CN112085666B - Image complement method based on restarting strategy and approximate alternation punishment algorithm - Google Patents
Image complement method based on restarting strategy and approximate alternation punishment algorithm Download PDFInfo
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Abstract
The invention discloses an image complement method based on a restarting strategy and an approximate alternation punishment algorithm, which comprises the steps of obtaining natural image data to be complemented and inputting the natural image data to a low-rank total variation repair model; the low-rank total variation repair model is solved in an iteration mode through an approximate alternation punishment algorithm, and when the iteration times reach multiples of N, a variable in the approximate alternation punishment algorithm is reset through a restarting strategy until the iteration is carried out to the preset maximum iteration times; and outputting the complemented natural image data obtained by solving. The low-rank total variation repair model established by the method considers image repair from multiple dimensions, solves the low-rank total variation repair model by adopting the approximate alternation punishment algorithm, is applied to natural image repair, has high solving speed, simultaneously uses a re-strategy, namely iterates to a certain number of times, and re-gives an initial value to parameters, so that the performance of the approximate alternation punishment algorithm is obviously improved, and the better repair effect is achieved while the repair efficiency is improved.
Description
Technical Field
The application belongs to the technical field of image processing, and particularly relates to an image complement method based on a restarting strategy and an approximate alternation punishment algorithm, in particular to complement of a low-rank full variation natural image.
Background
With the progress of acquisition technology, a large number of high-order tensor data sets are established in numerous fields such as computer vision, neuroscience, remote sensing and recommendation systems. As a generalization of matrices and vectors, kolda et al state that a multi-dimensional array can be represented by tensors, which can effectively represent interactions of multi-dimensional data with multiple factors. For example, a color image may be represented as a tensor in three decimal states, where the three dimensions are height, width, and color channel, respectively. However, tensors built in real applications may contain missing values due to the huge cost of information loss or acquisition of complete data. Therefore, the completion of missing values (called tensor completion problem) is an important research topic.
Tensor completion is a typical ill-posed inverse problem, meaning that the solution to the problem is not unique. To solve this problem, a series of prior conditions, such as a local smooth prior condition, a sparse prior condition, a low rank prior condition, etc., need to be introduced. In recent years, low rank priors have become increasingly important in solving the matrix and tensor completion problems. Unlike the matrix, however, the rank of the tensor is not unique, how to minimize the rank of the tensor is an NP-hard, so minimizing the kernel norm of the tensor is considered a standard approach to tensor low rank priors, simplifying an NP-hard into a convex optimization problem by minimizing the kernel norm of the tensor.
Li et al propose that in the field of visual data repair, low rank priors effectively utilize the sparsity, similarity, repeatability, and other characteristics of visual data, but cannot effectively mine the smooth and segmented structure of the visual data in the spatial dimension. Without special consideration of such local structure, a good repair effect may not be achieved. It is therefore proposed to add Total Variation (TV) regularization terms to the tensor complement model as a low-rank a priori complement structure so that a locally piecewise smooth structure of the visual data can be exploited. Tensor patch models based on Low-rank total variation (Low-Rank Tensor Completion with Total Variation, LRTV) are thus proposed and applied to visual data patching.
Solving the LRTV model, as a convex optimization problem, can be solved with many classical convex optimization algorithms, such as the alternate direction multiplier method (Alternating Direction Method of Multipliers, ADMM), the original dual split method (PDS), etc. However, the existing algorithms such as ADMM, PDS and the like are low in calculation efficiency and poor in repairing effect when solving the LRTV model.
Disclosure of Invention
The image complement method based on the restarting strategy and the approximate alternation punishment algorithm is high in image complement restoration efficiency and good in restoration effect.
In order to achieve the above purpose, the technical scheme adopted by the application is as follows:
an image complement method based on a restart strategy and an approximate alternation penalty algorithm, the image complement method based on the restart strategy and the approximate alternation penalty algorithm comprises the following steps:
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, < >>Is an index matrix, index matrix->0 in (2) 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 support set omega represents an element set which is not lost, alpha and beta respectively represent parameters of a low-rank term and a total variation regularization term, and the value of delta is defined according to input natural image data to be complemented ∈>A loss rate of the element, the loss rateSymbol: = representation is defined as.
The following provides several alternatives, but not as additional limitations to the above-described overall scheme, and only further additions or preferences, each of which may be individually combined for the above-described overall scheme, or may be combined among multiple alternatives, without technical or logical contradictions.
Preferably, the step 2 is to iteratively solve the low-rank total variation repair model by using an approximate alternation penalty algorithm, and reset variables in the approximate alternation penalty algorithm by using a restart strategy when the iteration number reaches a multiple of N, until the iteration number reaches a preset maximum iteration number, and includes:
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:
wherein X and Y each represent tensorsAnd tensor->Is a matrix form of (a);
step 2.2, let g (Y) denotef (X) represents-> Representation of
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 2.8, judging whether the iteration number iter is an integer multiple of N, if so, resetting the variable in the approximate alternation penalty 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 updatingAs the complement natural image data obtained by solving; otherwise, returning to the step 2.4 to continue iteration.
Preferably, the step 2.4 calculationComprising the following steps:
in the method, in the process of the invention,ρ iter is a penalty parameter, and proj, constraint term for objective function κ (. Cndot.) represents the projection operator on the convex set, and prox represents the approximation operator of the objective function, which is 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 root ofMatrix W and diagonal matrix E are scaled according to singular value contraction operators 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.
Preferably, the step 2.5 calculatesComprising the following steps:
wherein g represents g (Y),representation of the function->Middle variable->Gradient is calculated, and->
Solving by using a rapid gradient descent methodk takes on the value 1 or 2 or 3.
Preferably, 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.
According to the image complement method based on the restarting strategy and the approximate alternation punishment algorithm, the image restoration is considered from multiple dimensions by the established low-rank total variation restoration model, the approximate alternation punishment algorithm is adopted to solve the low-rank total variation restoration model so as to be applied to natural image restoration, the solving speed is high, meanwhile, the restarting strategy is used, namely iteration is carried out for a certain number of times, the parameters are assigned with initial values again, so that the performance of the approximate alternation punishment algorithm is remarkably improved, and the better restoration effect is achieved while the restoration efficiency is improved.
Drawings
Fig. 1 is a flowchart of an image complement method based on a restart strategy and an approximate alternation penalty algorithm of the present application.
Detailed Description
The following description of the technical solutions in the embodiments of the present application will be made clearly and completely with reference to the drawings in the embodiments of the present application, and it is apparent that the described embodiments are only some embodiments of the present application, not all embodiments. All other embodiments, which can be made by one of ordinary skill in the art without undue burden from the present disclosure, are within the scope of the present disclosure.
Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs. The terminology used herein in the description of the present application is for the purpose of describing particular embodiments only and is not intended to be limiting of the application.
In one embodiment, an image complement method based on a restart strategy and an approximate alternation 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 is understood to be a natural landscape image with repeated scene elements, which is a non-linearised image, obtainable by an image/video acquisition device, such as a camera or the like.
The approximate alternating penalty algorithm (Proximal Alternating Penalty Algorithm, PAPA) used in this embodiment uses a novel combination of classical quadratic penalty method, alternating minimization method, the acceleration method of the nydrov and the parameter adaptive strategy method to effectively solve the unsmooth constrained convex optimization problem, and achieves a well-known non-traversal senseConvergence, iter is the number of iterations.
The quadratic penalty is a classical optimization framework for handling constraint problems, which is often inefficient when used alone, but can achieve good results when combined with an alternating strategy. The key point of the PAPA algorithm is a parameter adaptive strategy in the acceleration scheme of the nydrov, which can accelerate the convergence speed of the PAPA algorithm, and can automatically update penalty parameters and other parameters without manual adjustment. Meanwhile, the embodiment combines a strategy (Restarting strategy) of Restarting an approximate center point (RS), namely iterating to a certain number of times, and reassigning an initial value to the parameter, so that the PAPA algorithm is prevented from being trapped into local optimum, and the performance of the PAPA algorithm can be obviously improved by combining the Restarting strategy.
As shown in fig. 1, the image complement method based on the restart strategy and the approximate alternation penalty algorithm of the present embodiment includes:
step 1, acquiring natural image data to be complementedAnd inputting a low-rank total variation restoration 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. The natural images may be rearranged into a three-dimensional tensor based on the three channels of RGB to facilitate subsequent image restoration.
The low-rank Total Variation repair model (LRTV repair model) adds Total Variation (TV) regularization term to the tensor complement model as a low-rank a priori complement structure so that a local piecewise smooth structure of visual data can be utilized.
The low-rank total variation repair model provided in the embodiment includes:
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->The present embodiment uses a kernel norm to represent low rank terms, which determinesMeaning->And Y is (k) Representing tensor->M x n matrix, lambda of the k-th dimension of (2) k Representing tensor->The kernel norm Y of the m x n matrix of the k-th dimension of (c) (k) || * Weight coefficient of>Representing tensor->Is defined as +.>w k Representing tensor->The fully-variable regularization term Y of the k-th dimension m x n matrix (k) || tv And Y (k) || tv Is defined as follows:
in which y i,j Representation matrix Y (k) The 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, < >>Is an index matrix, index matrix->0 in (2) 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 support set omega represents an element set which is not lost, alpha and beta respectively represent parameters of a low-rank term and a total variation regularization term, and the value of delta is defined according to input natural image data to be complemented ∈>The loss rate of the medium elements is regulated, and generally, the higher the loss rate is, the larger the delta value is; the lower the loss rate, the smaller the delta value. For example, when 80% of pixels (elements) are lost in natural image data, δ takes 10 -3 Wherein the loss rate->Symbol: = representation is defined as.
And 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.
According to the embodiment, the low-rank total variation repair model is solved by adopting the approximate alternation punishment algorithm and combining with the restarting strategy, so that the solving efficiency can be remarkably improved, and the repair effect of a natural image can be effectively improved. One solution process provided in this embodiment is as follows:
step 2.1, according to a solving rule (comprising a constraint condition, two variables are needed in an objective function) of an approximate alternation punishment algorithm, converting the low-rank total variation repair model into the following form, and taking the converted form as the objective function:
wherein X and Y each represent tensorsAnd tensor->Is a matrix form of (c).
Step 2.2, let g (Y) denotef (X) represents-> Representation of
Step 2.3, initializingα，β，ρ 0 Vector w, vector λ, iteration number iter=0. Wherein tensor->Comprises->Tensor->Comprises->Tensor->Comprises->Wherein k is 1 to 3. And in this embodiment set +.>The low rank term parameter alpha is set to 0.009, the parameter beta of the total variation regularization term is set to 1.2, ρ 0 Is set to 0.5, w, lambda is a one-dimensional vector, and initial values are respectively set to [0.4,0.4,0.2 ]]And [1,1 ]]，λ k And w k Represents the kth element of vectors lambda and w.
It should be noted that, in the above parameter setting provided for the present embodiment, in the actual use process, the setting of the initial value of the parameter may be adjusted according to the loss rate of the natural image or the image restoration requirement.
Step 2.4, calculatingComprising the following steps:
in the method, in the process of the invention,the formula uses a classical quadratic penalty method, ρ iter For penalty parameter, quadratic penalty function-> Proj, constraint term for objective function κ (. Cndot.) represents the projection operator on the convex set, which refers to the set where the solution space of the low-rank total variation repair model is located, since the low-rank total variation repair model is a convex optimization problem.
prox represents an approximation operator (proximal operator) of the objective function, which is defined as follows:
prox γf (x) The presence of a unique value v is indicated as an approximation operator of the function f at the variable x.
According to the definition of the approximation operator, the method canThe rewriting is as follows:
where f represents f (X), the present embodiment introduces the lemma 1 to solveNamely solving the formulas (4), (5) and (6):
lemma 1: order theIs a given matrix, the singular value decomposition for matrix W of rank r is defined as follows:
in the formula, diag ({ sigma) i } 1≤i≤r ) Representing diagonal matrix, i.e. E r Is a diagonal matrix, the diagonal element corresponding to the i rows is sigma i At the same time 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)} (11)
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.
Step 2.5, calculatingComprising the following steps:
wherein g representsRepresentation of the function->Middle variable->Gradient is calculated, and->
Solving by using a rapid gradient descent methodk takes on the value 1 or 2 or 3. The rapid gradient descent method used in the embodiment is a method in the prior art, for example, the rapid gradient descent method proposed by Beck et al is used for solving an approximation operator of a TV regularization term, and the solving process is not repeated.
Step 2.6, updatingComprising the following steps:
step 2.7, update ρ iter+1 Comprising:
ρ iter+1 :＝(iter+2)ρ 0 (17)
step 2.8, judging whether the iteration number iter is an integer multiple of N, if so, resetting the variable in the approximate alternation penalty algorithm by using a restarting strategy, wherein the reset variable is ρ iter ＝ρ 0 ，Otherwise, step 2.9 is performed. Wherein N can be complemented according to the requirementThe pixel loss rate of the natural image of (a) or the desired restoration effect is adjusted, and is preferably set to 50 in this embodiment.
Step 2.9, judging whether the iteration number item reaches the preset maximum iteration number, if so, ending the iteration and updatingAs the complement natural image data obtained by kappa solution; otherwise, returning to the step 2.4 to continue iteration.
Step 3, outputting the complemented natural image data obtained by solving, namely outputting the latest
The natural image restoration has a certain challenge, the natural image can be effectively restored through the LRTV model, but the general algorithm based on original-dual splitting such as ADMM, PDS and the like is used, so that the solving efficiency is too low and the restoration effect is poor, therefore, the application provides the method for solving the LRTV model through a restarting strategy and an approximate alternation punishment algorithm and applying the method to the natural image restoration so as to obtain better solving efficiency and restoration effect.
It should be noted that in this embodiment, there are representations of common letters and greek letters, and representations of upper-case letters and lower-case letters, where each representation mode has different meanings, for example, the meanings expressed by k and k are not the same; as another exampleX, x also has different meanings.
The restoration experiment research is carried out by taking the restarting strategy and the approximate alternation punishment algorithm (PAPA-RS algorithm) and the existing ADMM and PDS algorithm as experimental objects.
The repair object is three types of natural images with 60%,70% and 80% pixel loss, 3 natural images are set in each type, the low-rank total variation repair model in the application is taken as an LRTV model, and the LRTV model is solved by using a PAPA-RS algorithm, an ADMM algorithm and a PDS algorithm respectively. Each algorithm runs on the same performance computer device and runs in the same environment and number of times, etc.
The experiment is carried out for many times on each natural image in each class, and the average value is taken as final test data. The test data are recorded as shown in tables 1 and 2. Table 1 shows the PSNR (peak signal to noise ratio) values for three image repairs of lena, baboon, house by three algorithms of PAPA-RS, ADMM, PDS at 60%,70%,80% loss rate, and table 2 shows the time taken for three image repairs of lena, baboon, house by three algorithms of PAPA-RS, ADMM, PDS at 60%,70%,80% loss rate.
TABLE 1 PSNR values
| Graph 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
| Graph 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 tables 1 and 2, it can be seen that, under the condition that the loss rate of the natural image is the same, the PSNR value after the PAPA-RS algorithm is repaired is the highest, and the repairing time is obviously shortened; aiming at the restoration of natural images with different loss rates, the PAPA-RS algorithm of the application still has the highest PSNR value and the shortest restoration time in 3 algorithms.
Experimental results show that the natural image complement method provided by the invention can effectively repair natural images losing a large number of pixel points, and the solving efficiency is obviously higher than that of algorithms such as ADMM, PDS and the like.
In another embodiment, a computer device is provided, which may be a terminal, and the internal structure of which may include a processor, a memory, a network interface, a display screen, and an input device connected by a system bus. Wherein the processor of the computer device is configured to provide computing and control capabilities. The memory of the computer device includes a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system and a computer program. The internal memory provides an environment for the operation of the operating system and computer programs in the non-volatile storage media. The network interface of the computer device is used for communicating with an external terminal through a network connection. The computer program, when executed by a processor, implements the image complement method based on the restart strategy and the approximate alternation penalty algorithm. The display screen of the computer equipment can be a liquid crystal display screen or an electronic ink display screen, and the input device of the computer equipment can be a touch layer covered on the display screen, can also be keys, a track ball or a touch pad arranged on the shell of the computer equipment, and can also be an external keyboard, a touch pad or a mouse and the like.
It should be understood that, although the steps in the flowchart of fig. 1 are shown in sequence as indicated by the arrows, the steps are not necessarily performed in sequence as indicated by the arrows. The steps are not strictly limited to the order of execution unless explicitly recited herein, and the steps may be executed in other orders. Moreover, at least some of the steps in fig. 1 may include multiple sub-steps or stages that are not necessarily performed at the same time, but may be performed at different times, nor do the order in which the sub-steps or stages are performed necessarily performed in sequence, but may be performed alternately or alternately with at least a portion of other steps or sub-steps of other steps.
The technical features of the above-described embodiments may be arbitrarily combined, and all possible combinations of the technical features in the above-described embodiments are not described for brevity of description, however, as long as there is no contradiction between the combinations of the technical features, they should be considered as the scope of the description.
The above examples merely represent a few embodiments of the present application, which are described in more detail and are not to be construed as limiting the scope of the invention. It should be noted that it would be apparent to those skilled in the art that various modifications and improvements could be made without departing from the spirit of the present application, which would be within the scope of the present application. Accordingly, the scope of protection of the present application is to be determined by the claims appended hereto.
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.
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