CN112085666B - Image complement method based on restarting strategy and approximate alternation punishment algorithm - Google Patents

Abstract

The invention discloses an image completion method based on a restart strategy and an approximate alternating penalty algorithm, which includes obtaining natural image data to be completed and inputting a low-rank total variation repair model; using an approximate alternating penalty algorithm to iteratively solve the low-rank full variation repair model Variational repair model, and when the number of iterations reaches a multiple of N, a restart strategy is used to reset the variables in the approximate alternating penalty algorithm until the iteration reaches the preset maximum number of iterations; the completed natural image data obtained from the solution is output. The low-rank total variation repair model established in this application considers image repair from multiple dimensions, and uses an approximate alternating penalty algorithm to solve the low-rank total variation repair model, so as to be applied to natural image repair. The solution speed is fast, and at the same time, it uses re- The strategy is to iterate to a certain number of times and reassign initial values to the parameters to significantly improve the performance of the approximate alternating penalty algorithm and achieve better repair results while improving repair efficiency.

Description

An image completion method based on restart strategy and approximate alternating penalty algorithm

Technical field

This application belongs to the field of image processing technology, and specifically relates to an image completion method based on a restart strategy and an approximate alternating penalty algorithm, especially for the completion of low-rank total variation natural images.

Background technique

With the advancement of acquisition technology, a large number of high-order tensor data sets have been established in many fields such as computer vision, neuroscience, remote sensing, and recommendation systems. As a generalization of matrices and vectors, Kolda et al. pointed out that multidimensional arrays can be represented by tensors, which can effectively represent the interaction of multidimensional data with multiple factors. For example, a color image can be represented as a third-order decimal tensor, where the three dimensions are height, width and color channel. However, tensors constructed in real-world applications may contain missing values due to information loss or the huge cost of obtaining complete data. Therefore, completing missing values (called the tensor completion problem) has become an important research topic.

Tensor completion is a typical ill-posed inverse problem, which means that the solution to the problem is not unique. To solve this problem, a series of prior conditions need to be introduced, such as local smooth prior conditions, sparse prior conditions, low-rank prior conditions, etc. In recent years, low-rank priors have become increasingly important in solving matrix and tensor completion problems. However, unlike matrices, the rank of a tensor is not unique. How to minimize the rank of a tensor is an NP-hard problem. Therefore, minimizing the nuclear norm of a tensor is regarded as a standard method for tensor low-rank prior. By minimizing The tensor core norm simplifies an NP difficult problem into a convex optimization problem.

Li et al. proposed that in the field of visual data repair, although low-rank prior effectively utilizes the sparseness, similarity, and repeatability of visual data, it cannot effectively mine the smoothness and segmentation structure of visual data in the spatial dimension. . Without special consideration of such local structures, a good restoration may not be achieved. Therefore, it is proposed to add the total variation (TV) regular term to the tensor completion model as a supplementary structure to the low-rank prior, so that the local piecewise smooth structure of visual data can be exploited. This paper proposes a tensor completion model based on Low-Rank Tensor Completion with Total Variation (LRTV) and applies it to visual data repair.

Solving the LRTV model, as a convex optimization problem, can be solved by many classic convex optimization algorithms, such as the Alternating Direction Method of Multipliers (ADMM), Primal-dual Splitting Algorithm (PDS) )wait. However, existing ADMM, PDS and other algorithms for solving LRTV models have low computational efficiency and poor repair effects.

Contents of the invention

The purpose of this application is to provide an image completion method based on a restart strategy and an approximate alternating penalty algorithm, which has high image completion and repair efficiency and good repair effects.

In order to achieve the above purpose, the technical solutions adopted by this application are:

An image completion method based on a restart strategy and an approximate alternating penalty algorithm. The image completion method based on a restart strategy and an approximate alternating penalty algorithm includes:

Step 1. Obtain the natural image data to be completed Enter the low-rank total variation repair model, where m and n represent the width and height of the natural image respectively, and 3 represents the three RGB channels of the natural image data;

Step 2: Use the approximate alternating penalty algorithm to iteratively solve the low-rank total variation repair model, and use the restart strategy to reset the variables in the approximate alternating penalty algorithm when the number of iterations reaches a multiple of N until the number of iterations reaches the preset maximum number of iterations;

Step 3. Output the completed natural image data obtained by solving the problem;

Among them, the low-rank total variation repair model includes:

The low-rank total variation restoration model for natural image completion restoration is defined as follows:

In the formula, is a tensor, representing the input natural image data to be completed,/> is a tensor, representing the output completed natural image data,/> Represents tensor/> The low-rank term of , which is defined as And Y _(k) represents the tensor/> The m×n matrix of the kth dimension, λ _k represents the tensor/> The weight coefficient of the nuclear norm of the k-th dimension m×n matrix,/> Represents tensor/> The total variation regular term of , which is defined as w _k represents tensor/> The weight coefficient of the total variation regularization term of the k-th dimension m×n matrix, and ||Y _(k) || _tv is defined as follows:

In the formula, yi _,j represents the element corresponding to the i-th row and j-th column of matrix Y _(k) ;

||·|| _F is the Frobenius norm, which is defined as Q _i,j is the element value corresponding to the i-th row and j-th column of matrix Q,/> is an index matrix, index matrix/> The 0 in represents the input natural image data/> The missing element in , 1 represents the input natural image data/> Among the observable elements, Ω is the support set Ω, which represents the set of elements that have not been lost. α and β represent the parameters of the low-rank term and the total variation regular term respectively. The value of δ is based on the input natural image data to be completed/ > depends on the element loss rate, the loss rate Symbol:= means defined as.

Several optional methods are also provided below, but they are not used as additional limitations on the above-mentioned overall plan. They are only further additions or preferences. On the premise that there are no technical or logical contradictions, each optional method can be independently implemented for the above-mentioned overall plan. Combination can also be a combination between multiple optional methods.

Preferably, step 2 uses an approximate alternating penalty algorithm to iteratively solve the low-rank total variation repair model, and when the number of iterations reaches a multiple of N, a restart strategy is used to reset the variables in the approximate alternating penalty algorithm until the number of iterations reaches Preset maximum number of iterations, including:

Step 2.1. According to the solution rules of the approximate alternating penalty algorithm, transform the low-rank total variation repair model into the following form, and use the transformed form as the objective function:

In the formula, X and Y respectively represent tensors and tensor/> matrix form;

Step 2.2, let g(Y) represent f(X) represents/> express

Step 2.3, initialization α, β, ρ ₀ , vector w, vector λ, iteration number iter=0, λ _k and w _k represent the k-th element of vectors λ and w;

Step 2.4, Calculate

Step 2.5, Calculate

Step 2.6, update

Step 2.7, update ρ _iter+1 ;

Step 2.8. Determine whether the number of iterations iter is an integer multiple of N. If so, use the restart strategy to reset the variables in the approximate alternating penalty algorithm. The reset variables are ρ _iter =ρ ₀ , Otherwise, go to step 2.9;

Step 2.9. Determine whether the number of iterations iter reaches the preset maximum number of iterations. If so, end the iteration and add the latest as the completed natural image data obtained from the solution; otherwise, return to step 2.4 to continue iteration.

As a preference, step 2.4 calculates include:

In the formula, ρ _iter is the penalty parameter, and is the constraint term of the objective function, proj _κ (·) represents the projection operator on the convex set, and prox represents the approximate operator of the objective function, which is defined as follows:

According to the definition of approximation operator, it can be Rewritten as:

In the formula, f represents f(X), and Lemma 1 is introduced to solve it.

Lemma 1: Let is a given matrix, then the singular value decomposition of the matrix W with rank r is defined as follows:

W＝UE _r V ^T ,E _r =diag({σ _i } _1≤i≤r )

In the formula, diag({σ _i } _1≤i≤r ) represents a diagonal matrix, the diagonal element corresponding to the i row is σ _i , and σ _i is the i-th singular value of the matrix W, and U is the left singular matrix , V ^T is the right singular matrix;

And the singular value contraction operator will obey the following formula:

Therefore, the matrix W and the diagonal matrix E _r are expressed according to the singular value shrinkage operator as:

_Jξ (W)＝ _UJξ (W) ^VT ， _Jξ ( _Er )＝diag{max(( _σi -ξ),0)}

ξ is the input threshold. Through Lemma 1, we can get the formula The solution is:

In the formula, the value of k is 1 or 2 or 3.

As a preference, step 2.5 calculates include:

In the formula, g represents g(Y), Represents the function/> Medium variable/> Find the gradient, and/>

Solve using fast gradient descent method The value of k is 1 or 2 or 3.

As a preference, the step 2.6 updates include:

The described step 2.7 updates ρ _iter+1 , including:

ρ _iter+1 :＝(iter+2)ρ ₀

In the formula, iter is the number of iterations.

This application provides an image completion method based on the restart strategy and the approximate alternating penalty algorithm. The established low-rank total variation repair model considers image repair from multiple dimensions, and uses the approximate alternating penalty algorithm to solve the low-rank total variation repair model. , to be applied to natural image repair, the solution speed is fast, and a re-strategy is used, that is, iterating to a certain number of times, re-assigning initial values to the parameters, so as to significantly improve the performance of the approximate alternating penalty algorithm, and achieve better results while improving the repair efficiency. repair effect.

Description of the drawings

Figure 1 is a flow chart of the image completion method based on the restart strategy and the approximate alternating penalty algorithm of this application.

Detailed ways

The technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present application. Obviously, the described embodiments are only some of the embodiments of the present application, rather than all of the embodiments. Based on the embodiments in this application, all other embodiments obtained by those of ordinary skill in the art without creative efforts fall within the scope of protection of this application.

Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the technical field to which this application belongs. The terms used herein in the description of the present application are only for the purpose of describing specific embodiments and are not intended to limit the present application.

In one embodiment, an image completion method based on a restart strategy and an approximate alternating penalty algorithm is provided, which can complete the restoration of natural images with high efficiency and high restoration degree.

The natural image referred to in this embodiment should be understood as a natural landscape image with repeated scene elements, which is a non-lined image and can be obtained through an image/video collection device, such as a camera.

The approximate alternating penalty algorithm (PAPA) used in this embodiment mainly uses a novel combination of the classic quadratic penalty method, alternating minimization method, Nesterov's acceleration method and parameter adaptive strategy method to effectively solve the problem non-smooth constrained convex optimization problems, and obtains the famous Convergence, iter is the number of iterations.

The quadratic penalty method is a classic optimization framework for dealing with constrained problems. This method is usually inefficient when used alone, but when combined with an alternating strategy, it can achieve good results. The key to the PAPA algorithm is the parameter adaptation strategy in Nesterov's acceleration scheme, which can speed up the convergence of the PAPA algorithm and automatically update the penalty parameters and other parameters without manual adjustment. At the same time, this embodiment combines the strategy of "restarting the approximate center point (Restarting, RS)" (restart strategy), that is, iterating to a certain number of times and re-assigning initial values to the parameters to avoid the PAPA algorithm falling into local optimality. Combined with the restart strategy The performance of the PAPA algorithm can be significantly improved.

As shown in Figure 1, the image completion method based on the restart strategy and the approximate alternating penalty algorithm in this embodiment includes:

Step 1. Obtain the natural image data to be completed Enter the low-rank total variation repair model, where m and n represent the width and height of the natural image respectively, and 3 represents the three RGB channels of the natural image data. The natural image can be rearranged into a three-dimensional tensor according to the three RGB channels to facilitate subsequent image repair.

The low-rank total variation inpainting model (LRTV inpainting model) adds the total variation (TV) regular term to the tensor completion model as a supplementary structure to the low-rank prior so that local parts of the visual data can be utilized Segment smooth structure.

The low-rank total variation repair model provided in this embodiment includes:

The low-rank total variation restoration model for natural image completion restoration is defined as follows:

In the formula, is a tensor, representing the input natural image data to be completed,/> is a tensor, representing the output completed natural image data,/> Represents tensor/> The low-rank term of , this embodiment uses the nuclear norm to represent the low-rank term, which is defined as/> And Y _(k) represents the tensor/> The m×n matrix of the kth dimension, λ _k represents the tensor/> The nuclear norm of the k-th dimension m×n matrix ||Y _(k) || The weight coefficient of _* ,/> Represents tensor/> The total variation regular term of , which is defined as/> w _k represents tensor/> The weight coefficient of the total variation regularization term ||Y _(k) || _tv of the k-th dimension m×n matrix, and ||Y _(k) || _tv is defined as follows:

In the formula, yi _,j represents the element corresponding to the i-th row and j-th column of matrix Y _(k) .

||·|| _F is the Frobenius norm, which is defined as Q _i,j is the element value corresponding to the i-th row and j-th column of matrix Q,/> is an index matrix, index matrix/> The 0 in represents the input natural image data/> The missing element in , 1 represents the input natural image data/> Among the observable elements, Ω is the support set Ω, which represents the set of elements that have not been lost. α and β represent the parameters of the low-rank term and the total variation regular term respectively. The value of δ is based on the input natural image data to be completed/ > It is adjusted based on the element loss rate. Generally, the higher the loss rate, the greater the value of δ; the lower the loss rate, the smaller the value of δ. For example, when 80% of pixels (elements) are lost in natural image data, δ is 10 ^-3 , where the loss rate/> Symbol:= means defined as.

Step 2: Use the approximate alternating penalty algorithm to iteratively solve the low-rank total variation repair model, and use the restart strategy to reset the variables in the approximate alternating penalty algorithm when the number of iterations reaches a multiple of N until the number of iterations reaches the preset maximum Number of iterations.

This embodiment uses an approximate alternating penalty algorithm combined with a restart strategy to solve the low-rank total variation repair model, which can not only significantly improve the solution efficiency, but also effectively improve the repair effect of natural images. A solution process provided in this embodiment is as follows:

Step 2.1. According to the solution rules of the approximate alternating penalty algorithm (including a constraint, there must be two variables in the objective function), transform the low-rank total variation repair model into the following form, and use the transformed form as the objective function. :

In the formula, X and Y respectively represent tensors and tensor/> matrix form.

Step 2.2, let g(Y) represent f(X) represents/> express

Step 2.3, initialization α, β, ρ ₀ , vector w, vector λ, iteration number iter=0. where tensor/> Contains/> Tensor/> Contains/> Tensor/> Contains/> Among them, k is 1 to 3. And in this embodiment, it is set/> The parameter α of the low-rank term is set to 0.009, the parameter β of the total variation regular term is set to 1.2, ρ ₀ is set to 0.5, w and λ are one-dimensional vectors, and the initial values are set to [0.4, 0.4, 0.2] and [1 respectively. ,1,1], λ _k and w _k represent the k-th element of the vectors λ and w.

It should be noted that the above is a parameter setting provided in this embodiment. During actual use, the setting of the initial value of the parameter can be adjusted according to the loss rate of natural images or image repair requirements.

Step 2.4, Calculate include:

In the formula, This formula uses the classic quadratic penalty method, ρ _iter is the penalty parameter, and the quadratic penalty function/> is the constraint term of the objective function, proj _κ (·) represents the projection operator on the convex set. Since the low-rank total variation repair model is a convex optimization problem, the convex set here refers to the solution space of the low-rank total variation repair model. collection.

prox represents the approximate operator of the target function (proximal operator), which is defined as follows:

prox _γf (x) means that there is a unique value v that is an approximation operator of function f at variable x.

According to the definition of approximation operator, it can be Rewritten as:

In the formula, f represents f(X). In this embodiment, Lemma 1 is introduced to solve the problem. That is, solve formulas (4), (5), and (6):

Lemma 1: Let is a given matrix, then the singular value decomposition of the matrix W with rank r is defined as follows:

In the formula, diag({σ _i } _1≤i≤r ) represents a diagonal matrix, that is, E _r is a diagonal matrix, the diagonal element corresponding to its i row is σ _i , and σ _i is the i-th element of matrix W Singular values, U is the left singular matrix, V ^T is the right singular matrix;

And the singular value contraction operator will obey the following formula:

Therefore, the matrix W and the diagonal matrix E _r are expressed according to the singular value shrinkage operator as:

J _ξ (W)＝UJ _ξ (W)V ^T , J _ξ (E _r )＝diag{max((σ _i -ξ),0)} (11)

ξ is the input threshold. Through Lemma 1, we can get the formula The solution is:

In the formula, the value of k is 1 or 2 or 3.

Step 2.5, Calculate include:

In the formula, g represents Represents the function/> Medium variable/> Find the gradient, and/>

Solve using fast gradient descent method The value of k is 1 or 2 or 3. The fast gradient descent method used in this embodiment is a method in the prior art. For example, the fast gradient descent method proposed by Beck et al. is used to solve the approximate operator of the TV regular term. The solution process will not be described again.

Step 2.6, update include:

Step 2.7, update ρ _iter+1 , including:

ρ _iter+1 :＝(iter+2)ρ ₀ (17)

Step 2.8. Determine whether the number of iterations iter is an integer multiple of N. If so, use the restart strategy to reset the variables in the approximate alternating penalty algorithm. The reset variables are ρ _iter =ρ ₀ , Otherwise, proceed to step 2.9. N can be adjusted according to the pixel loss rate of the natural image to be completed or the desired repair effect. In this embodiment, it is preferably set to 50.

Step 2.9. Determine whether the number of iterations iter reaches the preset maximum number of iterations. If so, end the iteration and add the latest as the completed natural image data obtained by finding the κ solution; otherwise, return to step 2.4 to continue iteration.

Step 3. Output the completed natural image data obtained by solving the problem, that is, output the latest

The restoration of natural images is challenging. The LRTV model can be used to effectively restore natural images. However, the general algorithms based on primal-dual splitting, such as ADMM and PDS, have low solution efficiency and poor restoration effect. Therefore, this application proposes to use a restart strategy and an approximate alternating penalty algorithm to solve the LRTV model, and apply it to the restoration of natural images to obtain better solution efficiency and restoration effect.

It should be noted that in this embodiment, there are representations of ordinary letters and Greek letters, and representations of uppercase letters and lowercase letters. Each representation has a different meaning. For example, κ and k have different meanings; another example is The meanings expressed by X and x are also different.

This embodiment is based on the restart strategy and approximate alternating penalty algorithm (PAPA-RS algorithm) and the existing ADMM and PDS algorithms as the experimental objects to conduct repair experimental research.

The repair objects are three types of natural images with a loss of 60%, 70%, and 80% of pixels. Each type is set with 3 natural images. The low-rank total variation repair model in this application is used as the LRTV model, and PAPA-RS is used respectively. algorithm, ADMM algorithm, and PDS algorithm to solve the LRTV model. Each algorithm runs on computer equipment with the same performance, and the running environment and times are the same.

This experiment was run multiple times for each natural image in each category, and the average was taken as the final experimental data. Record the test data as shown in Table 1 and Table 2. Table 1 shows the PSNR (peak signal-to-noise ratio) values of the three images of lena, baboon, and house repaired by PAPA-RS, ADMM, and PDS under the loss rates of 60%, 70%, and 80%. The time it takes for PAPA-RS, ADMM, and PDS to repair the three images of lena, baboon, and house under the loss rate of %, 70%, and 80%.

Table 1 PSNR values

image name: Loss rate PAPA-RS ADMM PDS lena 60% 30.83 29.02 29.17 lena 70% 28.54 27.32 27.64 lena 80% 26.12 25.04 25.12 baboon 60% 24.07 23.01 23.57 baboon 70% 22.51 21.92 21.75 baboon 80% 20.13 19.52 19.38 house 60% 31.24 30.93 30.67 house 70% 29.47 28.65 28.74 house 80% 26.38 25.28 25.17

Table 2 Repair time

image name: Loss rate PAPA-RS ADMM PDS lena 60% 5.76s 31s 21.20s lena 70% 9.18s 29.85s 35.27s lena 80% 13.81s 32.81s 44.72s baboon 60% 5.95s 33.91s 24.54s baboon 70% 9.32s 32.50s 32.24s baboon 80% 14.51s 33.01s 41.97s house 60% 6.64s 28.47s 26.98s house 70% 10.36s 31.14s 38.06s house 80% 15.01s 31.32s 45.54s

According to the experimental data in Table 1 and Table 2, it can be seen that when the loss rate of natural images is the same, the PSNR value after repair by the PAPA-RS algorithm of this application is the highest, and the repair time is significantly shortened; for different loss rates For the repair of natural images, the PAPA-RS algorithm of this application still has the highest PSNR value and the shortest repair time among the three algorithms.

Experimental results show that the natural image completion method proposed by the present invention can effectively repair natural images that have lost a large number of pixels, and its solution efficiency is significantly higher than algorithms such as ADMM and PDS.

In another embodiment, a computer device is provided. The computer device may be a terminal, and its internal structure may include a processor, a memory, a network interface, a display screen, and an input device connected through a system bus. Wherein, the processor of the computer device is used to provide computing and control capabilities. The memory of the computer device includes non-volatile storage media and internal memory. The non-volatile storage medium stores operating systems and computer programs. This internal memory provides an environment for the execution of operating systems and computer programs in non-volatile storage media. The network interface of the computer device is used to communicate with external terminals through a network connection. When the computer program is executed by the processor, it implements the above image completion method based on the restart strategy and the approximate alternating penalty algorithm. The display screen of the computer device may be a liquid crystal display or an electronic ink display. The input device of the computer device may be a touch layer covered on the display screen, or may be a button, trackball or touch pad provided on the computer device shell. , it can also be an external keyboard, trackpad or mouse, etc.

It should be understood that although various steps in the flowchart of FIG. 1 are shown in sequence as indicated by arrows, these steps are not necessarily executed in the order indicated by arrows. Unless explicitly stated in this article, there is no strict order restriction on the execution of these steps, and these steps can be executed in other orders. Moreover, at least some of the steps in Figure 1 may include multiple sub-steps or multiple stages. These sub-steps or stages are not necessarily executed at the same time, but may be executed at different times. The execution of these sub-steps or stages The sequence is not necessarily sequential, but may be performed in turn or alternately with other steps or sub-steps of other steps or at least part of the stages.

The technical features of the above-described embodiments can be combined in any way. To simplify the description, not all possible combinations of the technical features in the above-described embodiments are described. However, as long as there is no contradiction in the combination of these technical features, All should be considered to be within the scope of this manual.

The above-described embodiments only express several implementation modes of the present application, and their descriptions are relatively specific and detailed, but they should not be construed as limiting the scope of the invention patent. It should be noted that, for those of ordinary skill in the art, several modifications and improvements can be made without departing from the concept of the present application, and these all fall within the protection scope of the present application. Therefore, the protection scope of this patent application should be determined by the appended claims.

Claims (5)

1. The image complement method based on the restarting strategy and the approximate alternation punishment algorithm is characterized by comprising the following steps of:

step 1, acquiring natural image data to be complementedInputting a low-rank total variation repair model, wherein m and n respectively represent the width and the height of a natural image, and 3 represents three RGB channels of natural image data;

step 2, iteratively solving the low-rank total variation repair model by using an approximate alternation punishment algorithm, and resetting variables in the approximate alternation punishment algorithm by using a restarting strategy when the iteration times reach multiples of N until the iteration times reach a preset maximum iteration times;

step 3, outputting the complemented natural image data obtained by solving;

wherein, the low-rank total variation repair model comprises:

the low-rank total variation repair model for natural image complement repair is defined as follows:

in the method, in the process of the invention,for tensors, representing the input natural image data to be complemented, < >>Representing the output complement natural image data as tensors, < >>Representing tensor->Is defined asAnd Y is _(k) Representing tensor->M x n matrix, lambda of the k-th dimension of (2) _k Representing tensor->Weight coefficient of the kernel norm of the m x n matrix of the k-th dimension, +.>Representing tensor->Is defined asw _k Representing tensor->Weight coefficients of the full variation regularization term of the m×n matrix of the k-th dimension, and ||y _(k) || _tv Is defined as follows:

in which y _i,j Representation matrix Y _(k) Elements corresponding to the ith row and the jth column;

||·|| _F is the Frobenius norm, defined asQ _i,j For the element values corresponding to the ith row and the jth column of the matrix Q, < >>θ∈{0,1} ^m×n×3 For an index matrix, 0 in the index matrix θ represents input natural image data +.>Is a missing element of 1,1 represents the input natural image data +.>Wherein omega is a support set omega, the non-lost element set is represented, alpha and beta respectively represent parameters of a low-rank term and a total variation regularization term, and the value of delta is based on inputNatural image data to be complemented +.>The loss rate of the element is dependent on the loss rate +.>Symbol: = representation is defined as.

2. The image complement method based on the restart strategy and the approximate alternation penalty algorithm according to claim 1, wherein the step 2 of iteratively solving the low-rank total variation repair model by using the approximate alternation penalty algorithm, and resetting variables in the approximate alternation penalty algorithm by using the restart strategy when the iteration number reaches a multiple of N until the iteration number reaches a preset maximum iteration number comprises:

2.1, converting the low-rank total variation repair model into the following form according to a solving rule of an approximate alternation punishment algorithm, and taking the converted form as an objective function:

s.t.X＝Y

wherein X and Y each represent tensorsAnd tensor->Is a matrix form of (a);

step 2.2, let g (Y) denotef (X) represents-> Representation->

Step 2.3, initializingα，β，ρ ₀ Vector w, vector λ, iteration number iter=0, λ _k And w _k The kth element representing vectors λ and w;

step 2.4, calculating

Step 2.5, calculating

Step 2.6, updating

Step 2.7, update ρ _iter+1 ；

Step 28, judging whether the iteration number iter is an integer multiple of N, if so, resetting the variable in the approximate alternation punishment algorithm by using a restarting strategy, wherein the reset variable is ρ _iter ＝ρ ₀ ，Otherwise, executing the step 2.9;

step 2.9, judging whether the iteration number item reaches the preset maximum iteration number, if so, ending the iteration and updatingNumber of natural images after completion obtained as a result of solvingAccording to the above; otherwise, returning to the step 2.4 to continue iteration.

3. The image complement method based on the restart strategy and the approximate alternation penalty algorithm according to claim 2, wherein the step 2.4 calculatesComprising the following steps:

in the method, in the process of the invention,ρ _iter is a penalty parameter, and as a constraint term for the objective function, projκ (·) represents the projection operator on the convex set, _prox an approximation operator representing an objective function, defined as follows:

according to the definition of the approximation operator, the method canThe rewriting is as follows:

where f represents f (X), introducing a lemma 1 to solve

Lemma 1: order theIs a given matrix, the singular value decomposition for matrix W of rank r is defined as follows:

W＝UE _r V ^T ,E _r ＝diag({σ _i } _1≤i≤r )

in the formula, diag ({ sigma) _i } _1≤i≤r ) Representing a diagonal matrix with a diagonal element sigma corresponding to row i _i And sigma _i Is the ith singular value of matrix W, U is the left singular matrix, V ^T Is a right singular matrix;

and the singular value contraction operator will obey the following formula:

thus, the matrix W and the diagonal matrix E are scaled according to the singular value contraction operator _r Expressed as:

J _ξ (W)＝UJ _ξ (W)V ^T ，J _ξ (E _r )＝diag{max((σ _i -ξ),0)}

xi is the threshold value of the input, and the equation can be obtained by the lemma 1The solution of (2) is:

wherein k takes on the value of 1 or 2 or 3.

4. The image complement method based on the restart strategy and the approximate alternation penalty algorithm according to claim 3, wherein the step 2.5 calculatesComprising the following steps:

wherein g represents g (Y), representation of the function->Medium variableGradient is calculated, and->

Solving by using a rapid gradient descent methodk takes on the value 1 or 2 or 3.

5. The image complement method based on the restart strategy and the approximate alternation penalty algorithm according to claim 2, wherein the step 2.6 updatesComprising the following steps:

said step 2.7 updates ρ _iter+1 Comprising:

ρ _iter+1 :＝(iter+2)ρ ₀

where iter is the number of iterations.