CN107133930A - Ranks missing image fill method with rarefaction representation is rebuild based on low-rank matrix - Google Patents

Abstract

The invention belongs to computer vision field, pixel column row missing image is accurately filled to realize.The present invention is adopted the technical scheme that, the ranks missing image fill method with rarefaction representation is rebuild based on low-rank matrix, and step is that introduce low-rank priori based on low-rank matrix reconstruction theory enters row constraint to latent image；Simultaneously, it is contemplated that each row of row missing image can be by row dictionary rarefaction representation, and every a line of row missing image can be by row dictionary rarefaction representation, therefore introduces separable two-dimentional sparse prior based on sparse representation theory；So as to based on above-mentioned joint low-rank and separable two-dimentional sparse prior, specifically be expressed as solving constrained optimization equation with the image completion problem that ranks are lacked, so as to realize that ranks missing image is filled.Present invention is mainly applied to computer vision processing occasion.

Description

Line-column missing image filling method based on low-rank matrix reconstruction and sparse representation

Technical Field

The invention belongs to the field of computer vision. In particular to a line and column missing image filling method based on low-rank matrix reconstruction and sparse representation.

Background

The problem of recovering an unknown complete matrix from a portion of known pixels of the matrix has attracted considerable attention in recent years. Such problems are often encountered in many application areas of computer vision and machine learning, such as image inpainting, recommendation systems, and background modeling.

There have been many research efforts on methods to solve the image filling problem. Due to the ill-conditioned nature of the matrix filling problem, current matrix filling methods generally consider potential matrices to be low-rank or nearly low-rank, and then fill in missing pixel values through low-rank matrix reconstruction. Such as Singular Value Thresholding (SVT), Augmented Lagrange Multiplier (ALM), and accelerated nearest neighbor gradient (APG). However, these existing filling algorithms all use the low-rank characteristic of the image to fill in missing pixel values, which is effective for the case where pixels are randomly missing and each row and each column of the image has an observed value, but when there is a pixel missing in an entire row and an entire column of the image, the existing algorithms cannot solve the image filling problem. The matrix filling problem due to the large number of missing row and column pixels cannot be solved under the condition of only using the low rank characteristic for constraint. In practical applications, such as image transmission, seismic data acquisition, etc., the image matrix is likely to suffer from some row-column missing degradation. Therefore, it is necessary to design a filling algorithm capable of effectively filling the missing matrix rows and columns.

At present, aiming at the defects of the matrix filling method only using the low-rank characteristic, the academic community introduces sparse constraint on column vectors on the basis of the defect, and the recovery of image row loss is realized. However, due to the insufficiency of the prior condition, the problem of simultaneous missing of matrix rows and columns is still unsolved. Therefore, the low-rank separable two-dimensional sparse prior is introduced into the model, so that the matrix with missing rows and columns is accurately filled.

Disclosure of Invention

The invention aims to make up the defects of the prior art, namely, the accurate filling of the missing pixel row-column image is realized. The method adopts the technical scheme that a row-column missing image filling method based on low-rank matrix reconstruction and sparse representation comprises the steps of introducing low-rank prior to constraining potential images based on a low-rank matrix reconstruction theory; meanwhile, considering that each column of the row-missing image can be sparsely represented by a column dictionary and each row of the column-missing image can be sparsely represented by a row dictionary, separable two-dimensional sparse prior is introduced based on a sparse representation theory; therefore, based on the combined low-rank and separable two-dimensional sparse prior, the image filling problem with row and column missing is specifically expressed as a solution constraint optimization equation, and row and column missing image filling is achieved.

The image filling problem with row and column deletion is specifically expressed as a concrete step of solving a constraint optimization equation, and the concrete step is detailed as follows:

1) the image filling problem with row-column deficiency is specifically formulated to solve the following constrained optimization equation:

where tr (-) is the trace of the matrix, representing a low rank prior term;represents the dot multiplication operation of two matrixes, | ·| non-woven phosphor₁Representing a norm of the matrix, wherein two norm terms respectively represent separable two-dimensional sparse prior terms; omega is the observation space, representing a known pixel, P, within the observation matrix D with a missing row and column_Ω(. cndot.) is a projection operator, which represents the value of the variable projected into the spatial domain Ω, a is a filled matrix, Σ ═ diag ([ σ ])₁,σ₂,...,σ_n]) Representing a diagonal matrix consisting of the singular values of A in non-increasing order, W_a、W_bAnd W_cWeight matrices, γ, representing weighted low rank terms and separable two-dimensional sparse terms, respectively_B，γ_CRegularization coefficients, phi, respectively representing separable two-dimensional sparse terms_cAnd phi_rRespectively representing the trained column dictionary and row dictionary, the corresponding coefficient matrixes are respectively represented by B and C, and E represents the missing in the observation matrix DA pixel;

and converting the constrained optimization problem (1) into an unconstrained optimization problem by adopting an augmented Lagrange multiplier method (ALM) to solve, wherein an augmented Lagrange equation is as follows:

wherein Y is₁、Y₂And Y₃Representing the Lagrange multiplier matrix, mu₁、μ₂And mu₃Is a penalty factor that is a function of,<·,·>representing the inner product of two matrices, | · | > non-woven phosphor_FA Frobenius (Frobenius) norm representing a matrix;

the solving process is that a column dictionary and a row dictionary phi are trained_cAnd phi_rInitializing the weight matrix W_a、W_bAnd W_cAlternately updating coefficient matrixes B and C, restoring matrix A, missing pixel matrix E and Lagrange multiplier matrix Y₁、Y₂And Y₃Penalty factor μ₁、μ₂And mu₃And a weight matrix W_a、W_bAnd W_cUntil the algorithm converges, at which point the result of the iteration A^(l)Is the final solution a of the original problem.

Specifically, the dictionary Φ is trained_cAnd phi_r: training out-of-column and row dictionaries Φ using online learning algorithms on high quality image datasets_cAnd phi_r。

Specifically, the weight matrix W is initialized_a、W_bAnd W_c: when the weighting times is l, and l is 0, the weighting matrix is set Andthe initial values are all assigned to 1, indicating that there is no re-weighting for the first iteration.

Specifically, equation (2) is converted into the following sequence for iterative solution by using an alternating direction method ADM:

in the above formulaAndrespectively representing the values of variables B, C, A and E, p, at which the objective function is to be minimized₁、ρ₂And ρ₃Is a multiple factor, k is the number of iterations; then, the iterative solution is carried out according to the following steps:

1) solving for B^k+1: finding B using an accelerated neighbor gradient algorithm^k+1；

Removing the terms irrelevant to B in the objective function for solving B in the formula (3) to obtain the following equation:

using Taylor expansion method to construct a second order function to approximate the above formula, then solving the original equation for the second order function to makeAnd introducing a variable Z, and finally solving the following steps:

wherein soft (·,) is a shrink operator,gradient of (a), L_fIs a constant with a value ofVariable Z_jThe update rule of (2) is as follows:

wherein, t_jIs a set of constant sequences, j is the number of iterations of the variable;

2) solving for C^k+1: using an accelerated neighbor gradient algorithm to find C^k+1；

Removing the terms irrelevant to C in the objective function for solving C in the formula (3) to obtain the following equation:

using Taylor expansion method to construct a second order function to approximate the above formula, then solving the original equation for the second order function to makeReintroducing variablesFinally, the following can be obtained:

wherein soft (·,) is a shrink operator,is composed ofGradient of (a), L_fIs a constant with a value ofVariables ofThe update rule of (2) is as follows:

wherein, t_jIs a set of constant sequences, j is the number of iterations of the variable;

3) solving for A^k+1: solving for A using Singular Value Thresholding (Singular Value Thresholding) SVT^k+1；

Removing terms independent of A in the objective function for solving A in the formula (3), and obtaining the terms through a formula:

wherein,to Q^k+1Solving by using a singular value threshold method to obtain:

wherein H^k+1，V^k+1Are each Q^k+1The left singular matrix and the right singular matrix;

4) solving for E^k+1：E^k+1The solution of (a) consists of two parts;

in the observation space Ω, the value of E is 0; outside the observation space omega, i.e. complementary spaceAnd solving by using a first-order derivative, and combining the two parts to obtain a final solution of the E:

5) repeating the steps 1), 2), 3) and 4) until the algorithm is converged, wherein the iterative result A is obtained^k+1、Σ^k+1、B^k+1、C^k+1And E^k+1Is the result A of the original problem without re-weighting^(l)、Σ^(l)、B^(l)、C^(l)And E^(l)Where, l is the number of reweighting times;

6) updating the weight matrix W_a、W_bAnd W_c；

In order to counteract the influence of the signal amplitude on the kernel norm term and the one norm term, a re-weighting scheme is introduced according to the currently estimated singular value matrix sigma^(l)Coefficient matrix B^lAnd C^lIteratively updating the weight matrix W using an inverse scaling rule_a、W_bAnd W_c：

WhereinIs the position coordinates of a pixel in the image and is an arbitrarily small positive number.

7) Repeating the steps 1) -7) until the algorithm is converged, and iterating the stepsResults of (A)^(l)Is the final solution a of the original problem.

The invention has the technical characteristics and effects that:

aiming at the row and column missing image filling problem, the method realizes the solution of the row and column missing image filling problem by introducing separable two-dimensional sparse prior. The invention has the following characteristics:

1. algorithms such as an augmented Lagrange multiplier method (ALM), an Alternate Direction Method (ADM), an accelerated nearest neighbor gradient algorithm, a singular value threshold method and the like are used for solving the subproblems, and the advantages of the existing algorithms are integrated.

2. Sparse representation of columns and rows of an image using a column dictionary and a row dictionary is more efficient than conventional block dictionaries.

3. The low-rank matrix reconstruction theory and the sparse representation theory are combined, dictionary learning is introduced into a traditional low-rank matrix reconstruction model, and combined low-rank information and separable two-dimensional sparsity prior are provided, so that images with missing rows and columns can be accurately filled.

4. By performing low-rank and sparse combined constraint on the missing damaged graph, the filling performance is improved, and not only the row and column missing can be filled, but also the random missing can be filled more accurately.

Drawings

The above advantages of the present invention will become apparent and readily appreciated from the following description of the embodiments taken in conjunction with the accompanying drawings, in which:

FIG. 1 is a flow chart of the present invention;

FIG. 2 is an original truth image without dropout;

fig. 3 is a damaged image with rows and columns and random defects, the black color indicates the missing pixels, and the total defect rates from left to right are: (1) 10% missing; (2) 20% missing; (3) 30% missing; (4) 50% missing;

FIG. 4 is a graph of the filling results of the deletion plots for four deletion rates using the method of the present invention: (1) 10% missing fill result, PSNR 40.79; (2) 20% missing filling result, PSNR 37.32; (3) 30% missing padding result, PSNR 35.69; (4) the result of 50% missing fill, PSNR is 32.23.

Detailed Description

The method for filling a row-column missing image based on low-rank matrix reconstruction and sparse representation according to the present invention is described in detail below with reference to the embodiments and the accompanying drawings.

According to the invention, low-rank matrix reconstruction and sparse representation are combined, a dictionary learning model is introduced on the basis of a traditional low-rank matrix reconstruction model, and the problem that the existing algorithm cannot realize column-row missing image filling is solved by adopting the constraint of a combined low-rank and separable two-dimensional sparse prior condition for missing images. The method comprises the following steps:

1) the low-rank characteristic of a natural image is considered, and low-rank prior is introduced based on a low-rank matrix reconstruction theory to constrain potential images; meanwhile, considering that each column of the row-missing image can be sparsely represented by a column dictionary and each row of the column-missing image can be sparsely represented by a row dictionary, separable two-dimensional sparse prior is introduced based on a sparse representation theory; therefore, based on the combined low-rank and separable two-dimensional sparse prior, the image filling problem with row and column deletion is specifically expressed as solving the following constraint optimization equation:

where tr (-) is the trace of the matrix, representing a low rank prior term;represents the dot multiplication operation of two matrixes, | ·| non-woven phosphor₁Representing a norm of the matrix, wherein two norm terms respectively represent separable two-dimensional sparse prior terms; omega is the observation space, representing a known pixel, P, within the observation matrix D with a missing row and column_Ω(. cndot.) is a projection operator, which represents the value of the variable projected into the spatial domain Ω, a is a filled matrix, Σ ═ diag ([ σ ])₁,σ₂,...,σ_n]) Representing a diagonal matrix consisting of the singular values of A in non-increasing order, W_a、W_bAnd W_cWeight matrices, γ, representing weighted low rank terms and separable two-dimensional sparse terms, respectively_B，γ_CRegularization coefficients, phi, respectively representing separable two-dimensional sparse terms_cAnd phi_rRespectively representing the trained column dictionary and row dictionary, wherein the corresponding coefficient matrixes are respectively represented by B and C, and E represents the missing pixels in the observation matrix D;

11) the invention adopts an augmented Lagrange multiplier method (ALM) to convert a constrained optimization problem (1) into an unconstrained optimization problem for solving, wherein an augmented Lagrange equation is as follows:

wherein Y is₁、Y₂And Y₃Representing the Lagrange multiplier matrix, mu₁、μ₂And mu₃Is a penalty factor that is a function of,<·,·>representing the inner product of two matrices, | · | > non-woven phosphor_FA Frobenius (Frobenius) norm representing a matrix;

12) the solving process is that a column dictionary and a row dictionary phi are trained_cAnd phi_rInitializing the weight matrix W_a、W_bAnd W_cAlternately updating coefficient matrixes B and C, restoring matrix A, missing pixel matrix E and Lagrange multiplier matrix Y₁、Y₂And Y₃Penalty factor μ₁、μ₂And mu₃And a weight matrix W_a、W_bAnd W_c；

2) Training dictionary phi_cAnd phi_r: training out-of-column and row dictionaries Φ using online learning algorithms on high quality image datasets_cAnd phi_r；

21) Constructing a column dictionary phi_cSo that the matrix A can be sparsely represented by a column dictionary, i.e. satisfying A-phi_cB, wherein B is a coefficient matrix and is sparse; constructing a line dictionary phi_rSo that the transpose of the matrix A can be sparsely represented by the row dictionary, i.e. satisfying A^Τ＝Φ_rC, where C is a coefficient matrix and is sparse. The invention trains out a column dictionary and a row dictionary phi on a Kodak image set by using an Online Learning algorithm_cAnd phi_r。

22) The relevant parameters of the training dictionary are set as follows: the number of rows and the dictionary phi of the matrix A to be reconstructed_cThe dimension m of the middle element is equal, i.e. the number of rows of A and phi_cThe number of rows is m; column number and dictionary of A_rDimension n of the middle element is equal, i.e. column number of A and phi_rThe number of rows of (c) is n. Trained dictionary phi_cAnd phi_rAre overcomplete dictionaries, i.e., the number of columns of the dictionary must be greater than the number of rows.

3) Initializing the weight matrix W_a、W_bAnd W_c；

When the weighting times is l, and l is 0, the weighting matrix is setAndthe initial values are all assigned to 1, indicating that there is no re-weighting for the first iteration.

4) The iterative solution is performed by converting equation (2) into the following sequence using the Alternating Direction Method (ADM):

in the above formulaAndrespectively representing the values of variables B, C, A and E, p, at which the objective function is to be minimized₁、ρ₂And ρ₃Is a multiple factor, k is the number of iterations; setting initial values of all parameters, and then carrying out iterative solution according to the methods of the steps 5), 6), 7) and 8) to obtain a result without re-weighting.

5) Solving for B^k+1: finding B using an accelerated neighbor gradient algorithm^k+1。

51) Removing the terms irrelevant to B in the objective function for solving B in the formula (3) to obtain the following equation:

by taylor expansion, a second order function is constructed to approximate the above formula, and then the original equation is solved for the second order function. Order toThe variable Z is reintroduced, defining the function:

wherein,is a gradient of f (Z), L_fIs a constant with a value ofIs used for protectingAll Z's are confirmed to have F (Z). ltoreq.Q (B, Z).

52) Through the above transformation, equation (4) is transformed to solve Q (B, Z)_j) By formulation, the following form is obtained:

wherein,variable Z_jThe update rule of (2) is as follows:

wherein, t_jIs a set of constant sequences and j is the number of iterations of the variable. Using the shrink operator to solve:

where soft (·, ·) is the shrink operator.

6) Solving for C^k+1: using an accelerated neighbor gradient algorithm to find C^k+1。

61) Removing the terms irrelevant to C in the objective function for solving C in the formula (3) to obtain the following equation:

using Taylor expansion, a second order function is constructed to approximate the above equation, and then the original equation is solved for the second order function. Order toReintroducing variablesThe following function is defined:

wherein,is composed ofGradient of (a), L_fIs a constant with a value ofFor ensuring to allAre all provided with

62) Through the above transformation, equation (9) is transformed to solveBy formulation, the following form is obtained:

wherein,variables ofThe update rule of (2) is as follows:

wherein, t_jIs a set of constant sequences and j is the number of iterations of the variable. Using the shrink operator to solve:

where soft (·, ·) is the shrink operator.

7) Solving for A^k+1: solving for A using Singular Value Thresholding (Singular Value Thresholding) SVT^k+1。

Removing terms independent of A in the objective function for solving A in the formula (3) to obtain:

the above formula is rewritten using the recipe as:

wherein,to Q^k+1Solving by using a singular value threshold method to obtain:

wherein H^k+1，V^k+1Are each Q^k+1The left singular matrix and the right singular matrix;

8) solving for E^k+1：E^k+1The solution of (a) consists of two parts.

81) In the observation space Ω, the value of E is 0; i.e. P_Ω(E)＝0。

82) Outside the observation space omega, i.e. complementary spaceIn respect of E^k+1Is shown as follows:

using first order derivation to solve, obtain

83) The solutions inside and outside the spatial domain Ω are combined to be the final solution of E:

9) repeating the steps 5), 6), 7) and 8) until the algorithm converges, and then iterating the result A^k+1、Σ^k+1、B^k+1、C^k+1And E^k+1Is the result A of the original problem without re-weighting^(l)、Σ^(l)、B^(l)、C^(l)And E^(l). Here, l is the number of reweighting times.

10) Updating the weight matrix W_a、W_bAnd W_c。

In order to counteract the influence of the signal amplitude on the kernel norm term and the one norm term, a re-weighting scheme is introduced according to the currently estimated singular value matrix sigma^(l)Coefficient matrix B^lAnd C^lThe amplitude of (2) is overlapped by inverse proportion principleUpdating the weight matrix W instead_a、W_bAnd W_c：

WhereinIs the position coordinates of a pixel in the image and is an arbitrarily small positive number.

11) Repeating the above steps 4), 5), 6), 7), 8), 9), 10) until the algorithm converges, at which time the result A of the iteration^(l)Is the final solution a of the original problem.

The method combines low-rank matrix reconstruction with a sparse representation theory, introduces a dictionary learning model on the basis of a traditional low-rank matrix reconstruction model, and solves the problem that the prior art cannot process by adopting the constraint of a combined low-rank and separable two-dimensional sparse prior condition for the missing image, namely, the filling of the row-column missing image is realized (an experimental flow chart is shown in figure 1). The detailed description of the embodiments in conjunction with the drawings is as follows:

1) the experiment used a 321 × 481 pixel image randomly selected from the BSDS500 dataset (as shown in fig. 2) as the original image on which 4 defect images were constructed with respective defect rates of 10%, 20%, 30% and 50% and were tested (as shown in fig. 3), including row and column defects and random defects. The invention adopts a dictionary with the atom size fixed as 100, so that the graph to be filled is firstly divided into a plurality of 100 multiplied by 100 image blocks in a window sliding mode from top to bottom and from left to right. The step size of the sliding window is 90 pixels. The 100 × 100 image blocks are sequentially filled, and finally combined to obtain a filling map with an original size of 321 × 481. When the first image block is filled, the first image block is expressed by a matrix D, and the problem of filling the current image block with row and column missing is specifically expressed by solving the following constrained optimization equation:

where tr (-) is the trace of the matrix, representing a low rank prior term;represents the dot multiplication operation of two matrixes, | ·| non-woven phosphor₁Representing a norm of the matrix, wherein two norm terms respectively represent separable two-dimensional sparse prior terms; omega is the observation space, representing a known pixel, P, within the observation matrix D with a missing row and column_Ω(. cndot.) is a projection operator, which represents the value of the variable projected into the spatial domain Ω, a is a filled matrix, Σ ═ diag ([ σ ])₁,σ₂,...,σ_n]) Representing a diagonal matrix consisting of the singular values of A in non-increasing order, W_a、W_bAnd W_cWeight matrices, γ, representing weighted low rank terms and separable two-dimensional sparse terms, respectively_B，γ_CRegularization coefficients, phi, respectively representing separable two-dimensional sparse terms_cAnd phi_rRespectively representing the trained column dictionary and row dictionary, wherein the corresponding coefficient matrixes are respectively represented by B and C, and E represents the missing pixels in the observation matrix D;

11) the invention adopts an augmented Lagrange multiplier method (ALM) to convert a constrained optimization problem (1) into an unconstrained optimization problem for solving, wherein an augmented Lagrange equation is as follows:

wherein Y is₁、Y₂And Y₃Representing the Lagrange multiplier matrix, mu₁、μ₂And mu₃Is a penalty factor that is a function of,<·,·>representing the inner product of two matrices, | · | > non-woven phosphor_FA Frobenius (Frobenius) norm representing a matrix;

12) the solving process is that a column dictionary and a row dictionary phi are trained_cAnd phi_rInitializing the weight matrix W_a、W_bAnd W_cAlternately updating coefficient matrixes B and C, restoring matrix A, missing pixel matrix E and Lagrange multiplier matrix Y₁、Y₂And Y₃Penalty factor μ₁、μ₂And mu₃And a weight matrix W_a、W_bAnd W_c；

2) Training dictionary phi_cAnd phi_r: training out-of-column and row dictionaries Φ using online learning algorithms on high quality image datasets_cAnd phi_r；

21) Constructing a column dictionary phi_cSo that the matrix A can be sparsely represented by a column dictionary, i.e. satisfying A-phi_cB, wherein B is a coefficient matrix and is sparse; constructing a line dictionary phi_rSo that the transpose of the matrix A can be sparsely represented by the row dictionary, i.e. satisfying A^Τ＝Φ_rThe present invention trains out a column dictionary and a row dictionary Φ using the Online Learning algorithm to randomly select 230000 pixel columns of size 100 × 1 as training data over all images in the Kodak image set_cAnd phi_r。

22) The relevant parameters of the training dictionary are set as follows: number of rows and dictionary phi of reconstruction matrix A_cThe dimension m of the middle element is equal, i.e. the number of rows of A and phi_cThe number of rows in (1) is m, and m is 100 in the experiment. Column number and dictionary of A_rDimension n of the middle element is equal, i.e. column number of A and phi_rThe number of rows is n, and n is 100 in the experiment. Trained dictionary phi_cAnd phi_rAre overcomplete dictionaries, i.e., the number of columns of the dictionary must be greater than the number of rows. In the experiment, the number of columns of the row dictionary and the column dictionary is 400, and then the dictionary phi_cAnd phi_rAll specification of (2) is 100 × 400.

3) Initializing the weight matrix W_a、W_bAnd W_c；

When the weighting times is l, and l is 0, the weighting matrix is setAndthe initial values are all assigned to 1, indicating that there is no re-weighting for the first iteration.

4) The iterative solution is performed by converting equation (2) into the following sequence using the Alternating Direction Method (ADM):

in the above formulaAndrespectively representing the values of variables B, C, A and E, p, at which the objective function is to be minimized₁、ρ₂And ρ₃Is a multiple factor, k is the number of iterations; setting initial values of all parameters, and then carrying out iterative solution according to the methods of the steps 5), 6), 7) and 8) to obtain a result without re-weighting. Initial values were set in the experiment as: l is 0; k is 1; rho₁＝ρ₂＝ρ₃＝1.1；A¹＝B¹＝C¹＝E¹＝0。

5) Solving for B^k+1: finding B using an accelerated neighbor gradient algorithm^k+1。

51) Removing the terms irrelevant to B in the objective function for solving B in the formula (3) to obtain the following equation:

by taylor expansion, a second order function is constructed to approximate the above formula, and then the original equation is solved for the second order function. Order toThe variable Z is reintroduced, defining the function:

wherein,is a gradient of f (Z), L_fIs a constant with a value ofTo ensure that there is F (Z). ltoreq.Q (B, Z) for all Z.

52) Through the above transformation, equation (4) is transformed to solve Q (B, Z)_j) By formulation, the following form is obtained:

wherein,variable Z_jThe update rule of (2) is as follows:

wherein, t_jIs a set of constant sequences and j is the number of iterations of the variable. After the conversion, the initial values of the parameters are set as follows: j is 1; t is t₁＝1；Z₁0. The convergence can be solved as follows:

where soft (·, ·) is the shrink operator.

6) Solving for C^k+1: using an accelerated neighbor gradient algorithm to find C^k+1。

61) Removing the terms irrelevant to C in the objective function for solving C in the formula (3) to obtain the following equation:

using Taylor expansion, a second order function is constructed to approximate the above equation, and then the original equation is solved for the second order function. Order toReintroducing variablesThe following function is defined:

wherein,is composed ofGradient of (a), L_fIs a constant with a value ofFor ensuring to allAre all provided with

62) Through the above transformation, equation (9) is transformed to solveBy formulation, the following form is obtained:

wherein,variables ofThe update rule of (2) is as follows:

wherein, t_jIs a set of constant sequences and j is the number of iterations of the variable. After the conversion, the initial values of the parameters are set as follows: j is 1;

t₁＝1；the convergence can be solved as follows:

where soft (·, ·) is the shrink operator.

7) Solving for A^k+1: SVT Using Singular Value Thresholding (Singular Value Thresholding)Solving for A^k+1。

Removing terms independent of A in the objective function for solving A in the formula (3) to obtain:

the above formula is rewritten using the recipe as:

wherein,to Q^k+1Solving by using a singular value threshold method to obtain:

wherein H^k+1，V^k+1Are each Q^k+1The left singular matrix and the right singular matrix;

8) solving for E^k+1：E^k+1The solution of (a) consists of two parts.

81) In the observation space Ω, the value of E is 0; i.e. P_Ω(E)＝0。

82) Outside the observation space omega, i.e. complementary spaceIn respect of E^k+1Is shown as follows:

solving using first order derivativesTo obtain

83) The solutions inside and outside the spatial domain Ω are combined to be the final solution of E:

9) repeating the steps 5), 6), 7) and 8) until the algorithm converges, and then iterating the result A^k+1、Σ^k+1、B^k+1、C^k+1And E^k+1Is the result A of the original problem without re-weighting^(l)、Σ^(l)、B^(l)、C^(l)And E^(l). Here, l is the number of reweighting times.

10) Updating the weight matrix W_a、W_bAnd W_c。

In order to counteract the influence of the signal amplitude on the kernel norm term and the one norm term, a re-weighting scheme is introduced according to the currently estimated singular value matrix sigma^(l)Coefficient matrix B^lAnd C^lIteratively updating the weight matrix W using an inverse scaling rule_a、W_bAnd W_c：

WhereinIs the position coordinates of a pixel in the image and is an arbitrarily small positive number. In the experiment, the value was 0.001.

11) Repeating the above steps 4), 5), 6), 7), 8), 9), 10) until the algorithm converges, at which time the result A of the iteration^(l)Is the final solution a of the original problem.

12) Sequentially processing the rest image blocks obtained in the step 1) until the image blocks are completely filled, and then combining the image blocks into a final filling graph (as shown in fig. 4). And during combination, taking the average value of the pixel points which are filled for multiple times and are divided and overlapped as a final value.

The experimental results are as follows: the invention adopts PSNR (peak signal-to-noise ratio) as the measurement measure of the image filling result, and the unit is dB:

wherein I is the filled image, I₀For a real image without missing, w is the width of the image, h is the height of the image, (x, y) represents the pixel value of the x-th row and y-th column of the image, Σ represents the summation operation, | · | is an absolute value. The experiment takes n as 8, and the filling results of 4 pictures with different degree of row-column deletion used for testing in the experiment are marked in figure 4.

Claims (5)

1. A row-column missing image filling method based on low-rank matrix reconstruction and sparse representation is characterized by comprising the steps of introducing low-rank prior to constrain potential images based on a low-rank matrix reconstruction theory; meanwhile, considering that each column of the row-missing image can be sparsely represented by a column dictionary and each row of the column-missing image can be sparsely represented by a row dictionary, separable two-dimensional sparse prior is introduced based on a sparse representation theory; therefore, based on the combined low-rank and separable two-dimensional sparse prior, the image filling problem with row and column missing is specifically expressed as a solution constraint optimization equation, and row and column missing image filling is achieved.

2. The method for column-row missing image filling based on low rank matrix reconstruction and sparse representation as claimed in claim 1, wherein the step of specifically describing the image filling problem with column-row missing as solving the constrained optimization equation is detailed as:

1) the image filling problem with row-column deficiency is specifically formulated to solve the following constrained optimization equation:

where tr (-) is the trace of the matrix, representing a low rank prior term;represents the dot multiplication operation of two matrixes, | ·| non-woven phosphor₁Representing a norm of the matrix, wherein two norm terms respectively represent separable two-dimensional sparse prior terms; omega is the observation space, representing a known pixel, P, within the observation matrix D with a missing row and column_Ω(. cndot.) is a projection operator, which represents the value of the variable projected into the spatial domain Ω, a is a filled matrix, Σ ═ diag ([ σ ])₁,σ₂,...,σ_n]) Representing a diagonal matrix consisting of the singular values of A in non-increasing order, W_a、W_bAnd W_cWeight matrices, γ, representing weighted low rank terms and separable two-dimensional sparse terms, respectively_B，γ_CRegularization coefficients, phi, respectively representing separable two-dimensional sparse terms_cAnd phi_rRespectively representing the trained column dictionary and row dictionary, wherein the corresponding coefficient matrixes are respectively represented by B and C, and E represents the missing pixels in the observation matrix D;

and converting the constrained optimization problem (1) into an unconstrained optimization problem by adopting an augmented Lagrange multiplier method (ALM) to solve, wherein an augmented Lagrange equation is as follows:

wherein Y is₁、Y₂And Y₃Representing the Lagrange multiplier matrix, mu₁、μ₂And mu₃Is a penalty factor that is a function of,<·,·>representing the inner product of two matrices, | · | > non-woven phosphor_FA Frobenius (Frobenius) norm representing a matrix;

the solving process is that a column dictionary and a row dictionary phi are trained_cAnd phi_rInitializing the weight matrix W_a、W_bAnd W_cAlternately updating coefficient matrixes B and C, restoring matrix A, missing pixel matrix E and Lagrange multiplier matrix Y₁、Y₂And Y₃Penalty factor μ₁、μ₂And mu₃And a weight matrix W_a、W_bAnd W_cUntil the algorithm converges, at which point the result of the iteration A^(l)Is the final solution a of the original problem.

3. The method of row-column missing image filling based on low rank matrix reconstruction and sparse representation as claimed in claim 2, characterized in that in particular the dictionary Φ is trained_cAnd phi_r: training out-of-column and row dictionaries Φ using online learning algorithms on high quality image datasets_cAnd phi_r。

4. The method of row-column missing image filling based on low rank matrix reconstruction and sparse representation as claimed in claim 2, characterized in that specifically the weight matrix W is initialized_a、W_bAnd W_c: when the weighting times is l, and l is 0, the weighting matrix is set、Andthe initial values are all assigned to 1, indicating that there is no re-weighting for the first iteration.

5. The method of row-column missing image filling based on low rank matrix reconstruction and sparse representation as claimed in claim 2, characterized in that in particular the iterative solution is performed using the alternating direction method ADM to convert equation (2) into the following sequence:

<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msup> <mi>B</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>arg</mi> <munder> <mi>min</mi> <mi>B</mi> </munder> <mo>{</mo> <msub> <mi>L</mi> <mrow> <msubsup> <mi>&amp;mu;</mi> <mn>1</mn> <mi>k</mi> </msubsup> <mo>,</mo> <msubsup> <mi>&amp;mu;</mi> <mn>2</mn> <mi>k</mi> </msubsup> <mo>,</mo> <msubsup> <mi>&amp;mu;</mi> <mn>3</mn> <mi>k</mi> </msubsup> </mrow> </msub> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mi>k</mi> </msup> <mo>,</mo> <mi>B</mi> <mo>,</mo> <msup> <mi>C</mi> <mi>k</mi> </msup> <mo>,</mo> <msup> <mi>E</mi> <mi>k</mi> </msup> <mo>,</mo> <msubsup> <mi>Y</mi> <mn>1</mn> <mi>k</mi> </msubsup> <mo>,</mo> <msubsup> <mi>Y</mi> <mn>2</mn> <mi>k</mi> </msubsup> <mo>,</mo> <msubsup> <mi>Y</mi> <mn>3</mn> <mi>k</mi> </msubsup> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mi>C</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>arg</mi> <munder> <mi>min</mi> <mi>C</mi> </munder> <mo>{</mo> <msub> <mi>L</mi> <mrow> <msubsup> <mi>&amp;mu;</mi> <mn>1</mn> <mi>k</mi> </msubsup> <mo>,</mo> <msubsup> <mi>&amp;mu;</mi> <mn>2</mn> <mi>k</mi> </msubsup> <mo>,</mo> <msubsup> <mi>&amp;mu;</mi> <mn>3</mn> <mi>k</mi> </msubsup> </mrow> </msub> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mi>k</mi> </msup> <mo>,</mo> <msup> <mi>B</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mi>C</mi> <mo>,</mo> <msup> <mi>E</mi> <mi>k</mi> </msup> <mo>,</mo> <msubsup> <mi>Y</mi> <mn>1</mn> <mi>k</mi> </msubsup> <mo>,</mo> <msubsup> <mi>Y</mi> <mn>2</mn> <mi>k</mi> </msubsup> <mo>,</mo> <msubsup> <mi>Y</mi> <mn>3</mn> <mi>k</mi> </msubsup> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mi>A</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>arg</mi> <munder> <mi>min</mi> <mi>A</mi> </munder> <mo>{</mo> <msub> <mi>L</mi> <mrow> <msubsup> <mi>&amp;mu;</mi> <mn>1</mn> <mi>k</mi> </msubsup> <mo>,</mo> <msubsup> <mi>&amp;mu;</mi> <mn>2</mn> <mi>k</mi> </msubsup> <mo>,</mo> <msubsup> <mi>&amp;mu;</mi> <mn>3</mn> <mi>k</mi> </msubsup> </mrow> </msub> <mrow> <mo>(</mo> <mi>A</mi> <mo>,</mo> <msup> <mi>B</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>C</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>E</mi> <mi>k</mi> </msup> <mo>,</mo> <msubsup> <mi>Y</mi> <mn>1</mn> <mi>k</mi> </msubsup> <mo>,</mo> <msubsup> <mi>Y</mi> <mn>2</mn> <mi>k</mi> </msubsup> <mo>,</mo> <msubsup> <mi>Y</mi> <mn>3</mn> <mi>k</mi> </msubsup> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mi>E</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>arg</mi> <munder> <mi>min</mi> <mrow> <msub> <mi>P</mi> <mi>&amp;Omega;</mi> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </munder> <mo>{</mo> <msub> <mi>L</mi> <mrow> <msubsup> <mi>&amp;mu;</mi> <mn>1</mn> <mi>k</mi> </msubsup> <mo>,</mo> <msubsup> <mi>&amp;mu;</mi> <mn>2</mn> <mi>k</mi> </msubsup> <mo>,</mo> <msubsup> <mi>&amp;mu;</mi> <mn>3</mn> <mi>k</mi> </msubsup> </mrow> </msub> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>B</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>C</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mi>E</mi> <mo>,</mo> <msubsup> <mi>Y</mi> <mn>1</mn> <mi>k</mi> </msubsup> <mo>,</mo> <msubsup> <mi>Y</mi> <mn>2</mn> <mi>k</mi> </msubsup> <mo>,</mo> <msubsup> <mi>Y</mi> <mn>3</mn> <mi>k</mi> </msubsup> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>Y</mi> <mn>1</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>Y</mi> <mn>1</mn> <mi>k</mi> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;mu;</mi> <mn>1</mn> <mi>k</mi> </msubsup> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>-</mo> <msub> <mi>&amp;Phi;</mi> <mi>r</mi> </msub> <msup> <mi>B</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>Y</mi> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>Y</mi> <mn>2</mn> <mi>k</mi> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;mu;</mi> <mn>2</mn> <mi>k</mi> </msubsup> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow> <mi>k</mi> <mo>+</mo> <msup> <mn>1</mn> <mi>T</mi> </msup> </mrow> </msup> <mo>-</mo> <msub> <mi>&amp;Phi;</mi> <mi>r</mi> </msub> <msup> <mi>C</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>Y</mi> <mn>3</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>Y</mi> <mn>3</mn> <mi>k</mi> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;mu;</mi> <mn>3</mn> <mi>k</mi> </msubsup> <mrow> <mo>(</mo> <mi>D</mi> <mo>-</mo> <msup> <mi>A</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>-</mo> <msup> <mi>E</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>&amp;mu;</mi> <mn>1</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <msub> <mi>&amp;rho;</mi> <mn>1</mn> </msub> <msubsup> <mi>&amp;mu;</mi> <mn>1</mn> <mi>k</mi> </msubsup> <mo>,</mo> <msubsup> <mi>&amp;mu;</mi> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <msub> <mi>&amp;rho;</mi> <mn>2</mn> </msub> <msubsup> <mi>&amp;mu;</mi> <mn>2</mn> <mi>k</mi> </msubsup> <mo>,</mo> <msubsup> <mi>&amp;mu;</mi> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <msub> <mi>&amp;rho;</mi> <mn>3</mn> </msub> <msubsup> <mi>&amp;mu;</mi> <mn>3</mn> <mi>k</mi> </msubsup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>

in the above formulaAndrespectively representing the values of variables B, C, A and E, p, at which the objective function is to be minimized₁、ρ₂And ρ₃Is a multiple factor, k is the number of iterations; then, the iterative solution is carried out according to the following steps:

1) solving for B^k+1: finding B using an accelerated neighbor gradient algorithm^k+1；

Removing the terms irrelevant to B in the objective function for solving B in the formula (3) to obtain the following equation:

using Taylor expansion method to construct a second order function to approximate the above formula, then solving the original equation for the second order function to makeAnd introducing a variable Z, and finally solving the following steps:

<mrow> <msup> <mi>B</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mi>B</mi> <mrow> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>k</mi> </msubsup> <mo>=</mo> <mi>s</mi> <mi>o</mi> <mi>f</mi> <mi>t</mi> <mrow> <mo>(</mo> <msub> <mi>U</mi> <mrow> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mfrac> <msub> <mi>&amp;gamma;</mi> <mi>B</mi> </msub> <msub> <mi>L</mi> <mi>f</mi> </msub> </mfrac> <msub> <mi>W</mi> <mi>b</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>

wherein soft (·,) is a shrink operator, is a gradient of f (Z), L_fIs a constant with a value ofVariable Z_jThe update rule of (2) is as follows:

<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>t</mi> <mrow> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <msqrt> <mrow> <mn>4</mn> <msubsup> <mi>t</mi> <mi>j</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mn>1</mn> </mrow> </msqrt> <mo>)</mo> </mrow> <mo>/</mo> <mn>2</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>Z</mi> <mrow> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msubsup> <mi>B</mi> <mrow> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>k</mi> </msubsup> <mo>+</mo> <mfrac> <mrow> <msub> <mi>t</mi> <mi>j</mi> </msub> <mo>-</mo> <mn>1</mn> </mrow> <msub> <mi>t</mi> <mrow> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mfrac> <mrow> <mo>(</mo> <msubsup> <mi>B</mi> <mrow> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>k</mi> </msubsup> <mo>-</mo> <msubsup> <mi>B</mi> <mi>j</mi> <mi>k</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>

wherein, t_jIs a set of constant sequences, j is the number of iterations of the variable;

2) solving for C^k+1: using an accelerated neighbor gradient algorithm to find C^k+1；

Removing the terms irrelevant to C in the objective function for solving C in the formula (3) to obtain the following equation:

using Taylor expansion method to construct a second order function to approximate the above formula, then solving the original equation for the second order function to makeReintroducing variablesFinally, the following can be obtained:

<mrow> <msup> <mi>C</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mi>C</mi> <mrow> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>k</mi> </msubsup> <mo>=</mo> <mi>s</mi> <mi>o</mi> <mi>f</mi> <mi>t</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>U</mi> <mo>~</mo> </mover> <mrow> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mfrac> <msub> <mi>&amp;gamma;</mi> <mi>C</mi> </msub> <msub> <mi>L</mi> <mi>f</mi> </msub> </mfrac> <msub> <mi>W</mi> <mi>c</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>

wherein soft (·,) is a shrink operator, is composed ofGradient of (a), L_fIs a constant with a value ofVariables ofThe update rule of (2) is as follows:

<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>t</mi> <mrow> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <msqrt> <mrow> <mn>4</mn> <msubsup> <mi>t</mi> <mi>j</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mn>1</mn> </mrow> </msqrt> <mo>)</mo> </mrow> <mo>/</mo> <mn>2</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>Z</mi> <mo>~</mo> </mover> <mrow> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msubsup> <mi>C</mi> <mrow> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>k</mi> </msubsup> <mo>+</mo> <mfrac> <mrow> <msub> <mi>t</mi> <mi>j</mi> </msub> <mo>-</mo> <mn>1</mn> </mrow> <msub> <mi>t</mi> <mrow> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mfrac> <mrow> <mo>(</mo> <msubsup> <mi>C</mi> <mrow> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>k</mi> </msubsup> <mo>-</mo> <msubsup> <mi>C</mi> <mi>j</mi> <mi>k</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>

wherein, t_jIs a group ofConstant sequence, j is the number of iterations of the variable;

3) solving for A^k+1: solving for A using Singular Value Thresholding (Singular Value Thresholding) SVT^k+1；

Removing terms independent of A in the objective function for solving A in the formula (3), and obtaining the terms through a formula:

wherein,to Q^{k +1}Solving by using a singular value threshold method to obtain:

<mrow> <msup> <mi>A</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>H</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mi>s</mi> <mi>o</mi> <mi>f</mi> <mi>t</mi> <mrow> <mo>(</mo> <msup> <mi>&amp;Sigma;</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mfrac> <mn>1</mn> <mrow> <msubsup> <mi>&amp;mu;</mi> <mn>1</mn> <mi>k</mi> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;mu;</mi> <mn>2</mn> <mi>k</mi> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;mu;</mi> <mn>3</mn> <mi>k</mi> </msubsup> </mrow> </mfrac> <msub> <mi>W</mi> <mi>a</mi> </msub> <mo>)</mo> </mrow> <msup> <mi>V</mi> <mrow> <mi>k</mi> <mo>+</mo> <msup> <mn>1</mn> <mi>T</mi> </msup> </mrow> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>

wherein H^k+1，V^k+1Are each Q^k+1The left singular matrix and the right singular matrix;

4) solving for E^k+1：E^k+1The solution of (a) consists of two parts;

in the observation space Ω, the value of E is 0; outside the observation space omega, i.e. complementary spaceAnd solving by using a first-order derivative, and combining the two parts to obtain a final solution of the E:

<mrow> <msup> <mi>E</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>P</mi> <mi>&amp;Omega;</mi> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>P</mi> <mover> <mi>&amp;Omega;</mi> <mo>&amp;OverBar;</mo> </mover> </msub> <mrow> <mo>(</mo> <mi>D</mi> <mo>-</mo> <msup> <mi>A</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mfrac> <msubsup> <mi>Y</mi> <mn>3</mn> <mi>k</mi> </msubsup> <msubsup> <mi>&amp;mu;</mi> <mn>3</mn> <mi>k</mi> </msubsup> </mfrac> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>

5) repeating the steps 1), 2), 3) and 4) until the algorithm is converged, wherein the iterative result A is obtained^k+1、Σ^k+1、B^k+1、C^k+1And E^k+1Is the result A of the original problem without re-weighting^(l)、Σ^(l)、B^(l)、C^(l)And E^(l)Where, l is the number of reweighting times;

6) updating a weight matrixW_a、W_bAnd W_c；

In order to counteract the influence of the signal amplitude on the kernel norm term and the one norm term, a re-weighting scheme is introduced according to the currently estimated singular value matrix sigma^(l)Coefficient matrix B^lAnd C^lIteratively updating the weight matrix W using an inverse scaling rule_a、W_bAnd W_c：

<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>W</mi> <mi>a</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mover> <mi>i</mi> <mo>^</mo> </mover> <mo>,</mo> <mover> <mi>i</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <msubsup> <mi>&amp;sigma;</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>)</mo> </mrow> </msubsup> <mo>+</mo> <mi>&amp;epsiv;</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>W</mi> <mi>b</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mover> <mi>i</mi> <mo>^</mo> </mover> <mo>,</mo> <mover> <mi>j</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mo>|</mo> <msup> <mi>B</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <mover> <mi>i</mi> <mo>^</mo> </mover> <mo>,</mo> <mover> <mi>j</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> <mo>|</mo> <mo>+</mo> <mi>&amp;epsiv;</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>W</mi> <mi>c</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mover> <mi>i</mi> <mo>^</mo> </mover> <mo>,</mo> <mover> <mi>j</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mo>|</mo> <msup> <mi>C</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <mover> <mi>i</mi> <mo>^</mo> </mover> <mo>,</mo> <mover> <mi>j</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> <mo>|</mo> <mo>+</mo> <mi>&amp;epsiv;</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>

WhereinIs the position coordinates of a pixel in the image and is an arbitrarily small positive number.

7) Repeating the steps 1) -7) until the algorithm is converged, and then obtaining an iterative result A^(l)Is the final solution a of the original problem.