CN105303530A - Fabric image mole stripe elimination method based on low-rank sparse matrix decomposition - Google Patents

Abstract

The invention belongs to the digital image processing field and is for eliminating mole stripe patterns in fabric images. The fabric image mole stripe elimination method based on low-rank sparse matrix decomposition is characterized in that, local self-similarity of fabric textures and energy concentration distribution characteristics of the mole stripe in a frequency domain are utilized, a low rank sparse matrix decomposition model is constructed, the fabric textures are separated from the mole stripe pattern, and thereby a problem of mole stripes in the fabric image can be solved. The fabric image mole stripe elimination method based on low-rank sparse matrix decomposition is mainly applied to digital image processing.

Description

Fabric image moire elimination method based on low-rank sparse matrix decomposition

Technical Field

The invention belongs to the field of digital image processing, and particularly relates to a method for eliminating moire of a fabric image based on low-rank sparse matrix decomposition.

Background

Moire phenomena are highly prevalent in modern digital imaging processes, such as: a scanner scans a halftone print, a digital camera takes a picture of a screen or a scene containing a regular pattern (a cloth line on clothing, a window or grid on a building). Moire patterns tend to have large areas and obvious color cast in images, seriously affect image quality and image analysis results, and attract wide attention of people.

So far, image moir é removal algorithms are mainly directed to halftone scanned images. Among the types of moire in the image, the moire in the halftone image is a relatively simple moire pattern, and its shape, distribution, etc. are closely related to the printer apparatus without being affected by the original image. These algorithms are mainly classified into two broad categories, filtering-based methods and dictionary learning-based methods. Another relatively simple image moire is the moire phenomenon caused by interference of rays during the imaging process of a radiation microscope, and nearly parallel moire fringes are distributed throughout the whole image. For such moire, the most common current method of elimination is to select a suitable median gaussian filter, but inevitably produces a slight ringing effect at the local edge locations.

However, since the moire pattern in the fabric image is relatively complex: 1) the stripe distribution has no obvious rule, and the color cast is obvious; 2) moire patterns are similar to fabric textures, thus making it more difficult to eliminate such Moire patterns. To eliminate such moir é, most manufacturers choose to embed an optical low-pass filter in the digital camera to suppress the occurrence of moir é at the expense of image sharpness. The other solution is to design a special CFA (color Filter array) interpolation algorithm, and estimate aliasing information of a red channel and a blue channel and lost high-frequency information by using high-frequency information captured by a green channel to avoid the occurrence of a moire phenomenon; the method is severely limited by the shot image, and the complex interpolation algorithm increases the internal processing load of the camera, so that the method is not a reliable method for eliminating moire fringes. At present, the most effective post-processing method for eliminating moire fringes generated in the imaging process of a digital camera is to use Photoshop software, manually select a moire region, remove the color of the moire fringes in the region, and then repair a brightness layer by using chrominance and saturation information to eliminate gray moire fringes. In a word, no efficient processing algorithm for the moire phenomenon in the fabric image is provided at present, and the moire interference can be eliminated while the image sharpness is kept.

Disclosure of Invention

The invention aims to make up the defects of the prior art, namely, the elimination of the moire pattern in the fabric image is realized. The method adopts the technical scheme that a low-rank sparse matrix decomposition-based fabric image moire elimination method utilizes local self-similarity of fabric textures and energy concentration distribution characteristics of moire patterns in a frequency domain to construct a low-rank sparse matrix decomposition model and separate the fabric textures from moire patterns, so that the problem of eliminating moire in a fabric image is solved. The method comprises the following steps:

1) overlapping images I with u/2 as step interval_MDivided into image blocks of size u × u, then each image block y ' is processed to achieve separation of the fabric texture x ' from the moir pattern m ', i.e.:

y′＝x′+m′(1)

2) constructing a sparse constraint term: by utilizing the energy concentration distribution characteristic of the moire pattern in the frequency domain, the moire pattern is assumed to be sparsely represented by a Discrete Cosine Transform (DCT) dictionary, and the formula is as follows:

min||α||₁constraint m ═ D α Ω (2)

Wherein min represents solving the corresponding minimum value, | | · |. non-calculation₁Representing a matrixThe norm of the number of the first-order-of-arrival,is the constructed DCT dictionary and is,representing a real space, N, p representing the size of the spatial dimension,representing a two-dimensional real number space of size N × p and N ═ u²,p>>N, the symbol ∈ denotes belonging, m is the vector form of the moire image block m' to be separated, α is the coefficient matrix of the sparse representation of moire m under the dictionary, Ω is the constraint on the significant positions of the coefficients α, i.e. the values of the coefficients α at the non-significant positions are all set to zero;

3) constructing a low rank constraint term: the fabric texture image block x' is converted into k × k subblocks and converted into corresponding vector forms, the subblock vectors are arranged into a matrix C, and local self-similarity characteristics of the fabric texture in a space domain are utilized to add low-rank constraints as follows:

min||C||_*(3)

wherein | · | purple sweet_*Denotes the kernel norm of the matrix, C ═ C₁,c₂,…,c_n]WhereinIs the vector form of the ith sub-block, n is the number of sub-blocks, k²The size of the dimensions of the space is represented,denotes a size of k²One-dimensional real number space, i.e. vector; the construction equation for matrix C is as follows:

$C=S(I_n\⊗x)---\left(4\right)$

wherein, the symbolIs the kronecker operator, I_nIs onen × n diagonal matrix, x fabric texture image block x' vector form, S ═ S₁,s₂,…,s_n]Is a matrix for extracting sub-blocks of an image block, satisfies c_i＝s_ix,i＝1,2,…,n；

4) Combining the sparse constraint term of the moire pattern and the low-rank constraint term of the fabric texture similar block matrix by using a factor lambda to construct a final low-rank sparse matrix decomposition equation, wherein the specific formula is as follows:

min‖C‖_*+λ‖α‖₁

constraint conditions $\left\{\begin{array}{c}y=x+m\\ m=D\mathrm{\α},\mathrm{\α}\∈\mathrm{\Ω}\\ C=S(I_n\⊗x)\end{array}\right.---\left(5\right)$

Where λ is a weighting factor, balancing the weight between the two terms. In order to reduce the error of the convergence result caused by a plurality of constraints, the invention combines the constraints in the formula (5) to obtain the following decomposition model:

min‖C‖_*+λ‖α‖₁

constraint conditions $C=S(I_n\⊗(y-D\mathrm{\α})),\mathrm{\α}\∈\mathrm{\Ω}---\left(6\right)$

Recent research results on low-rank matrix recovery and sparse representation show that adding weights to the minimization model can improve the low rank and sparsity of the obtained result, so that a re-weighting strategy is introduced into the formula (6) to obtain the following equation:

mintr(W_CοΣ)+λ||W_αοα||₁

constraint conditions $C=S(I_n\⊗(y-D\mathrm{\α})),\mathrm{\α}\∈\mathrm{\Ω}---\left(7\right)$

Wherein tr (-) denotes the trace of the matrix，Is a diagonal matrix formed by the singular values of the matrix C, diag (·) represents the diagonal matrix,is a singular value of a matrix, W_CAnd W_αThe weight matrixes corresponding to the low-rank term and the sparse term are respectively, and the symbol o represents the dot product operation of the two matrixes;

5) and finally solving the equation by using an augmented Lagrange method, and after a plurality of iterations, achieving convergence to finally obtain the needed low-rank matrix C and the coefficient vector alpha of the Moire pattern m sparsely represented under the DCT dictionary, thereby obtaining the fabric texture x and the Moire pattern m:

$x={\left({\Σ}_{i=1}^n{s}_i^T{s}_i\right)}^{-1}{\Σ}_{i=1}^n{s}_i^T{c}_i---\left(19\right)$

m＝Dα(20)

wherein,is a summation formula (·)^-1Represents the inverse of the matrix within the brackets;

6) and (3) performing the above processing on all image blocks in the image area of the moire pattern to finally obtain the fabric texture image I without the moire pattern and the separated moire component image M.

Further specifically, the step of constructing the sparse constraint term includes 21) in some cases, the red channel and the blue channel of the color image have moire patterns, and the green channel does not include any moire component, so that a plurality of moire image blocks of the red channel and the blue channel and a plurality of moire-free image blocks at corresponding positions of the green channel are selected, statistical distribution maps of DCT coefficients corresponding to the moire image blocks and the moire-free image blocks are respectively obtained, and a conclusion is obtained: the Moire pattern is concentrated in energy distribution at middle and low frequency positions in a frequency domain, so that all coefficients of alpha corresponding to the high frequency position are set to zero, namely omega is the middle and low frequency position of a DCT domain;

22) to prevent erroneous classification of the position of wrinkles, edges, shadows, etc. in the fabric image as moire components, usingThe norm gradient minimization method smoothes the image block y' to obtain an image block containing wrinkles, edges, shadows, etc. but not containing moire and fabric textures, and further constrains the effective position of the coefficient α, i.e., Ω, according to its energy distribution in the DCT domain.

5) And finally solving the equation by using an augmented Lagrange method, wherein the method comprises the following specific steps:

51) convert equation (7) to lagrangian functional form:

l (C, α, Q) is the augmented Lagrangian function, where μ is the penalty factor, Q is the Lagrangian multiplier matrix,<·，·>representing the inner product of two matrices, | · | > non-woven phosphor_FA Flobenius norm representing a matrix;

52) the iterative solution equation of equation (8) is as follows:

$\left\{\begin{array}{c}{C}^{k+1,\mathrm{\&tau;}}={\mathrm{argmin}}_C\left\{L\left(C,{\mathrm{\&alpha;}}^{k,\mathrm{\&tau;}},Q^{k,\mathrm{\&tau;}}\right)\right\}\\ {\mathrm{\&alpha;}}^{k+1,\mathrm{\&tau;}}={\mathrm{argmin}}_{\mathrm{\&alpha;}}\left\{L\left(C^{k+1,\mathrm{\&tau;}},\mathrm{\&alpha;},Q^{k,\mathrm{\&tau;}}\right)\right\}\\ Q^{k+1,\mathrm{\&tau;}}=Q^{k,\mathrm{\&tau;}}+{\mathrm{\&mu;}}^{k,\mathrm{\&tau;}}\left(C^{k+1,\mathrm{\&tau;}}-S\left(I_n\&CircleTimes;\left(y-{\mathrm{D\&alpha;}}^{k+1,\mathrm{\&tau;}}\right)\right)\right)\\ {\mathrm{\&mu;}}^{k+1,\mathrm{\&tau;}}={\mathrm{\&rho;\&mu;}}^{k,\mathrm{\&tau;}}\\ W_{C,g}^{\mathrm{\&tau;}+1}=\frac{1}{{\mathrm{\&sigma;}}_g^{\mathrm{\&tau;}}+\mathrm{\&xi;}},W_{\mathrm{\&alpha;},g}^{\mathrm{\&tau;}+1}=\frac{1}{\left|{\mathrm{\&alpha;}}_g^{\mathrm{\&tau;}}\right|+\mathrm{\&xi;}}\end{array}\right.---\left(9\right)$

argmin in the above formula_X{. is the value of variable X at the time of minimizing the objective function, ρ is a multiple factor, superscripts k, k +1 and τ, τ +1 is the number of iterations, ξ is an arbitrarily small positive real number, and subscript g represents the position coordinates of the matrix elements;

53) solving for C^k+1，τSolving for C in the equation (9)^k+1,τThe C-independent term in the objective function of (1) yields:

the above formula is rewritten using the recipe as:

wherein:

$P^{k,\mathrm{\&tau;}}=S(I_n\&CircleTimes;(y-{\mathrm{D\&alpha;}}^{k,\mathrm{\&tau;}}\left)\right)-\frac{Q^{k,\mathrm{\&tau;}}}{{\mathrm{\&mu;}}^{k,\mathrm{\&tau;}}}$

using singular value threshold decomposition for equation (10) yields:

$C^{k+1,\mathrm{\&tau;}}=U^{k,\mathrm{\&tau;}}soft({\mathrm{\&Sigma;}}^{k,\mathrm{\&tau;}},\frac{W_C^{\mathrm{\&tau;}}}{{\mathrm{\&mu;}}^{k,\mathrm{\&tau;}}}){\left(V^{k,\mathrm{\&tau;}}\right)}^T---\left(11\right)$

wherein U is^k,τ，V^k,τAre respectively P^k,τLeft and right singular matrices of, soft (,) is a contraction calculationA seed;

54) solving α using an accelerated neighbor gradient algorithm^k+1,τThe solution α in equation (9) is removed^k+1，τThe α -independent term in the objective function of (a) yields the following equation:

using Taylor expansion method, construct a second order function to approximate the above equation, and then solve the original equation for the second order function:

order to $f\left(\mathrm{\&alpha;}\right)=<Q^{k,\mathrm{\&tau;}},C^{k+1,\mathrm{\&tau;}}-S(I_n\&CircleTimes;(y-{\mathrm{D\&alpha;}}^{\mathrm{\&tau;}}))>+\frac{{\mathrm{\&mu;}}^{k,\mathrm{\&tau;}}}{2}{\left|\right|C^{k+1,\mathrm{\&tau;}}-S(I_n\&CircleTimes;(y-{\mathrm{D\&alpha;}}^{\mathrm{\&tau;}}))\left|\right|}_F^2,$ Reintroducing variable β defines the function:

wherein,is a gradient of f (β), L_fIs a constant value ofUsed for ensuring that F (β) is less than or equal to T (α) for all β;

after the above conversion, the formula (12) is converted into the solution T (α)_j) By formulation, the following form is obtained:

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

$\left\{\begin{array}{c}{\mathrm{\&beta;}}_{j+1}={\mathrm{\&alpha;}}_{j+1}^{k,\mathrm{\&tau;}}+\frac{t_{j+1}-1}{t_{j+1}}\left({\mathrm{\&alpha;}}_{j+1}^{k,\mathrm{\&tau;}}-{\mathrm{\&alpha;}}_j^{k,\mathrm{\&tau;}}\right)\\ t_{j+1}=\left(1+\sqrt{4{t_j}^2+1}\right)/2\end{array}\right.---\left(15\right)$

t_jis a set of constant sequences, j and j +1 are the number of iterations, from which it is solved:

${\mathrm{\&alpha;}}^{k+1,\mathrm{\&tau;}}={\mathrm{\&alpha;}}_{j+1}^{k,\mathrm{\&tau;}}=soft(U_{j+1},\frac{\mathrm{\&lambda;}}{L_f}{W}_{\mathrm{\&alpha;}}^{\mathrm{\&tau;}})---\left(16\right)$

55) updating a penalty factor mu and an iteration count parameter k:

μ^k+1，τ＝ρμ^k，τ,k＝k+1(17)

56) when k reaches the maximum iteration number, updating the weight factor and the iteration count parameter tau:

$W_{C,g}^{\mathrm{\&tau;}+1}=\frac{1}{{\mathrm{\&sigma;}}_g^{\mathrm{\&tau;}}+\mathrm{\&xi;}},W_{\mathrm{\&alpha;},g}^{\mathrm{\&tau;}+1}=\frac{1}{\left|{\mathrm{\&alpha;}}_g^{\mathrm{\&tau;}}\right|+\mathrm{\&xi;}},\mathrm{\&tau;}=\mathrm{\&tau;}+1---\left(18\right)$

57) when τ reaches the maximum iteration number of the reweighting matrix, a coefficient vector α of the low-rank matrix C and the moire pattern m which are expected to be sparsely represented in the DCT dictionary can be obtained, so that a fabric texture x and the moire pattern m are obtained:

$x={\left({\mathrm{\&Sigma;}}_{i=1}^n{s}_i^T{s}_i\right)}^{-1}{\mathrm{\&Sigma;}}_{i=1}^n{s}_i^T{c}_i---\left(19\right)$

m＝Dα(20)。

the invention has the technical characteristics and effects that:

the method avoids changing imaging equipment hardware, adopts a post-processing method, and separates the fabric texture from the moire pattern in the image by using a low-rank sparse matrix decomposition model, namely, the elimination of the moire of the fabric image is realized. Has the following characteristics:

1. the program is simple and easy to realize.

2. The elimination of moire in the fabric image is achieved.

3. Removing the moire pattern of the fabric image by adopting a low-rank coefficient matrix decomposition model: the elimination problem of the moire in the fabric image is attributed to the image segmentation problem, a low-rank sparse matrix decomposition model is constructed by utilizing the local self-similarity of the fabric texture and the energy concentration distribution characteristic of the moire in a frequency domain, and the fabric texture is separated from the moire pattern. Therefore, the difference between the fabric texture and the moire texture in the space domain and the frequency domain can be well utilized, and the fabric texture component and the moire texture component can be better distinguished.

4. The separation performance of the fabric texture and the moire pattern is improved by carrying out effective position constraint on the coefficient sparsely represented by the moire pattern under the DCT dictionary. This ensures that the positions of wrinkles, edges, shadows, etc. in the fabric image are not erroneously classified as moir é components.

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 method of the present invention;

FIG. 2 is a comparison graph of the statistical distribution of DCT coefficients for an image block with a Moire and an image block without a Moire: 1) is a statistical distribution map of the DCT coefficients of the image block with Moire fringes; 2) is the DCT coefficient statistical distribution map of the image block without Moire; 3) difference graphs of the two;

FIG. 3 is an image of a moir é textured fabric;

FIG. 4 is a comparison of fabric image moire removal results: 1) is the result of the Photoshop software processing the figure 3; 2) is the result of the processing of figure 3 by the method of the present invention.

Detailed Description

The method for eliminating the moire of the fabric image based on the low-rank sparse matrix decomposition is described in detail below by combining the embodiment and the attached drawings.

The method adopts a low-rank sparse matrix decomposition model to remove the moire patterns of the fabric image: the elimination problem of the moire in the fabric image is attributed to the image segmentation problem, a low-rank sparse matrix decomposition model is constructed by utilizing the local self-similarity of the fabric texture and the energy concentrated distribution characteristic of the moire in a frequency domain, and the fabric texture is separated from the moire pattern; by restricting the distribution of the moire patterns in the frequency domain, the separation performance of the fabric texture and the moire patterns is improved, and the positions of folds, edges, shadows and the like in the fabric image are prevented from being mistaken as the moire patterns. The method comprises the following steps:

1) constructing initial data:

11) taking a fabric image with moire patterns downloaded from Google as experimental data;

12) overlapping images I with u/2 as step interval_MDivided into image blocks of size u × u, then each image block y ' is processed to achieve separation of the fabric texture x ' from the moir pattern m ', i.e.:

y′＝x′+m′(1)

13) given the higher resolution of the Discrete Cosine Transform (DCT) domain than the Discrete Fourier Transform (DFT) domain, an N × p sized overcomplete DCT dictionary is constructed to sparsely represent the moir é pattern.

2) Constructing a sparse constraint term: by utilizing the energy concentration distribution characteristic of the moire pattern in the frequency domain, the moire pattern is assumed to be sparsely represented under the DCT dictionary, and the formula is as follows:

min||α||₁constraint m ═ D α Ω (2)

Wherein min represents solving the corresponding minimum value, | | · |. non-calculation₁Representing a matrixThe norm of the number of the first-order-of-arrival,is the constructed DCT dictionary and is,representing a real space, N, p representing the size of the spatial dimension,two-dimensional real number space of size N × p and N ═ u²,p>>N, the symbol ∈ denotes belonging, m is the vector form of the moire image block m' to be separated, α is the coefficient matrix of the sparse representation of moire m under the dictionary, Ω is the constraint on the significant positions of the coefficients α, i.e. the values of the coefficients α at the non-significant positions are all set to zero;

21) in some cases, the red and blue channels of the color image have moire patterns and the green channel does not contain any moire components. Therefore, a plurality of image blocks with moire patterns of the red channel and the blue channel and a plurality of image blocks without moire patterns at corresponding positions of the green channel are selected, a statistical distribution graph of DCT coefficients corresponding to the image blocks with moire patterns and the image blocks without moire patterns is obtained, and a conclusion is obtained: the moire pattern is energy concentrated in the frequency domain at the mid and low frequency locations. Therefore, all the coefficients of the high-frequency position corresponding to the alpha are set to zero, namely omega is the middle-low frequency position of the DCT domain;

22) to prevent erroneous classification of the position of wrinkles, edges, shadows, etc. in the fabric image as moire components, usingThe norm gradient minimization method smoothes the image block y' to obtain an image block containing wrinkles, edges, shadows, etc. but not containing moire and fabric textures, and further constrains the effective position of the coefficient α, i.e., Ω, according to its energy distribution in the DCT domain.

3) Constructing a low rank constraint term: dividing a fabric texture image block x' into k × k subblocks and converting the subblocks into corresponding vector forms, arranging the subblock vectors into a matrix C, and adding low-rank constraint by using the local self-similarity characteristic of the fabric texture in a spatial domain as follows:

min||C||_*(3)

wherein | · | purple sweet_*Denotes the kernel norm of the matrix, C ═ C₁，c₂,…,c_n]WhereinIs the vector form of the ith sub-block, n is the number of sub-blocks, k²The size of the dimensions of the space is represented,denotes a size of k²Is a one-dimensional real number space (i.e., vector). The construction equation for matrix C is as follows:

$C=S(I_n\&CircleTimes;x)---\left(4\right)$

wherein, the symbolIs the kronecker operator, I_nIs a diagonal matrix of n × n, x' vector form of x texture image block, S ═ S₁，s₂,…,s_n]Is a matrix for extracting sub-blocks of an image block, satisfies c_i＝s_ix,i＝1,2,…,n。

4) Combining the sparse constraint term of the moire pattern and the low-rank constraint term of the fabric texture similar block matrix by using a factor lambda to construct a final low-rank sparse matrix decomposition equation, wherein the specific formula is as follows:

min||C||_*+λ||α||₁

constraint conditions $\left\{\begin{array}{c}y=x+m\\ m=D\mathrm{\&alpha;},\mathrm{\&alpha;}\&Element;\mathrm{\&Omega;}\\ C=S(I_n\&CircleTimes;x)\end{array}\right.---\left(5\right)$

Where λ is a weighting factor, balancing the weight between the two terms. In order to reduce the error of the convergence result caused by a plurality of constraints, the invention combines the constraints in the formula (5) to obtain the following decomposition model:

min||C||_*+λ||α||₁

constraint conditions $C=S(I_n\&CircleTimes;(y-D\mathrm{\&alpha;})),\mathrm{\&alpha;}\&Element;\mathrm{\&Omega;}---\left(6\right)$

Recent research results on low-rank matrix recovery and sparse representation show that adding weight to the minimization model can improve the low rank and sparsity of the obtained result. Therefore, the present invention introduces a re-weighting strategy in equation (6), and obtains the following equation:

mintr(W_CοΣ)+λ||W_αοα||₁

constraint conditions $C=S(I_n\&CircleTimes;(y-D\mathrm{\&alpha;})),\mathrm{\&alpha;}\&Element;\mathrm{\&Omega;}---\left(7\right)$

Where tr (-) denotes the trace of the matrix,is a diagonal matrix formed by the singular values of the matrix C, diag (·) represents the diagonal matrix,is a singular value of a matrix, W_CAnd W_αThe weight matrices are respectively corresponding to the low-rank term and the sparse term, and the symbol o represents the dot product operation of the two matrices.

5) And finally solving the equation by using an augmented Lagrange method, wherein the method comprises the following specific steps:

51) convert equation (7) to lagrangian functional form:

l (C, α, Q) is the augmented Lagrangian function, where μ is the penalty factor, Q is the Lagrangian multiplier matrix,<·，·>representing the inner product of two matrices, | · | > non-woven phosphor_FA Flobenius norm representing a matrix;

52) the iterative solution equation of equation (8) is as follows:

$\left\{\begin{array}{c}{C}^{k+1,\mathrm{\&tau;}}={\mathrm{argmin}}_C\left\{L\left(C,{\mathrm{\&alpha;}}^{k,\mathrm{\&tau;}},Q^{k,\mathrm{\&tau;}}\right)\right\}\\ {\mathrm{\&alpha;}}^{k+1,\mathrm{\&tau;}}={\mathrm{argmin}}_{\mathrm{\&alpha;}}\left\{L\left(C^{k+1,\mathrm{\&tau;}},\mathrm{\&alpha;},Q^{k,\mathrm{\&tau;}}\right)\right\}\\ Q^{k+1,\mathrm{\&tau;}}=Q^{k,\mathrm{\&tau;}}+{\mathrm{\&mu;}}^{k,\mathrm{\&tau;}}\left(C^{k+1,\mathrm{\&tau;}}-S\left(I_n\&CircleTimes;\left(y-{\mathrm{D\&alpha;}}^{k+1,\mathrm{\&tau;}}\right)\right)\right)\\ {\mathrm{\&mu;}}^{k+1,\mathrm{\&tau;}}={\mathrm{\&rho;\&mu;}}^{k,\mathrm{\&tau;}}\\ W_{C,g}^{\mathrm{\&tau;}+1}=\frac{1}{{\mathrm{\&sigma;}}_g^{\mathrm{\&tau;}}+\mathrm{\&xi;}},W_{\mathrm{\&alpha;},g}^{\mathrm{\&tau;}+1}=\frac{1}{\left|{\mathrm{\&alpha;}}_g^{\mathrm{\&tau;}}\right|+\mathrm{\&xi;}}\end{array}\right.---\left(9\right)$

argmin in the above formula_X{. is the value of variable X at the time of minimizing the objective function, ρ is a multiple factor, superscripts k, k +1 and τ, τ +1 is the number of iterations, ξ is an arbitrarily small positive real number, and subscript g represents the position coordinates of the matrix elements;

53) solving for C^k+1,τSolving for C in the equation (9)^k+1,τThe C-independent term in the objective function of (1) yields:

the above formula is rewritten using the recipe as:

wherein:

$P^{k,\mathrm{\&tau;}}=S(I_n\&CircleTimes;(y-{\mathrm{D\&alpha;}}^{k,\mathrm{\&tau;}}\left)\right)-\frac{Q^{k,\mathrm{\&tau;}}}{{\mathrm{\&mu;}}^{k,\mathrm{\&tau;}}}$

using singular value threshold decomposition for equation (10) yields:

$C^{k+1,\mathrm{\&tau;}}=U^{k,\mathrm{\&tau;}}soft({\mathrm{\&Sigma;}}^{k,\mathrm{\&tau;}},\frac{W_C^{\mathrm{\&tau;}}}{{\mathrm{\&mu;}}^{k,\mathrm{\&tau;}}}){\left(V^{k,\mathrm{\&tau;}}\right)}^T---\left(11\right)$

wherein U is^k,τ，V^k,τAre respectively P^k,τLeft and right singular matrices, soft (·,) are the shrink operators.

54) Solving α using an accelerated neighbor gradient algorithm^k+1,τThe solution α in equation (9) is removed^k+1，τThe α -independent term in the objective function of (a) yields the following equation:

using Taylor expansion method, a second order function is constructed to approximate the above formula, and then the original equation is solved for the second order function.

Order to $f\left(\mathrm{\&alpha;}\right)=<Q^{k,\mathrm{\&tau;}},C^{k+1,\mathrm{\&tau;}}-S(I_n\&CircleTimes;(y-{\mathrm{D\&alpha;}}^{\mathrm{\&tau;}}))>+\frac{{\mathrm{\&mu;}}^{k,\mathrm{\&tau;}}}{2}{\left|\right|C^{k+1,\mathrm{\&tau;}}-S(I_n\&CircleTimes;(y-{\mathrm{D\&alpha;}}^{\mathrm{\&tau;}}))\left|\right|}_F^2,$ Reintroducing variable β defines the function:

wherein,is a gradient of f (β), L_fIs a constant value ofIs used to ensure that F (β) is less than or equal to T (α) for all β.

After the above conversion, the formula (12) is converted into the solution T (α)_j) By formulation, the following form is obtained:

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

$\left\{\begin{array}{c}{\mathrm{\&beta;}}_{j+1}={\mathrm{\&alpha;}}_{j+1}^{k,\mathrm{\&tau;}}+\frac{t_{j+1}-1}{t_{j+1}}\left({\mathrm{\&alpha;}}_{j+1}^{k,\mathrm{\&tau;}}-{\mathrm{\&alpha;}}_j^{k,\mathrm{\&tau;}}\right)\\ t_{j+1}=\left(1+\sqrt{4{t_j}^2+1}\right)/2\end{array}\right.---\left(15\right)$

t_jis a set of constant sequences, and j +1 are the number of iterations. Thus, the following results are obtained:

${\mathrm{\&alpha;}}^{k+1,\mathrm{\&tau;}}={\mathrm{\&alpha;}}_{j+1}^{k,\mathrm{\&tau;}}=soft(U_{j+1},\frac{\mathrm{\&lambda;}}{L_f}{W}_{\mathrm{\&alpha;}}^{\mathrm{\&tau;}})---\left(16\right)$

55) updating a penalty factor mu and an iteration count parameter k:

μ^k+1,τ＝ρμ^k,τ,k＝k+1(17)

56) when k reaches the maximum iteration number, updating the weight factor and the iteration count parameter tau:

$W_{C,g}^{\mathrm{\&tau;}+1}=\frac{1}{{\mathrm{\&sigma;}}_g^{\mathrm{\&tau;}}+\mathrm{\&xi;}},W_{\mathrm{\&alpha;},g}^{\mathrm{\&tau;}+1}=\frac{1}{\left|{\mathrm{\&alpha;}}_g^{\mathrm{\&tau;}}\right|+\mathrm{\&xi;}},\mathrm{\&tau;}=\mathrm{\&tau;}+1---\left(18\right)$

57) when tau reaches the maximum iteration number of the re-weighting matrix, the low rank matrix C and the coefficient matrix alpha of sparse representation of the Moire pattern m under the DCT dictionary which are expected by us can be obtained. Thus, the fabric texture x and the molar pattern m can be obtained:

$x={\left({\mathrm{\&Sigma;}}_{i=1}^n{s}_i^T{s}_i\right)}^{-1}{\mathrm{\&Sigma;}}_{i=1}^n{s}_i^T{c}_i---\left(19\right)$

m＝Dα(20)

wherein,is a summation formula (·)^-1The inverse of the matrix in brackets is shown.

6) And (3) performing the above processing on all image blocks in the image area of the moire pattern to finally obtain the fabric texture image I without the moire pattern and the separated moire component image M.

The invention provides an image moire elimination method for low-rank sparse matrix decomposition (as shown in the flow of fig. 1), which is described in detail in the following with reference to the accompanying drawings and embodiments:

1) constructing initial data:

11) taking a fabric image with moire patterns downloaded from Google as experimental data;

12) overlapping the images I at step intervals of 32_M(as shown in fig. 3) is divided into image blocks of size 64 × 64, and each image block y ' is then processed to achieve separation of the fabric texture x ' from the moir pattern m ', i.e.:

y′＝x′+m′(1)

13) considering that a Discrete Cosine Transform (DCT) domain has higher resolution than a Discrete Fourier Transform (DFT) domain, constructing an overcomplete DCT dictionary with the size of 4096 x 6400 to sparsely represent Moire patterns;

2) constructing a sparse constraint term: by utilizing the energy concentration distribution characteristic of the moire pattern in the frequency domain, the moire pattern is assumed to be sparsely represented under the DCT dictionary, and the formula is as follows:

min||α||₁constraint m ═ D α Ω (2)

Wherein min represents solving the corresponding minimum value, | | · |. non-calculation₁Representing a matrixThe norm of the number of the first-order-of-arrival,is the constructed DCT dictionary and is,representing a real space, 4096,6400 is the size of the spatial dimension,a two-dimensional real space of size 4096 × 6400 is represented, symbol ∈ represents belonging, m is the vector form of the moire image block m' to be separated, α is the coefficient matrix of sparse representation of moire m under a dictionary, Ω is the constraint on the effective position of the coefficient α, i.e. the values of the coefficient α at the non-effective position are all set to zero;

21) selecting a plurality of image blocks with moire patterns of the red channel and the blue channel and a plurality of image blocks without moire patterns at corresponding positions of the green channel, respectively obtaining a statistical distribution graph (shown in figure 2) of DCT coefficients of the image blocks with moire patterns and the image blocks without moire patterns, and obtaining a conclusion: the moire pattern is energy concentrated in the frequency domain at the mid and low frequency locations. Therefore, all the high-frequency position coefficients corresponding to alpha are set to zero, namely omega is the middle and low frequency position of the DCT domain;

22) to prevent erroneous classification of the position of wrinkles, edges, shadows, etc. in the fabric image as moire components, usingSmoothing the image block y' by a norm gradient minimization method to obtain an image block which contains folds, edges, shadows and the like and does not contain moire fringes and fabric textures, setting all values of a coefficient α at the energy distribution position to zero according to the energy distribution of the image block in a DCT domain, and further constraining the effective position of a coefficient α, namely limiting omega;

3) constructing a low rank constraint term: dividing a fabric texture image block x' into 8 x 8 sub-blocks and converting the sub-blocks into corresponding vector forms, arranging the sub-block vectors into a matrix C, and adding low-rank constraint as follows by utilizing the local self-similarity characteristic of the fabric texture in a spatial domain:

min||C||_*(3)

wherein | · | purple sweet_*Denotes the kernel norm of the matrix, C ═ C_1，c₂,…,c_n]WhereinIs the vector form of the ith sub-block, n-225 is the number of sub-blocks, 64 represents the size of the spatial dimension,representing a one-dimensional real space (i.e., a vector) of size 64. The construction equation for matrix C is as follows:

$C=S(I_n\&CircleTimes;x)---\left(4\right)$

wherein, the symbolIs the kronecker operator, I_nIs a diagonal matrix of 225 × 225, x texture image blocks x' vector form, S ═ S_1，s₂,…,s_n]Is a matrix for extracting sub-blocks of an image block, satisfies c_i＝s_ix,i＝1,2,…,225。

4) Combining the sparse constraint term of the moire pattern and the low-rank constraint term of the fabric texture similar block matrix by using a factor lambda to construct a final low-rank sparse matrix decomposition equation, wherein the specific formula is as follows:

min||C||_*+λ||α||₁

constraint conditions $\left\{\begin{array}{c}y=x+m\\ m=D\mathrm{\&alpha;},\mathrm{\&alpha;}\&Element;\mathrm{\&Omega;}\\ C=S(I_n\&CircleTimes;x)\end{array}\right.---\left(5\right)$

Where λ is a weighting factor, with a value of 0.015, which is used to balance the weight between the two terms. In order to reduce the error of the convergence result caused by a plurality of constraints, the invention combines the constraints in the formula (5) to obtain the following decomposition model:

min||C||_*+λ||α||₁

constraint conditions $C=S(I_n\&CircleTimes;(y-D\mathrm{\&alpha;})),\mathrm{\&alpha;}\&Element;\mathrm{\&Omega;}---\left(6\right)$

Recent research results on low-rank matrix recovery and sparse representation show that adding weight to the minimization model can improve the low rank and sparsity of the obtained result. Therefore, the present invention introduces a re-weighting strategy in equation (6), and obtains the following equation:

mintr(W_CοΣ)+λ||W_αοα||₁

constraint conditions $C=S(I_n\&CircleTimes;(y-D\mathrm{\&alpha;})),\mathrm{\&alpha;}\&Element;\mathrm{\&Omega;}---\left(7\right)$

Where tr (·) denotes a trace of the matrix, Σ ═ diag ([ σ ])₁,σ₂,…,σ₆₄]) Is a diagonal matrix formed by the singular values of the matrix C, diag (·) representing the diagonal matrix, σ₁,σ₂,…,σ₆₄Is a singular value of a matrix, W_CAnd W_αThe weight matrices are respectively corresponding to the low-rank term and the sparse term, and the symbol o represents the dot product operation of the two matrices.

5) And finally solving the equation by using an augmented Lagrange method, wherein the method comprises the following specific steps:

51) convert equation (7) to lagrangian functional form:

l (C, α, Q) is the augmented Lagrangian function, where μ is the penalty factor, Q is the Lagrangian multiplier matrix,<·，·>representing the inner product of two matrices, | · | > non-woven phosphor_FThe frobenius norm of the matrix is represented.

52) The iterative solution equation of equation (8) is as follows:

$\left\{\begin{array}{c}{C}^{k+1,\mathrm{\&tau;}}={\mathrm{argmin}}_C\left\{L\left(C,{\mathrm{\&alpha;}}^{k,\mathrm{\&tau;}},Q^{k,\mathrm{\&tau;}}\right)\right\}\\ {\mathrm{\&alpha;}}^{k+1,\mathrm{\&tau;}}={\mathrm{argmin}}_{\mathrm{\&alpha;}}\left\{L\left(C^{k+1,\mathrm{\&tau;}},\mathrm{\&alpha;},Q^{k,\mathrm{\&tau;}}\right)\right\}\\ Q^{k+1,\mathrm{\&tau;}}=Q^{k,\mathrm{\&tau;}}+{\mathrm{\&mu;}}^{k,\mathrm{\&tau;}}\left(C^{k+1,\mathrm{\&tau;}}-S\left(I_n\&CircleTimes;\left(y-{\mathrm{D\&alpha;}}^{k+1,\mathrm{\&tau;}}\right)\right)\right)\\ {\mathrm{\&mu;}}^{k+1,\mathrm{\&tau;}}={\mathrm{\&rho;\&mu;}}^{k,\mathrm{\&tau;}}\\ W_{C,g}^{\mathrm{\&tau;}+1}=\frac{1}{{\mathrm{\&sigma;}}_g^{\mathrm{\&tau;}}+\mathrm{\&xi;}},W_{\mathrm{\&alpha;},g}^{\mathrm{\&tau;}+1}=\frac{1}{\left|{\mathrm{\&alpha;}}_g^{\mathrm{\&tau;}}\right|+\mathrm{\&xi;}}\end{array}\right.---\left(9\right)$

argmin in the above formula_X{. is the value of the variable X at the time of minimizing the objective function, ρ is a multiple factor, superscripts k, k +1 and τ, τ +1 is the number of iterations, ξ is an arbitrarily small positive real number, and subscript g denotes the matrixThe position coordinates of the elements. Initial values were set in the experiment as: $k=1,\mathrm{\&tau;}=1,\mathrm{\&rho;}=1.2,\mathrm{\&mu;}=1.4,C^1=Q^1=0,{\mathrm{\&alpha;}}^1=0,W_C^1=1,W_{\mathrm{\&alpha;}}^1=1,\mathrm{\&xi;}={10}^{-5}.$

53) solving for C^k+1,τRemoving C in the formula (9)^k+1,τThe C-independent term in the objective function of (1) yields:

the formula is rewritten as follows:

wherein:

$P^{k,\mathrm{\&tau;}}=S(I_n\&CircleTimes;(y-{\mathrm{D\&alpha;}}^{k,\mathrm{\&tau;}}\left)\right)-\frac{Q^{k,\mathrm{\&tau;}}}{{\mathrm{\&mu;}}^{k,\mathrm{\&tau;}}}$

using singular value threshold decomposition for equation (10) yields:

$C^{k+1,\mathrm{\&tau;}}=U^{k,\mathrm{\&tau;}}soft({\mathrm{\&Sigma;}}^{k,\mathrm{\&tau;}},\frac{W_C^{\mathrm{\&tau;}}}{{\mathrm{\&mu;}}^{k,\mathrm{\&tau;}}}){\left(V^{k,\mathrm{\&tau;}}\right)}^T---\left(11\right)$

wherein U is^k,τ，V^k,τAre respectively P^k,τLeft and right singular matrices, soft (·,) are the shrink operators.

54) Solving α using an accelerated neighbor gradient algorithm^k+1,τRemoval of α in formula (9)^k+1,τThe α -independent term in the objective function of (a) yields the following equation:

using Taylor expansion method, a second order function is constructed to approximate the above formula, and then the original equation is solved for the second order function.

Order to $f\left(\mathrm{\&alpha;}\right)=<Q^{k,\mathrm{\&tau;}},C^{k+1,\mathrm{\&tau;}}-S(I_n\&CircleTimes;(y-{\mathrm{D\&alpha;}}^{\mathrm{\&tau;}}))>+\frac{{\mathrm{\&mu;}}^{k,\mathrm{\&tau;}}}{2}{\left|\right|C^{k+1,\mathrm{\&tau;}}-S(I_n\&CircleTimes;(y-{\mathrm{D\&alpha;}}^{\mathrm{\&tau;}}))\left|\right|}_F^2,$ Reintroducing variable β defines the function:

wherein,is a gradient of f (β), L_fIs a constant value ofIs used to ensure that F (β) is less than or equal to T (α) for all β.

Through the stepsConversion, equation (12) to solution T (α)_j) By formulation, the following form is obtained:

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

$\left\{\begin{array}{c}{\mathrm{\&beta;}}_{j+1}={\mathrm{\&alpha;}}_{j+1}^{k,\mathrm{\&tau;}}+\frac{t_{j+1}-1}{t_{j+1}}\left({\mathrm{\&alpha;}}_{j+1}^{k,\mathrm{\&tau;}}-{\mathrm{\&alpha;}}_j^{k,\mathrm{\&tau;}}\right)\\ t_{j+1}=\left(1+\sqrt{4{t_j}^2+1}\right)/2\end{array}\right.---\left(15\right)$

t_jis a set of constant sequences, and j +1 are the number of iterations. Setting initial values as follows: j is 1, t₁＝1,β₁＝1。

The convergence can be solved as follows:

${\mathrm{\&alpha;}}^{k+1,\mathrm{\&tau;}}={\mathrm{\&alpha;}}_{j+1}^{k,\mathrm{\&tau;}}=soft(U_{j+1},\frac{\mathrm{\&lambda;}}{L_f}{W}_{\mathrm{\&alpha;}}^{\mathrm{\&tau;}})---\left(16\right)$

55) updating a penalty factor mu and an iteration count parameter k:

μ^k+1，τ＝ρμ^k,τ,k＝k+1(17)

56) the iteration number is set to 40, and when the maximum iteration number, namely k is 40, the weight matrix and the iteration count parameter tau are updated:

$W_{C,g}^{\mathrm{\&tau;}+1}=\frac{1}{{\mathrm{\&sigma;}}_g^{\mathrm{\&tau;}}+\mathrm{\&xi;}},W_{\mathrm{\&alpha;},g}^{\mathrm{\&tau;}+1}=\frac{1}{\left|{\mathrm{\&alpha;}}_g^{\mathrm{\&tau;}}\right|+\mathrm{\&xi;}},\mathrm{\&tau;}=\mathrm{\&tau;}+1---\left(18\right)$

57) when the iteration number of the re-weighting matrix is set to 4, that is, τ is 4, we can obtain the desired coefficient vector α of the low rank matrix C and the sparse representation of the moire pattern m under the DCT dictionary. Thus, the fabric texture x and the molar pattern m can be obtained:

$x={\left({\mathrm{\&Sigma;}}_{i=1}^n{s}_i^T{s}_i\right)}^{-1}{\mathrm{\&Sigma;}}_{i=1}^n{s}_i^T{c}_i---\left(19\right)$

m＝Dα(20)

wherein,is a summation formula (·)^-1The inverse of the matrix in brackets is shown.

6) And (3) performing the above processing on all image blocks in the image area with the moire pattern to finally obtain a fabric texture image I (shown in fig. 4) without the moire pattern and a separated moire component image M.

Claims (3)

1. A fabric image moire elimination method based on low-rank sparse matrix decomposition is characterized in that a low-rank sparse matrix decomposition model is constructed by utilizing local self-similarity of fabric textures and the energy concentration distribution characteristics of moire patterns in a frequency domain, and the fabric textures are separated from moire patterns, so that the problem of eliminating moire in a fabric image is solved.

1) Overlapping images I with u/2 as step interval_MDivided into image blocks of u × u size, and then each image block y 'is processed to achieve the division of the fabric texture x' into the Moire pattern mAnd (2) separating, namely:

y′＝x′+m′(1)

2) constructing a sparse constraint term: by utilizing the energy concentration distribution characteristic of the moire pattern in the frequency domain, the moire pattern is assumed to be sparsely represented by a Discrete Cosine Transform (DCT) dictionary, and the formula is as follows:

min‖α‖₁constraint m ═ D α Ω (2)

Wherein min represents finding the corresponding minimum value, | |)₁The/1 norm of the matrix is represented,is the constructed DCT dictionary and is,representing a real space, N, p representing the size of the spatial dimension,representing a two-dimensional real number space of size N × p and N ═ u²P > N, the symbol ∈ denotes belonging, m is the vector form of the moire image block m' to be separated, α is the coefficient matrix of the sparse representation of moire m under the dictionary, Ω is the constraint on the significant positions of coefficients α, i.e. the values of coefficients α at the non-significant positions are all set to zero;

3) constructing a low rank constraint term: the fabric texture image block x' is converted into k × k subblocks and converted into corresponding vector forms, the subblock vectors are arranged into a matrix C, and local self-similarity characteristics of the fabric texture in a space domain are utilized to add low-rank constraints as follows:

min‖C‖_*(3)

wherein |_*Denotes the kernel norm of the matrix, C ═ C₁,c₂,…,c_n]WhereinIs the vector form of the ith sub-block, n is the number of sub-blocks, k²The size of the dimensions of the space is represented,denotes a size of k²One-dimensional real number space, i.e. vector; the construction equation for matrix C is as follows:

wherein, the symbolIs the kronecker operator, I_nIs a diagonal matrix of n × n, x' vector form of x texture image block, S ═ S₁,s₂,…,s_n]Is a matrix for extracting sub-blocks of an image block, satisfies c_i＝s_ix,i＝1,2,…,n；

4) Combining the sparse constraint term of the moire pattern and the low-rank constraint term of the fabric texture similar block matrix by using a factor lambda to construct a final low-rank sparse matrix decomposition equation, wherein the specific formula is as follows:

min‖C‖_*+λ‖α‖₁

constraint conditions

Where λ is a weighting factor, balancing the weight between the two terms. In order to reduce the error of the convergence result caused by a plurality of constraints, the invention combines the constraints in the formula (5) to obtain the following decomposition model:

min‖C‖_*+λ‖α‖₁

constraint conditions

Recent research results on low-rank matrix recovery and sparse representation show that adding weights to the minimization model can improve the low rank and sparsity of the obtained result, so that a re-weighting strategy is introduced into the formula (6) to obtain the following equation:

mintr(W_CоΣ)+λ‖W_αоα‖₁

constraint conditions

Where tr (-) denotes the trace of the matrix,is a diagonal matrix formed by the singular values of the matrix C, diag (·) represents the diagonal matrix,is a singular value of a matrix, W_CAnd W_αThe weight matrixes respectively correspond to the low-rank term and the sparse term, and the symbol o represents the dot multiplication operation of the two matrixes;

5) and finally solving the equation by using an augmented Lagrange method, and after a plurality of iterations, achieving convergence to finally obtain the needed low-rank matrix C and the coefficient vector alpha of the Moire pattern m sparsely represented under the DCT dictionary, thereby obtaining the fabric texture x and the Moire pattern m:

m＝Dα(20)

wherein,is a summation formula (·)^-1Represents the inverse of the matrix within the brackets;

6) and (3) performing the above processing on all image blocks in the image area of the moire pattern to finally obtain the fabric texture image I without the moire pattern and the separated moire component image M.

2. The method for eliminating the moire of the fabric image based on the low rank sparse matrix decomposition as claimed in claim 1, characterized in that the specific method comprises the following steps:

further specifically, the step of constructing the sparse constraint term includes 21) in some cases, the red channel and the blue channel of the color image have moire patterns, and the green channel does not include any moire component, so that a plurality of moire image blocks of the red channel and the blue channel and a plurality of moire-free image blocks at corresponding positions of the green channel are selected, statistical distribution maps of DCT coefficients corresponding to the moire image blocks and the moire-free image blocks are respectively obtained, and a conclusion is obtained: the Moire pattern is concentrated in energy distribution at middle and low frequency positions in a frequency domain, so that all coefficients of alpha corresponding to the high frequency position are set to zero, namely omega is the middle and low frequency position of a DCT domain;

22) in case of dividing the positions of folds, edges, shadows and the like in the fabric image into moire components by mistake, smoothing the image block y' by using an l0 norm gradient minimization method to obtain an image block which contains the folds, edges, shadows and the like and does not contain moire and fabric textures, and further constraining the effective position of the coefficient alpha, namely limiting omega according to the energy distribution of the image block in the DCT domain.

3. The method for eliminating the moire of the fabric image based on the low-rank sparse matrix decomposition as claimed in claim 1, wherein 5) the equation is finally solved by using an augmented Lagrangian method, and the method comprises the following specific steps:

51) convert equation (7) to lagrangian functional form:

l (C, α, Q) is the augmented Lagrangian function, where μ is the penalty factor, Q is the Lagrangian multiplier matrix,<·，·>representing the inner product of two matrices, | · | > non-woven phosphor_FA Flobenius norm representing a matrix;

52) the iterative solution equation of equation (8) is as follows:

argmin in the above formula_X{. is the value of variable X at the time of minimizing the objective function, ρ is a multiple factor, superscripts k, k +1 and τ, τ +1 is the number of iterations, ξ is an arbitrarily small positive real number, and subscript g represents the position coordinates of the matrix elements;

53) solving for C^k+1，τSolving for C in the equation (9)^k+1，τThe C-independent term in the objective function of (1) yields:

the above formula is rewritten using the recipe as:

wherein:

using singular value threshold decomposition for equation (10) yields:

wherein U is^k，τ，V^k，τAre respectively P^k，τLeft and right singular matrices of (1), soft (·,) being a contraction operator;

54) solving α using an accelerated neighbor gradient algorithm^k+1，τThe solution α in equation (9) is removed^k+1，τThe α -independent term in the objective function of (a) yields the following equation:

using Taylor expansion method, construct a second order function to approximate the above equation, and then solve the original equation for the second order function:

order to Reintroducing variable β defines the function:

wherein,is a gradient of f (β), L_fIs a constant value ofUsed for ensuring that F (β) is less than or equal to T (α) for all β;

after the above conversion, the formula (12) is converted into the solution T (α)_j) By formulation, the following form is obtained:

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

t_jis a set of constant sequences, j and j +1 are the number of iterations, from which it is solved:

55) updating a penalty factor mu and an iteration count parameter k:

μ^k+1，τ＝ρμ^k，τ，k＝k+1(17)

56) when k reaches the maximum iteration number, updating the weight factor and the iteration count parameter tau:

57) when τ reaches the maximum iteration number of the reweighting matrix, a coefficient vector α of the low-rank matrix C and the moire pattern m which are expected to be sparsely represented in the DCT dictionary can be obtained, so that a fabric texture x and the moire pattern m are obtained:

m＝Dα(20)。