An image restoration method based on tensor sequence decomposition

Abstract

The present invention discloses an image restoration method based on tensor column decomposition, which relates to the field of image processing technology, including using a multi-layer dimensionality increase scheme to convert a low-dimensional damaged image into a high-dimensional tensor; constraining the TT low rank of the high-dimensional tensor under the tensor column decomposition framework, and combining the differential term constraints to construct an image restoration model based on tensor column decomposition; based on the ADMM method, an iterative solution algorithm is given; the image restoration based on the tensor column under the dimensionality increase scheme can improve the restoration accuracy more than directly performing the image restoration based on the tensor column; the present invention provides an image multi-layer dimensionality increase scheme for mining the potential low rank of high-dimensional tensors of images, which expands the technical ideas of image restoration technology at the high-dimensional tensor level; the present invention can significantly improve the restoration accuracy of images and videos under various forms of damage.

Description

Technical Field

[0001] The present invention relates to the field of image processing technology, and in particular to an image restoration method based on tensor sequence decomposition. Background Art

[0002] Image or video data is prone to damage such as local data damage, salt and pepper noise, pixel loss, scratch damage, stripe damage, character overlay, etc. Image restoration refers to the process of filling lost pixels or restoring damaged areas of the image, which is an ill-posed inverse problem. In the present invention, images and videos are collectively referred to as images. Traditional image restoration technology usually constructs a mathematical model based on image prior information to solve the restoration problem. Commonly used prior information includes smoothness, transform domain sparsity, and matrix low rank. Because images have good local similarity, image restoration technology based on matrix low rank has excellent restoration performance when processing images such as salt and pepper noise, random scratches, and random pixel loss. When processing parallel line stripes and square-shaped damaged images, the restoration effect of the restoration technology based on direct matrix low rank is often poor.

[0003] In recent years, a variety of new tensor decomposition frameworks have emerged for processing high-dimensional data, and tensor analysis (including tensor decomposition and tensor filling theory) has received increasing attention. Images, such as RGB images and RGB videos, are inherently three-dimensional tensors. Traditional tensor CP decomposition or tensor Tucker decomposition methods, when applied to image restoration, directly treat the damaged image as a three-dimensional tensor for tensor filling, without deeply exploring the underlying low-rank prior information within the image. For high-dimensional tensors, new tensor decomposition tools such as tensor train (TT) decomposition are more suitable for describing the global information of the tensor than CP decomposition and TUCKER decomposition. However, due to the low dimensionality of RGB image data, direct image restoration using tensor train decomposition is ineffective.

[0004] To address the above problems, in order to improve the restoration accuracy of low-rank-based image restoration technology under various typical forms of image damage, the present invention deeply explores the low-rank prior information of the image's potential tensor and seeks an effective method to apply TT decomposition to image restoration problems.

[0005] There are two issues that need to be addressed:

[0006] 1. If the low-dimensional image data is constructed into a high-dimensional tensor, and the fact that the high-dimensional tensor has the low rank of the tensor sequence can be guaranteed, then the image restoration problem becomes a tensor filling problem that is suitable for tensor sequence decomposition;

[0007] 2. Based on the constructed high-dimensional tensor, how to build a mathematical model and how to solve the repair problem. Summary of the Invention

[0008] In view of the problem that traditional low-rank image and video restoration methods do not fully exploit the potential low-rank prior information at the high-dimensional tensor level of the data, resulting in poor restoration effects, the present invention provides an effective method for applying tensor column decomposition to the image restoration problem. First, in order to exploit the low-rank prior information of the potential tensor column of the image, a multi-layer data dimensionality increase scheme is designed to convert the image into a high-dimensional tensor suitable for tensor column decomposition without changing the image elements. Then, tensor column decomposition is used to perform image restoration modeling and optimize the algorithm solution. The purpose of the present invention is mainly achieved through the following scheme:

[0009] An image restoration method based on tensor train decomposition, characterized by comprising the following steps:

[0010] S1: Input the image data to be repaired, read the image data size as a×b×t, and obtain the pixel position set Ω to be repaired;

[0011] S2: Set a multi-layer tensor dimension increase operator Φ to increase the image data from a low-dimensional tensor to a high-dimensional tensor ΦX;

[0012] S3: Set the operator Tn to obtain the n-th TT matrix of the high-dimensional tensor ΦX;

[0013] S4: Based on the high-dimensional tensor ΦX and the operator Tn for obtaining the n-th TT matrix of the high-dimensional tensor ΦX, a mathematical restoration model for the image is constructed;

[0014] S5: Solve the mathematical restoration model of the image in step S4 based on the ADMM iterative optimization algorithm;

[0015] S6: Output the restored image, display, and store it.

[0016] Furthermore, the specific operation of step S2 includes the following steps:

[0017] S201: Determine whether the row length a and column length b of the image data to be repaired both satisfy the geometric multiplication relationship of 2. If not, expand the row length and column length by copying the boundary area of ​​the image so that a and b satisfy the geometric multiplication relationship of 2 at the minimum, that is, a=2 p b=2 q , where p and q are integers greater than 2; if satisfied, proceed directly to step S202;

[0018] S202: Initialize the number of dimensionality increase levels i=1;

[0019] S203: adding one dimension to each image data according to the quartering method;

[0020] S204: Judgment Is it equal to 4? If i is not equal to 4, set i=i+1 and execute steps S203-S204 to perform multi-layer dimensionality increase; is equal to or less than 4, then each image data obtained in step S203 is combined into a size of A high-dimensional tensor ΦX of represents i 4 consecutive ×;

[0021] S205: The above steps S201-S204 are defined as a tensor dimensionality increase operation operator Φ. After the above i layers of dimensionality increase, the dimension of the high-dimensional tensor ΦX finally obtained is N=i+2 or N=i+3.

[0022] Furthermore, the specific operation of the four-part extraction method in step S203 includes the following steps:

[0023] S2031: Extract pixel values ​​from every other row and column of the low-dimensional image data, and divide the pixel values ​​into four types: upper left, upper right, lower left, and lower right;

[0024] S2032: Recombine into four image data in order to obtain image data with an additional dimension.

[0025] Furthermore, the specific operation of step S3 includes the following steps:

[0026] S301: Set n=1, ..., N-1, and parallelize step S302;

[0027] S302: Modulo-(1,2,…,n) matrix tensor ΦX to obtain the nth TT matrix of ΦX, using A [n] Indicates that A [n] The size is m×h, where I1×I2×…×I N is the size of the tensor ΦX, and this step is defined as the operator T that obtains the n-th TT matrix of ΦX n .

[0028] Furthermore, the specific operation of step S4 includes the following steps:

[0029] S401: Use the nuclear norm to constrain the low rank of the tensor sequence of the high-dimensional tensor ΦX and construct a mathematical model for constraining the low rank of the tensor sequence:

[0030] satisfy

[0031] Where X represents the restored image, Y represents the image to be restored, ||·|| * represents the nuclear norm, and the weighted value α n Calculated by the following formula,

[0032] in

[0033] S402: Establish a mathematical model for jointly constraining the low rank and local smoothness of tensor sequences:

[0034] satisfy

[0035] Among them, β>0 is the introduced constraint balance coefficient, ||X|| TV is the total variation TV constraint term used to constrain local smoothness; D x 、D y 、D t Represent the difference operators along the row, column, and time frame directions, respectively.

[0036]

[0037]

[0038]

[0039] Unify D x 、D y 、D t Described as D, when X is an image, DX is (D x X,D y X), X is video, DX is (D x X,D y X,D t X);

[0040] S403: Introducing UV H Decomposition replaces the nuclear norm and optimizes model (2) to:

[0041]

[0042] satisfy

[0043] where Λ n is the introduced residual variable, n=1,2,…,N-1, ρ1>0 is the introduced residual balance coefficient, and model (3) is the mathematical restoration model of the image.

[0044] Furthermore, the specific operation of step S5 includes the following steps:

[0045] S501: Setting the algorithm convergence cutoff error η tol , limit the number of iterations s max ,Input the image data Y to be repaired and the pixel position set Ω to be repaired;

[0046] S502: Set the iteration variable s=1, initialize the constraint balance coefficient β and the residual balance coefficient ρ1; initialize the restored image X 0 , auxiliary variable Z 0 and residual variables All are zero tensors; initialize auxiliary tensors Can be calculated get;

[0047] S503: Convert the optimization model (3) into an unconstrained problem, and introduce an auxiliary variable Z so that Z=DX, introduce a residual variable L, and solve the model (3) into the following augmented Langrangian problem:

[0048]

[0049] in, ρ2>0 is the introduced residual balance coefficient;

[0050] S504: Obtain X in the sth iteration by solving the following problem s :

[0051]

[0052] S505: Obtain Z in the sth iteration by solving the following problem s :

[0053]

[0054] Use the shrinkage threshold algorithm to solve in one step, Where sign(·) represents the sign function, and .× represents the dot product operator;

[0055] S506: Obtain the s-th iteration by solving the following problem:

[0056]

[0057] Take the derivative to zero and solve in one step. Where I is a matrix of all 1s;

[0058] S507: Obtain the sth iteration by solving the following problem:

[0059]

[0060] Take the derivative to zero and solve in one step.

[0061] S508: Calculate and obtain Λ in the sth iteration s , L s :

[0062] L t =L t-1 +DX t -Z t ;

[0063] S509: Calculation And determine whether s>s max or η s >η tol ; If satisfied, the loop ends and the restored data X is output; if not satisfied, s=s+1, and steps S504-S509 are executed in a loop, where steps S504-S508 can be distributed and parallel calculated.

[0064] Furthermore, the specific operation of step S504 includes the following steps:

[0065] S5041: Take the derivative and take zero, and we get

[0066]

[0067] where Φ -1 、T n -1 、D -1 、F -1 are Φ and T respectively n , D, F's inverse operation operator, F represents Fourier transform;

[0068] S5042: Calculation renew

[0069] Compared with the prior art, the present invention has the following beneficial effects:

[0070] First, the present invention proposes a multi-layer dimensionality increase scheme that transforms low-dimensional data into high-dimensional tensors. This scheme can ensure the low rank of the tensor sequence of image data, so that low-dimensional image and video data can be optimized using tensor sequence decomposition tools.

[0071] Then, based on a multi-layer dimensionality increase scheme, a tensor column decomposition-based image restoration method was designed. This method can solve the problem of high-precision image restoration in the presence of salt and pepper noise, random scratches, random pixel loss, parallel line streaks, and block-shaped damage. It effectively overcomes the problem of poor restoration performance of direct matrix low-rank restoration techniques in these conditions, and significantly improves the restoration effect compared to traditional tensor Tucker decomposition and tensor CP decomposition methods. BRIEF DESCRIPTION OF THE DRAWINGS

[0072] Figure 1Take examples of the original image, original video, image to be restored, and video to be restored;

[0073] Figure 2 Give an example of a data quartering rearrangement scheme;

[0074] Figure 3 An example of a tensor dimensionality increase scheme for one layer of Lena image is given;

[0075] Figure 4 The pixel values ​​of the four small Lena images are arranged from large to small.

[0076] Figure 5 Give an example of a two-layer tensor dimension increase scheme for data;

[0077] Figure 6 The low-rank verification curve of the tensor TT matrix of the nine-dimensional tensor obtained after the three-dimensional image undergoes a six-layer dimensionality increase scheme;

[0078] Figure 7 for The third-order tensor represented by ;

[0079] Figure 8 The following is a comparison chart of the repair effects of the four methods. DETAILED DESCRIPTION

[0080] In order to enable those skilled in the art to better understand the technical solution of the present invention, the technical solution of the present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0081] An image restoration method based on tensor column decomposition includes the following steps: first, a multi-layer dimensionality increase scheme is proposed to convert low-dimensional image and video data into high-dimensional tensor form, and the low-rank property of the data tensor TT is mined; then, based on this, an image restoration model and solution algorithm based on tensor column decomposition are constructed; finally, experimental simulation verifies that the method of the present invention can improve the restoration accuracy more than the traditional low-rank restoration method, the differential constraint method, and the traditional tensor Tucker method; specifically, the image restoration method based on tensor column decomposition of the present invention includes the following steps:

[0082] S1: Input the image or video data to be repaired. The size of the read data is a×b×t (when the image data is a grayscale image, t=1; when the image data is an RGB color image, t=3, corresponding to the red, green, and blue channels respectively; when the image data is a video, t represents the number of time frames). Obtain the pixel position set Ω to be repaired; Figure 1 As shown in the figure, the original image, original video and the image and video to be repaired are examples;

[0083] S2: Set the tensor multi-layer dimensionality increase operation operator Φ to increase the dimension of the image or video data from a low-dimensional tensor to a high-dimensional tensor ΦX;

[0084] Specifically, a tensor multi-layer dimensionality increase scheme for image or video data aims to convert low-dimensional image or video data into a high-dimensional data form with TT low rank without changing the value of the data elements, that is, to increase the dimension of a low-dimensional (dimension of 2 or 3) tensor into a high-dimensional (dimension of > 3) tensor with TT low rank; the operation process of the tensor multi-layer dimensionality increase scheme includes the following steps:

[0085] S201: Data pre-adjustment; determine whether the row length a and column length b of the input image or video data both satisfy the geometric multiplication relationship of 2; if not, expand the row length and column length by copying the boundary area of ​​the image or video so that a and b at least satisfy the geometric multiplication relationship of 2, that is, a = 2 p b=2 q , where p and q are integers greater than 2; if satisfied, step S201 is ignored;

[0086] S202: Initialize the number of dimensionality increase levels i=1;

[0087] S203: adding a dimension to each image or video data according to the four-division extraction method, that is, dividing one image or video into four images or four videos;

[0088] like Figure 2 As shown in the figure, taking 3D image dimensionality increase as an example, the specific operation steps of the four-division extraction method are as follows:

[0089] S2031: Extract pixel values ​​from every other row and column of the 3D image data, so that four pixel values ​​are divided into upper left, upper right, lower left, and lower right;

[0090] S2032: Recombine into four data in order to obtain four-dimensional data.

[0091] like Figure 3 As shown in FIG, according to the four-division extraction method, an image can be divided into four small images, and the contents of these four images are visually similar.

[0092] like Figure 4 As shown in the figure, since the pixels of the four small images are adjacent in the original image, the curves of the pixel values ​​of the four images arranged from large to small are close to one curve; similarity leads to low rank, which is also the reason why the TT matrix obtained later has low rank.

[0093] S204: Judgment Is it equal to 4? If i is not equal to 4, set i=i+1 and execute steps S203-S204 to implement multi-layer dimensionality increase; is equal to or less than 4, then each image or video data obtained in step S203 is combined into a A high-dimensional tensor ΦX of Represents i 4 consecutive ×, if i=1 Represents 4, when i=2 Indicates 4×4;

[0094] S205: The above steps S201-S204 are defined as a tensor dimensionality increase operation operator Φ. After the above i layers of dimensionality increase, the dimension of the high-dimensional tensor ΦX finally obtained is N=i+2 (for grayscale images) or N=i+3 (for RGB color images or videos).

[0095] like Figure 5 As shown in the figure, a 2-dimensional tensor (matrix) of size 16×16 can be transformed into a 4-dimensional tensor of size 4×4×4×4 after i=2 layers of dimensionality increase, realizing tensor dimensionality increase while keeping the element value and the total number of elements unchanged.

[0096] S3: Set the operator T to obtain the n-th TT matrix of the high-dimensional tensor ΦX n ;

[0097] Specifically, based on the dimension-increased tensor ΦX, an operator T is designed to obtain the nth TT matrix of ΦX. n , where n = 1, 2, ..., N-1, the specific operation includes the following steps:

[0098] S301: Set n=1, ..., N-1, and parallelize step S302;

[0099] S302: Modulo-(1,2,…,n) matrix tensor ΦX to obtain the nth TT matrix of ΦX, using A [n] Indicates that A [n] The size is m×h, where I1×I2×…×I N is the size of the tensor ΦX; this step is defined as the operator T that obtains the n-th TT matrix of ΦX n .

[0100] Among them, the process of matrixing the tensor ΦX modulo (1,2,…,n) is to transform the tensor of size I1×I2×…×I N The elements in the tensor ΦX are sorted in rows according to the first 1 to n dimensions, and in columns according to the following n+1 to n dimensions, and finally arranged into a row with the number of rows (I1×…×I n ), the number of columns is (I n+1 ×…×I N ), that is, the modulo-(1,2,…,n) matrix of the tensor ΦX, also known as the nth TT matrix of the tensor ΦX.

[0101] like Figure 6As shown in the figure, an image of size 256×256×3 can be converted into a 9-dimensional tensor of size 4×4×4×4×4×4×4×4×3 after i=6 layers of dimensionality increase. Eight TT matrices can be obtained through the TT matrix operation operator. The normalized singular value curves of these eight TT matrices are represented by k=1, 2,…, 8 respectively. It can be seen that these TT matrices have low rank, that is, the tensor after dimensionality increase has the TT low rank property under the tensor column decomposition framework.

[0102] S4: Operation operator T based on the high-dimensional tensor ΦX and obtaining the n-th TT matrix of the high-dimensional tensor ΦX n ,construct mathematical restoration models for images or videos;

[0103] Specifically, based on the tensor dimensionality increase scheme, a mathematical restoration model for images or videos is constructed. This model constrains the low-rank property of the tensor sequence decomposition and jointly constrains the local smoothness of the image or video. The specific operation includes the following steps:

[0104] S401: Use the nuclear norm to constrain the low rank of the tensor TT after dimensionality increase, and construct a mathematical model to constrain the low rank of the tensor sequence:

[0105] satisfy

[0106] Where X represents the restored image, Y represents the image to be restored, ||·|| * represents the nuclear norm, T n Represents the operator that converts the tensor into the nth TT matrix, n = 1, 2, ..., N-1, and the size of the tensor ΦX is I1×I2×...×I N , N represents the number of dimensions of the tensor ΦX, and the weight value α n Calculated by the following formula,

[0107] in

[0108] S402: To further improve the model optimization accuracy, combined with local smoothness, a mathematical model that jointly constrains the low rank and local smoothness of tensor sequences is established:

[0109] satisfy

[0110] Among them, β>0 is the introduced constraint balance parameter, ||X|| TV It is a total variation TV (total variation) constraint term used to constrain local smoothness; generally constrain image data using ||X|| TV =||(D x X,D yX)||1, constrain video data usage||X|| TV =||(D x X,D y X,D t X)||1, where ||·||1 represents the sum of the absolute values ​​of the elements, D x 、D y 、D t Represent the difference operators along the row, column, and time frame directions, respectively.

[0111]

[0112]

[0113]

[0114] For the convenience of description, D x 、D y 、D t are described as D, X is an image, DX is (D x X,D y X), X is the video DX (D x X,D y X,D t X);

[0115] S403: To optimize the time efficiency of model solution and avoid time-consuming SVD decomposition, UV is introduced. H Decomposition replaces the nuclear norm, and model (2) is optimized as follows:

[0116]

[0117] satisfy

[0118] where Λ n is the introduced residual variable, n=1,2,…,N-1, ρ1>0 is the introduced residual balance coefficient, and model (3) is the mathematical restoration model of the image or video.

[0119] S5: Solve the mathematical restoration model of the image or video in step S4 using an iterative algorithm based on the ADMM optimization method. The specific operation includes the following steps:

[0120] S501: Setting the algorithm convergence cutoff error η tol , limit the number of iterations s max ,Input the image data Y to be repaired and the pixel position set Ω to be repaired;

[0121] S502: Set the iteration variable s=1, initialize the constraint balance coefficient β and the residual balance coefficient ρ1; initialize the restored image X0 , auxiliary variable Z 0 and residual variables All are zero tensors; initialize auxiliary tensors Can be calculated get;

[0122] S503: To facilitate the solution, the optimization model (3) is converted into an unconstrained problem, and an auxiliary variable Z is introduced so that Z = DX. The residual variable L is introduced, and the solution of the model (3) is changed to solving the following augmented Langrangian problem:

[0123]

[0124] in, ρ2>0 is the introduced residual balance coefficient;

[0125] S504: Obtain X in the sth iteration by solving the following problem s :

[0126]

[0127] S505: Obtain Z in the sth iteration by solving the following problem s :

[0128]

[0129] The shrinkage threshold algorithm can be used to solve it in one step. Where sign(·) represents the sign function, and .× represents the dot product operator;

[0130] S506: Obtain the s-th iteration by solving the following problem:

[0131]

[0132] Take the derivative to zero and solve in one step. Where I is a matrix of all 1s;

[0133] S507: Obtain the sth iteration by solving the following problem:

[0134]

[0135] Take the derivative to zero and solve in one step.

[0136] S508: Calculate and obtain Λ in the sth iteration s , L s :

[0137] Lt =L t-1 +DX t -Z t ;

[0138] S509: Calculation And determine whether s>s max or η s >η tol If the condition is satisfied, the loop ends and the restored data X is output; if not, s=s+1 and steps S504-S509 are executed in a loop, where steps S504-S508 can be distributed and parallelized;

[0139] Furthermore, when solving the problem in step S504, the specific operation includes the following steps:

[0140] S5041: Take the derivative and take zero, and we get

[0141]

[0142] where Φ -1 、 D -1 、F -1 They are Φ, T n , D, F's inverse operation operator;

[0143] When X is an image, F represents two-dimensional Fourier transform (2D-FFT), When X is video,

[0144] F represents three-dimensional Fourier transform (3D-FFT), Indicates Figure 7 The three-dimensional tensor shown;

[0145] in,

[0146]

[0147]

[0148]

[0149] S5042: Calculation renew

[0150] The specific algorithm for solving model (2) is shown in Table 1.

[0151] Table 1 Iterative optimization algorithm

[0152]

[0153]

[0154] S6: Output the restored image or video for display and storage.

[0155] Figure 8 Shown are the image to be repaired, the image repaired by the matrix low-rank method, the image repaired by the tensor Tucker decomposition method, the image repaired by the Bayesian tensor CP decomposition method, and the image repaired by the method of the present invention; after comparison and other experimental simulations, it is verified that the method of the present invention can further improve the repair accuracy, especially overcome the problem that the traditional method has poor repair effect under stripe damage.

[0156] The basic principles, main features, and advantages of the present invention are shown and described above. Those skilled in the art should understand that the present invention is not limited to the foregoing embodiments. The foregoing embodiments and descriptions are merely illustrative of the principles of the present invention. Various changes and modifications may be made to the present invention without departing from the spirit and scope of the present invention. Such changes and modifications are intended to fall within the scope of the present invention. The scope of protection claimed in the present invention is defined by the appended claims and their equivalents.

Claims

1. An image restoration method based on tensor train decomposition, characterized in that: The following steps are involved: S1: Input the image data to be repaired, read the image data size as a×b×t, and obtain the pixel position set Ω to be repaired; S2: Setting up multi-layer tensor dimension-increasing operators , the image data is increased from a low-dimensional tensor to a high-dimensional tensor ; S3: Setting up to get high-dimensional tensors The operator of the nth TT matrix ; S4: Based on high-dimensional tensors and obtain high-dimensional tensors The operator of the nth TT matrix , constructing a mathematical restoration model for the image, the specific operation includes the following steps: S401: Constraining High-Dimensional Tensors Using the Nuclear Norm The low rank property of tensor sequences is constructed to construct a mathematical model for constraining the low rank property of tensor sequences: (1) in X represents the restored image, Y represents the image to be repaired, represents the nuclear norm, weighted value Calculated by the following formula, ； S402: Establish a mathematical model for jointly constraining the low rank and local smoothness of tensor sequences: ,satisfy (2) in, is the introduced constraint balance coefficient, is the total variation TV constraint term used to constrain local smoothness; Represent the difference operators along the row, column, and time frame directions, respectively. Unification will When described as D and X is an image Right now , X is the video time Right now ; S403: Introduction Decomposition replaces the nuclear norm and optimizes model (2) to: （3） in is the introduced residual variable, , is the introduced residual balance coefficient, and model (3) is the mathematical restoration model of the image; S5: Solve the mathematical restoration model of the image in step S4 based on the ADMM iterative optimization algorithm; S6: Output the restored image, display, and store it.

2. The image restoration method based on tensor train decomposition according to claim 1, characterized in that: The specific operation of step S2 includes the following steps: S201: Determine whether the row length a and column length b of the image data to be repaired both satisfy the geometric multiplication relationship of 2. If not, expand the row length and column length by copying the boundary area of ​​the image so that a and b satisfy the geometric multiplication relationship of 2 at least, that is, , where p and q are integers greater than 2; if satisfied, proceed directly to step S202; S202: Initialize the dimension-increasing level number i=1; S203: adding one dimension to each image data according to the quartering method; S204: Judgment Is it equal to 4? If not equal to 4, set , execute steps S203-S204 to perform multi-layer dimensionality increase; if is equal to or less than 4, then each image data obtained in step S203 is combined into a size of High-dimensional tensor ,in represents i 4 consecutive ×; S205: Define the above steps S201-S204 as a tensor dimension increase operation operator , after the above i layers of dimensionality increase, the high-dimensional tensor obtained is The number of dimensions is or .

3. The image restoration method based on tensor train decomposition according to claim 2, characterized in that: The specific operation of the four-part extraction method in step S203 includes the following steps: S2031: Extract pixel values ​​from every other row and column of the low-dimensional image data, and divide the pixel values ​​into four types: upper left, upper right, lower left, and lower right; S2032: Recombine into four image data in order to obtain image data with an additional dimension.

4. The image restoration method based on tensor train decomposition according to claim 3, characterized in that: The specific operation of step S3 includes the following steps: S301: Settings , parallel computing step S302; S302: Model Matrixized Tensor get The nth TT matrix is express, Size , , ,in is a tensor The size of , this step is defined as getting The operator of the nth TT matrix .

5. The image restoration method based on tensor train decomposition according to claim 4, characterized in that: The specific operation of step S5 includes the following steps: S501: Set algorithm convergence cutoff error , limit the number of iterations , input the image data to be repaired Y , the pixel position set Ω to be repaired; S502: Setting iteration variables , initialize the constraint balance coefficient , residual balance coefficient ; Initialize the repaired image , auxiliary variables and residual variables , All are zero tensors; initialize auxiliary tensors 、 Can be calculated get; S503: Convert the optimization model (3) into an unconstrained problem and introduce the auxiliary variable Z so that , introducing the residual variable L, solving model (3) becomes solving the following augmented Langrangian problem, （4） in, is the introduced residual balance coefficient; S504: Obtain the s-th iteration by solving the following problem: : （5） S505: Obtain the s-th iteration by solving the following problem: : （6） Use the shrinkage threshold algorithm to solve in one step, ,in represents the symbolic function, Represents the dot product operator; S506: Obtain the s-th iteration by solving the following problem: : （7） Take the derivative to zero and solve in one step. ,in is a matrix of all 1s; S507: Obtain the sth iteration by solving the following problem: : （8） Take the derivative to zero and solve in one step. ; S508: Calculate and obtain the sth iteration 、 : ； S509: Calculation , and judge whether it satisfies or ; If satisfied, the loop ends and the recovery data is output If not satisfied, , steps S504-S509 are executed cyclically, wherein steps S504-S508 can be distributed and parallel calculated.

6. The image restoration method based on tensor train decomposition according to claim 5, characterized in that: The specific operation of step S504 includes the following steps: S5041: Take the derivative and get zero, in They are The inverse operator of represents Fourier transform; S5042: Calculation ,renew .

Images