of the present invention is to construct a third-order tensor of two-dimensional wavelet subband coefficients; to use the tensor nuclear norm to perform low-rank constraints on the wavelet coefficient tensor to improve the estimation accuracy of the image wavelet coefficients; to use the alternating direction iterative algorithm to solve the wavelet tensor low-rank reconstruction model, and to apply this method to the restoration of degraded images.

The beneficial effects of the present invention are as follows: a layer of redundant wavelet decomposition is utilized to fully exploit the smooth features and detail information of the image; a third-order tensor is used to represent the two-dimensional wavelet subband coefficients, which well preserves the intra-scale structure and the inter-scale correlation, and a tensor nuclear norm is used to impose a low-rank constraint on the third-order tensor, thereby improving the estimation accuracy of the image wavelet coefficients. Therefore, the final restored result image not only has a good overall visual effect, but also retains a large amount of detail information such as contours, edges, textures, etc. inside the image, making the entire estimation result closer to the true value.

The present invention is mainly verified by simulation experiment method, and all steps and conclusions are verified to be correct on MATLAB9.5.

BRIEF DESCRIPTION OF THE DRAWINGS

Fig. 1 is a flowchart of the workflow of the present invention;

FIG2 is an original Lena image used in the simulation of the present invention in an image repair scenario;

Figure 3 is a test image after the Lena original image is covered with text in the image repair scenario;

FIG4 is the reconstruction result of FIG3 using the SALSA method in the image repair scenario;

FIG5 is the reconstruction result of FIG3 using the SKR method in the image repair scenario;

FIG6 is a reconstruction result of FIG3 using the BPFA method in an image repair scenario;

FIG7 is the reconstruction result of FIG3 using the IRCNN method in the image inpainting scenario;

FIG8 is a reconstruction result of FIG3 using the IDBP method in an image repair scenario;

FIG9 is a reconstruction result of FIG3 using the method of the present invention in an image repair scenario;

FIG10 is an original image of House used in the simulation of the present invention in the image deblurring scenario;

Figure 11 is a test image of the original House image destroyed by the Gaussian blur kernel in the image deblurring scenario;

FIG12 is the reconstruction result of FIG11 using the SALSA method in the image deblurring scenario;

FIG13 is a reconstruction result of FIG11 using the SA-DCT method in an image deblurring scenario;

FIG14 is a reconstruction result of FIG11 using the IDD-BM3D method in an image deblurring scenario;

FIG15 is the reconstruction result of FIG11 using the IRCNN method in the image deblurring scenario;

FIG16 is the reconstruction result of FIG11 using the IDBP method in the image deblurring scenario;

FIG17 is a reconstruction result of FIG11 using the method of the present invention in an image deblurring scenario;

FIG18 is an original image of C.man used in the simulation of the present invention in the image denoising scenario;

FIG19 is a test image after adding noise to the original image of C.man in the image denoising scenario;

FIG20 is the reconstruction result of FIG19 using the NLM method in the image denoising scenario;

FIG21 is a reconstruction result of FIG19 using the SA-DCT method in an image denoising scenario;

FIG22 is the reconstruction result of FIG19 using the OWT-SURE-LET method in the image denoising scenario;

FIG23 is the reconstruction result of FIG19 using the IRCNN method in the image denoising scenario;

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

Detailed ways

1 , the present invention is an image restoration method based on wavelet tensor low-rank regularization, and the specific steps include the following:

Step 1: Perform a two-dimensional redundant wavelet decomposition on the image to obtain wavelet subband coefficients.

(1a) The input original spatial domain image x is passed through the upsampled low-pass and high-pass filters respectively to obtain two images with the same size as the original image;

(1b) The two images obtained in step (1a) are respectively passed through the up-sampled low-pass and high-pass filters, and finally four sub-band images with the same size as the original image are obtained, which can be expressed as:

α _i (x) = Φ _i x (i = 1, 2, 3, 4)

Where Φ _i represents the filter matrix. When i = 1, 2, 3, 4, α _i (x) represents the low-frequency two-dimensional wavelet subband coefficient α _L,L and the high-frequency two-dimensional wavelet subband coefficient α _L,H , α _H,L and α _H,H , respectively. The low-frequency two-dimensional wavelet subband coefficient reflects the overall smoothness of the image and is most similar to the original image. Therefore, it is also called the approximate coefficient or approximation coefficient. The high-frequency two-dimensional wavelet subband coefficient retains the local features of the image such as contour, edge, texture, etc., and is also called the detail coefficient.

Step 2: construct the wavelet subband coefficient tensor and use the tensor nuclear norm to constrain its low rank.

(2a) After obtaining the wavelet subband coefficients, in order to preserve the internal structure and correlation of each wavelet subband coefficient, the four two-dimensional wavelet subband coefficients are stacked in matrix form to form a third-order tensor by taking advantage of the tensor's ability to preserve the potential structure of multidimensional data. Combining α _i (x) = Φ _i x (i = 1, 2, 3, 4) can be expressed as:

where vec(·) represents the matrix vectorization operator, Represents the i-th two-dimensional wavelet subband coefficient slice;

(2b) From a visual perspective, although the pixel grayscale values of each wavelet subband coefficient are different, their texture and contour structures are highly similar, so it is believed that the wavelet coefficient tensor has an obvious low-rank characteristic; similar to the low-rank characteristic of the matrix nuclear norm used to constrain the matrix, the tensor nuclear norm is the sum of the tensor singular values and is a convex approximation of the tensor rank function. Similarly, the tensor nuclear norm can be used to impose low-rank constraints on the third-order wavelet coefficient tensor; in addition, high-order singular value decomposition, as a special case of Tucker decomposition, further imposes orthogonal constraints on the component matrices obtained after the decomposition, and is also considered to be a multilinear generalization of the matrix singular value decomposition. Therefore, according to It can be seen that the third-order tensor/> The nuclear norm of It can also be defined as/>

Step 3: Establish an image restoration model under the low-rank constraint of wavelet tensor and solve it.

(3a) Image restoration model under low-rank constraint of wavelet tensor:

Where y represents the degraded image, H represents the degradation matrix, represents the square of the vector's bi-norm, x represents the image to be restored, and λ is the regularization parameter; in order to solve the restoration model in (3a), an auxiliary variable is introduced/> Rewrite the restoration model as a multivariate minimization problem:

Where β is the penalty parameter, is the normalized Lagrange multiplier,/> represents the square of the Frobenius norm of the tensor; using the alternating direction iterative algorithm to solve it, the multivariate minimization problem can be decomposed into the image x to be restored and the wavelet coefficient tensor/> Two sub-problems of , and solve them alternately and iteratively:

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

in represents the i-th two-dimensional wavelet subband coefficient slice,/> Indicates/> The corresponding auxiliary variables, /> Indicates/> The corresponding normalized Lagrange multiplier, this subproblem is a least squares problem, and its closed solution can be obtained directly by matrix inversion:

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

according to Higher-order singular value decomposition of This sub-problem can be rewritten as about/> Multivariate minimization problem of U, V and W:

where (·) ^T represents the transpose of the matrix, Indicates/> The size of the identity matrix, I ₄ represents the size of the identity matrix of 4×4; the multivariate minimization problem can be decomposed into four sub-problems about each variable and solved alternately:

3c1) About The sub-problem of can be solved by the soft threshold algorithm:

where soft(·) is a point-wise soft threshold function:

3c2) The subproblem about U can be expressed as:

Since matrices V and W are both orthogonal matrices, the subproblem about U can be rewritten as:

The expand ₁ (·) indicates that the tensor is expanded according to Tucker mode -1, which means that all column fibers in the tensor are used as column vectors to form a matrix. Perform singular value decomposition:

Where P ₁ and Q ₁ are The left singular vector matrix and the right singular vector matrix of , Σ ₁ is the singular value matrix, and finally the closed solution of U can be obtained:

3c3) The subproblem about V can be expressed as:

The expand ₂ (·) indicates that the tensor is expanded according to Tucker mode -2, which means that all row fibers in the tensor are used as column vectors to form a matrix. Perform singular value decomposition:

Where P ₂ and Q ₂ are The left singular vector matrix and the right singular vector matrix of , Σ ₂ is the singular value matrix, and finally the closed solution of V can be obtained:

3c4) The subproblem about W can be expressed as:

The expand ₃ (·) indicates that the tensor is expanded according to Tucker mode-3, which means that all the tube fibers in the tensor are used as column vectors to form a matrix. Similarly, Perform singular value decomposition:

Among them, P ₃ and Q ₃ are The left singular vector matrix and the right singular vector matrix of , Σ ₃ is the singular value matrix, and finally the closed solution of W can be obtained:

3c5) Repeat steps 3c1) to 3c4) alternately to obtain the solution of the wavelet coefficient tensor

(4) Repeat step (3) until the variation between two adjacent reconstruction results is less than the iteration termination threshold or the maximum number of iterations is met.

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

1. Experimental conditions and contents

Experimental conditions: The experimental scenes include image inpainting, image deblurring and image denoising experiments. The experimental images use standard digital image test pictures, as shown in Figures 2, 10 and 18 respectively. The experimental result evaluation indicators use peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) to objectively evaluate the restoration results. The higher the PSNR and SSIM values, the better the restoration results are and the closer they are to the real image.

Experimental content: In the image inpainting scenario, the restoration results are compared with the method of the present invention using the SALSA method, SKR method, BPFA method, IRCNN method and IDBP method, which are currently representative methods in the field of image inpainting. In the image deblurring scenario, the restoration results are compared with the method of the present invention using the SALSA method, SKR method, BPFA method, IRCNN method and IDBP method, which are currently representative methods in the field of image deblurring. In the image denoising scenario, the restoration results are compared with the method of the present invention using the SALSA method, SKR method, BPFA method, IRCNN method and IDBP method, which are currently representative methods in the field of image denoising.

Experiment 1: In the image repair scenario, the image (Figure 3) covered with text in Figure 2 is restored using the method of the present invention, the SALSA method, the SKR method, the BPFA method, the IRCNN method and the IDBP method. The SALSA method is a method that uses total variation to quickly solve unconstrained optimization problems, and its restoration result is shown in Figure 4; the SKR method uses an adaptive kernel regression function method, which relies on the pixel position and intensity of the local structure of the image to characterize the spatial neighborhood pixels, and its restoration result is shown in Figure 5; the BPFA method is a method that uses dictionary learning under a non-parametric Bayesian framework to mine the spatial correlation of the image, and its restoration result is shown in Figure 6; the IRCNN method and the IDBP method are two reconstruction methods based on deep learning, and their restoration results are shown in Figures 7 and 8, respectively. In the experiment, the method of the present invention sets the regularization parameters β = 0.06, λ = 0.85, the maximum number of iterations T = 100, and the iteration termination threshold η = 1×10 ^-6 ; the final restoration result is shown in Figure 9.

From the restoration results of each method in Figures 4, 5, 6, 7, 8 and 9 and the magnified images of the local areas, it can be seen that by comparing the SALSA method, SKR method, BPFA method, IRCNN method and IDBP method with the method of the present invention, it can be seen that the restoration effect of the method of the present invention on the details is better than that of other compared methods.

Table 1 PSNR/SSIM indicators of different restoration methods

image SALSA Method SKR Method BPFA method IRCNN method IDBP Method Method of the present 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 gives the PSNR and SSIM indicators of various restoration methods, where the higher the PSNR and SSIM values are, the better the restoration effect is. It can be seen that in the image repair scenario, the PSNR and SSIM values of the method of the present invention are greatly improved compared with other methods, indicating that the restoration result of this method is closer to the real image, and this result is consistent with the restoration effect diagram.

Experiment 2: In the image deblurring scenario, the image (Figure 11) destroyed by the Gaussian blur kernel in Figure 10 is restored using the method of the present invention, the SALSA method, the SA-DCT method, the IDD-BM3D method, the IRCNN method and the IDBP method. The SALSA method is a method that uses total variation to quickly solve unconstrained optimization problems, and its restoration result is shown in Figure 12; the SA-DCT method is an image processing method that implements a hard threshold algorithm and Wiener filtering in a shape-adaptive discrete cosine transform domain, and its restoration result is shown in Figure 13; the IDD-BM3D method is a classic denoising algorithm commonly used for image restoration, known as an improved version of BM3D, and its restoration result is shown in Figure 14; the IRCNN method and the IDBP method are two reconstruction methods based on deep learning, and their restoration results are shown in Figures 15 and 16, respectively. In the experiment, the method of the present invention sets the regularization parameters β = 0.045, λ = 0.75, the maximum number of iterations T = 100, and the iteration termination threshold η = 1×10 ^-6 ; the final restoration result is shown in Figure 17.

From the restoration results of each method in Figures 12, 13, 14, 15, 16 and 17 and the enlarged images of the local areas, it can be seen that by comparing the SALSA method, SA-DCT method, IDD-BM3D method, IRCNN method and IDBP method with the method of the present invention, it can be seen that the restoration effect of the method of the present invention on the details is better than that of other compared methods.

Table 2 PSNR/SSIM indicators of different restoration methods

image SALSA Method SA-DCT method IDD-BM3D method IRCNN method IDBP Method Method of the present 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 gives the PSNR and SSIM indicators of various restoration methods, where the higher the PSNR and SSIM values are, the better the restoration effect is. It can be seen that in the image deblurring scenario, the PSNR and SSIM values of the method of the present invention are greatly improved compared with other methods, indicating that the restoration result of the method is closer to the real image, and this result is consistent with the restoration effect diagram.

Experiment 3: In the image denoising scenario, the image (Figure 19) after adding noise to Figure 18 is restored using the method of the present invention, the NLM method, the SA-DCT method, the OWT-SURE-LET method and the IRCNN method. The NLM method is a classic denoising method that weights the non-local information of the image and its restoration result is shown in Figure 20; the SA-DCT method is an image processing method that implements a hard threshold algorithm and Wiener filtering in the shape-adaptive discrete cosine transform domain, and its restoration result is shown in Figure 21; the OWT-SURE-LET method is a threshold denoising method using orthogonal wavelets, and its restoration result is shown in Figure 22; the IRCNN method is a reconstruction method based on deep learning, and its restoration results are shown in Figure 23. In the experiment, the method of the present invention sets the regularization parameters β = 0.055, λ = 0.8, the maximum number of iterations T = 100, and the iteration termination threshold η = 1×10 ^-6 ; the final restoration result is shown in Figure 24.

From the restoration results of each method in Figures 20, 21, 22, 23 and 24 and the enlarged images of the local areas, it can be seen that by comparing the NLM method, SA-DCT method, OWT-SURE-LET method and IRCNN method with the method of the present invention, it can be seen that the restoration effect of the method of the present invention on the details is better than that of other compared methods.

Table 3 PSNR/SSIM indicators of different restoration methods

image NLM Methods SA-DCT method OWT-SURE-LET Method IRCNN method Method of the present 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 gives the PSNR and SSIM indicators of various restoration methods, where the higher the PSNR and SSIM values are, the better the restoration effect is. It can be seen that in the image denoising scenario, the PSNR and SSIM values of the method of the present invention are greatly improved compared with other methods, indicating that the restoration result of the method is closer to the real image, and this result is consistent with the restoration effect diagram.

The above three experiments show that in the experimental scenarios of image repair, image deblurring and image denoising, the method of the present invention not only has obvious restoration effect, but also has rich content of the restored image and high objective evaluation index. It can be seen that the present invention is effective for restoring degraded images.

Claims (3)

1. An image restoration method based on wavelet tensor low-rank regularization, comprising the following steps: (1) Input a The image x to be restored is subjected to a one-layer two-dimensional redundant wavelet decomposition to obtain four two-dimensional wavelet subband coefficients; (2) The four two-dimensional wavelet subband coefficients obtained in (1) are stacked in matrix form to form a third-order tensor (3) Taking advantage of the strong correlation between different wavelet subband coefficients, the tensor nuclear norm is used to constrain the third-order tensor of wavelet coefficients to a low rank. The high-order singular value decomposition of can be expressed as: in represents the core tensor, U, V and W represent three orthogonal singular vector matrices, × ₁ , × ₂ and × ₃ represent Tucker mode-1 product, Tucker mode-2 product and Tucker mode-3 product respectively; based on this, the third-order tensor/> The nuclear norm of Defined as its core tensor/> The one-norm of It can be expressed as: (4) Based on constructing the wavelet coefficient tensor and defining the tensor nuclear norm constraint, an image restoration model under the low-rank constraint of the wavelet tensor is established: Where y represents the degraded image, H represents the degradation matrix, represents the square of the vector's bi-norm, x represents the image to be restored, and λ is the regularization parameter. To solve the restoration model, we first introduce auxiliary variables. Rewrite the restoration model as a multivariate minimization problem: Where β is the penalty parameter, is the normalized Lagrange multiplier,/> represents the square of the Frobenius norm of the tensor; using the alternating direction iterative algorithm to solve it, the multivariate minimization problem can be decomposed into the image x to be restored and the wavelet coefficient tensor/> Two sub-problems of , and solve them alternately and iteratively: (4a) For the variable x in the model, given Then the restoration model becomes a sub-problem of solving the image x to be restored: in represents the i-th two-dimensional wavelet subband coefficient slice,/> Indicates/> The corresponding auxiliary variables, /> Indicates/> The corresponding normalized Lagrange multiplier, this subproblem is a least squares problem, and its closed solution can be obtained directly by matrix inversion; (4b) After obtaining the image x to be restored, The sub-problem can be expressed as: according to Higher-order singular value decomposition of This sub-problem can be rewritten as about/> Multivariate minimization problem of U, V and W: where (·) ^T represents the transpose of the matrix, Indicates/> The size of the identity matrix, I ₄ represents the size of the identity matrix of 4×4; the multivariate minimization problem can be decomposed into sub-problems about each variable and solved alternately; (5) Repeat step (4) until the variation between two adjacent reconstruction results is less than the iteration termination threshold or the maximum number of iterations is met. 2. According to claim 1, an image restoration method based on wavelet tensor low-rank regularization is characterized in that in step (1), a layer of two-dimensional redundant wavelet decomposition is performed 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 coefficient reflects the overall smoothness characteristics of the image, and the high-frequency two-dimensional wavelet subband coefficient retains the local characteristics of the image such as contour, edge, texture, etc. 3. The image restoration method based on wavelet tensor low-rank regularization according to claim 1, characterized in that the model solution problem in step (4b) can be obtained by the following steps: 4b1) About The sub-problem of can be solved by the soft threshold algorithm: where soft(·) is a point-wise soft threshold function: 4b2) The subproblem about U can be expressed as: Since matrices V and W are both orthogonal matrices, the subproblem about U can be rewritten as: The expand ₁ (·) indicates that the tensor is expanded according to Tucker mode -1, which means that all column fibers in the tensor are used as column vectors to form a matrix. Perform singular value decomposition: Where P ₁ and Q ₁ are The left singular vector matrix and the right singular vector matrix of , Σ ₁ is the singular value matrix, and finally the closed solution of U can be obtained: 4b3) The subproblem about V can be expressed as: The expand ₂ (·) indicates that the tensor is expanded according to Tucker mode -2, which means that all row fibers in the tensor are used as column vectors to form a matrix. Perform singular value decomposition: Where P ₂ and Q ₂ are The left singular vector matrix and the right singular vector matrix of , Σ ₂ is the singular value matrix, and finally the closed solution of V can be obtained: 4b4) The subproblem about W can be expressed as: The expand ₃ (·) indicates that the tensor is expanded according to Tucker mode-3, which means that all the tube fibers in the tensor are used as column vectors to form a matrix. Similarly, Perform singular value decomposition: Among them, P ₃ and Q ₃ are The left singular vector matrix and the right singular vector matrix of , Σ ₃ is the singular value matrix, and finally the closed solution of W can be obtained: 4b5) Repeat steps 4b1) to 4b4) alternately to obtain the solution of the wavelet coefficient tensor .