CN105303530A

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

Info

Publication number: CN105303530A
Application number: CN201510639274.XA
Authority: CN
Inventors: 杨敬钰; 刘芳蕾
Original assignee: Tianjin University
Current assignee: Tianjin University
Priority date: 2015-09-30
Filing date: 2015-09-30
Publication date: 2016-02-03
Anticipated expiration: 2035-09-30
Also published as: CN105303530B

Abstract

The invention belongs to the field of digital image processing and aims to eliminate moiré patterns in fabric images. The technical solution adopted by the present invention is to construct a low-rank sparse matrix decomposition model by using the local self-similarity of fabric texture and the energy concentration and distribution characteristics of moiré in the frequency domain based on the low-rank sparse matrix decomposition method for eliminating moiré in fabric images. Solve the problem of eliminating moiré in fabric images by separating the fabric texture from the moiré pattern. The invention is mainly applied to digital image processing.

Description

Moiré Elimination Method of Fabric Image Based on Low Rank Sparse Matrix Decomposition

技术领域 technical field

本发明属于数字图像处理领域，具体讲，涉及基于低秩稀疏矩阵分解的织物图像摩尔纹消除方法。 The invention belongs to the field of digital image processing, and in particular relates to a method for eliminating moiré patterns of fabric images based on low-rank sparse matrix decomposition.

背景技术 Background technique

摩尔纹现象极易出现在现代数字成像过程中，比如：扫描仪扫描半色调印刷品，数码相机拍摄屏幕类或包含规则性图案(衣服上的布纹、建筑上的窗口或栅格)的场景。摩尔纹图案在图像中往往面积大，色偏明显，严重影响了图像质量和图像分析结果，引起人们的广泛关注。 The moiré phenomenon is very easy to appear in the modern digital imaging process, such as: scanner scanning halftone prints, digital cameras shooting screens or scenes containing regular patterns (cloth patterns on clothes, windows or grids on buildings). Moiré patterns often have a large area in the image, and the color shift is obvious, which seriously affects the image quality and image analysis results, and has attracted widespread attention.

目前为止，图像摩尔纹消除算法主要针对半色调扫描图像。在各类图像摩尔纹中，半色调图像中的摩尔纹是一类比较简单的摩尔纹图案，且它的形状、分布等与打印机设备密切相关而不受原图像的影响。这些算法主要分为两大类，基于滤波的方法和基于字典学习的方法。另一种比较简单的图像摩尔纹是射线显微镜成像过程中射线干扰导致的摩尔纹现象，近乎平行的摩尔纹条纹遍布整幅图像。对于该类摩尔纹，目前最常用的消除方法是选择合适的中值高斯滤波器，只不过不可避免地在局部边缘位置产生轻微的振铃效应。 So far, image moiré removal algorithms have mainly targeted halftone scanned images. Among various image moiré patterns, the moiré pattern in halftone images is a relatively simple type of moiré pattern, and its shape, distribution, etc. are closely related to the printer equipment and are not affected by the original image. These algorithms are mainly divided into two categories, filtering-based methods and dictionary-based learning methods. Another relatively simple image moiré is the moiré phenomenon caused by ray interference during the imaging process of the ray microscope. Nearly parallel moiré stripes spread across the entire image. For this type of moiré, the most commonly used elimination method at present is to select an appropriate median Gaussian filter, but it will inevitably produce a slight ringing effect at the local edge position.

但是，由于织物图像中的摩尔纹图案比较复杂：1)条纹分布没有明显规律，色偏明显；2)摩尔纹图案与织物纹理结构相似，因此消除这类摩尔纹更加的困难。为了消除这类摩尔纹，大部分厂商选择在数码相机中嵌入光学低通滤镜，以降低图像锐度为代价来抑制摩尔纹的产生。另一种解决方案是设计特殊的CFA(ColorFilterArray)插值算法，借助绿色通道捕获的高频信息估计红蓝通道的混叠信息和丢失的高频信息来避免摩尔纹现象的出现；该方法严重受限于拍摄图像，同时复杂的插值算法增加了相机内部处理的负担，并不是一种可靠的消除摩尔纹的方法。目前，消除数码相机成像过程产生的摩尔纹最有效的后处理方法是使用Photoshop软件，手动选择摩尔纹区域后，先去除该区域摩尔纹的彩色，然后利用色度和饱和度信息修补亮度层以消除灰色的摩尔纹条纹。总之，目前尚未提出对织物图像中摩尔纹现象的高效处理算法，能够既消除摩尔纹干扰，又保留图像锐度。 However, because the moiré pattern in the fabric image is more complicated: 1) the stripe distribution has no obvious rules, and the color shift is obvious; 2) the moiré pattern is similar to the texture structure of the fabric, so it is more difficult to eliminate this type of moiré pattern. In order to eliminate this kind of moiré, most manufacturers choose to embed an optical low-pass filter in the digital camera to suppress the generation of moiré at the cost of reducing image sharpness. Another solution is to design a special CFA (ColorFilterArray) interpolation algorithm, which uses the high-frequency information captured by the green channel to estimate the aliasing information and lost high-frequency information of the red and blue channels to avoid the appearance of moiré; It is limited to shooting images, and the complex interpolation algorithm increases the burden of internal processing of the camera, which is not a reliable method to eliminate moiré. At present, the most effective post-processing method to eliminate the moiré pattern produced by the digital camera imaging process is to use Photoshop software. After manually selecting the moiré pattern area, first remove the color of the moiré pattern in this area, and then use the chroma and saturation information to repair the brightness layer to Eliminates gray moiré streaks. In short, no efficient processing algorithm for moiré in fabric images has been proposed so far, which can eliminate moiré interference and preserve image sharpness.

发明内容 Contents of the invention

本发明意在弥补现有技术的不足，即实现对织物图像中摩尔纹图案的消除。本发明采取的技术方案是，基于低秩稀疏矩阵分解的织物图像摩尔纹消除方法，利用织物纹理的局部自相似性以及摩尔纹在频率域的能量集中分布特性，构建低秩稀疏矩阵分解模型，将织物纹理与摩尔纹图案分离，从而解决消除织物图像中摩尔纹的问题。具体方法包括以下步骤： The invention aims to make up for the deficiencies of the prior art, that is, to realize the elimination of the moiré pattern in the fabric image. The technical solution adopted by the present invention is to construct a low-rank sparse matrix decomposition model by using the local self-similarity of fabric texture and the energy concentration and distribution characteristics of moiré in the frequency domain based on the low-rank sparse matrix decomposition method for eliminating moiré in fabric images. Solve the problem of eliminating moiré in fabric images by separating the fabric texture from the moiré pattern. The specific method includes the following steps:

1)以u/2为步进间隔有重叠地将图像I_M划分成u×u大小的图像块，然后每个图像块y′进行处理，以实现织物纹理x′与摩尔纹图案m′的分离，即： 1) Divide the image I _M into image blocks of u×u size with u/2 as the step interval with overlapping, and then process each image block y' to realize the texture x' of the fabric and the moiré pattern m' separate, that is:

y′＝x′+m′(1) y'=x'+m'(1)

2)构建稀疏约束项：利用摩尔纹图案在频率域的能量集中分布特性，假设摩尔纹图案用离散余弦变换(DCT)字典稀疏表示，公式如下： 2) Constructing sparse constraint items: Utilizing the energy concentration and distribution characteristics of the moiré pattern in the frequency domain, assuming that the moiré pattern is sparsely represented by a discrete cosine transform (DCT) dictionary, the formula is as follows:

min||α||₁约束条件m＝Dα,α∈Ω(2) min||α|| ₁ constraint m=Dα,α∈Ω(2)

其中，min表示求对应的最小值，||·||₁表示矩阵的范数，是构建的DCT字典，表示实数空间，N,p表示空间维度的大小，表示大小为N×p的二维实数空间且N＝u²,p>>N，符号∈表示属于，m是待分离的摩尔纹图像块m′的向量形式，α是摩尔纹m在字典下稀疏表示的系数矩阵，Ω是对系数α有效位置的约束，即将非有效位置处的系数α的值全部置零； Among them, min means to seek the corresponding minimum value, ||·|| ₁ means the matrix norm, is the constructed DCT dictionary, Represents the real number space, N, p represents the size of the space dimension, Represents a two-dimensional real number space with a size of N×p and N=u ² , p>>N, the symbol ∈ represents belonging to, m is the vector form of the moiré image block m′ to be separated, and α is the moiré pattern m under the dictionary Sparse representation of the coefficient matrix, Ω is a constraint on the effective position of the coefficient α, that is, the value of the coefficient α at the non-effective position is all set to zero;

3)构建低秩约束项：将织物纹理图像块x′成k×k的子块并转换成对应的向量形式，将这些子块向量排成矩阵C，利用织物纹理在空间域的局部自相似特性，添加低秩约束如下： 3) Construct a low-rank constraint item: convert the fabric texture image block x′ into k×k sub-blocks and convert them into corresponding vector forms, arrange these sub-block vectors into a matrix C, and use the local self-similarity of the fabric texture in the spatial domain feature, adding the low-rank constraint as follows:

min||C||_*(3) min||C|| _* (3)

其中，||·||_*表示矩阵的核范数，C＝[c₁,c₂,…,c_n]，其中是第i个子块的向量形式，n是子块的个数，k²表示空间维度的大小，表示大小为k²的一维实数空间,即向量；矩阵C的构建方程如下： Among them, ||·|| _* represents the nuclear norm of the matrix, C=[c ₁ ,c ₂ ,…,c _n ], where is the vector form of the i-th sub-block, n is the number of sub-blocks, k ² represents the size of the spatial dimension, Represents a one-dimensional real number space of size k ² , that is, a vector; the construction equation of matrix C is as follows:

$C C = = S S (({I I}_{n no} &CircleTimes; &CircleTimes; x x)) - - - - - - ((44))$

其中，符号是克罗内克算子，I_n是一个n×n的对角线矩阵，x织物纹理图像块x′向量形式，S＝[s₁,s₂,…,s_n]是用来提取图像块子块的矩阵，满足c_i＝s_ix,i＝1,2,…,n； Among them, the symbol is the Kronecker operator, I _n is an n×n diagonal matrix, x fabric texture image block x′ vector form, S=[s ₁ ,s ₂ ,…,s _n ] is used to extract the image The matrix of blocks and sub-blocks satisfies c _i =s _i x,i=1,2,...,n;

4)将摩尔纹图案的稀疏约束项和织物纹理相似块矩阵的低秩约束项用一个因子λ联合在一起，构建最终的低秩稀疏矩阵分解方程，具体公式如下： 4) Combine the sparse constraint item of the moiré pattern and the low-rank constraint item of the fabric texture similar block matrix with a factor λ to construct the final low-rank sparse matrix decomposition equation, the specific formula is as follows:

min‖C‖_*+λ‖α‖₁ min‖C‖ _* +λ‖α‖ ₁

约束条件 $\{\begin{matrix} y = x + m \\ m = D α, α &Element; Ω \\ C = S (I_{n} &CircleTimes; x) \end{matrix} - - - (5)$ Restrictions $\{\begin{matrix} the y = x + m \\ m = D. α, α &Element; Ω \\ C = S (I_{no} &CircleTimes; x) \end{matrix} - - - (5)$

其中，λ是加权因子，平衡两项之间的权重。为了降低多个约束条件导致的收敛结果的误差，本发明将(5)式中的约束条件合并，得到如下的分解模型： Among them, λ is a weighting factor, which balances the weight between the two items. In order to reduce the error of the convergence result caused by multiple constraints, the present invention combines the constraints in formula (5) to obtain the following decomposition model:

min‖C‖_*+λ‖α‖₁ min‖C‖ _* +λ‖α‖ ₁

约束条件 $C = S (I_{n} &CircleTimes; (y - D α)), α &Element; Ω - - - (6)$ Restrictions $C = S (I_{no} &CircleTimes; (the y - D. α)), α &Element; Ω - - - (6)$

最近有关低秩矩阵恢复和稀疏表示的研究成果表明，给上述最小化模型添加权重能够提高所得结果的低秩性和稀疏性，因此，(6)式中引入了重加权策略，获得如下方程： Recent research results on low-rank matrix recovery and sparse representation show that adding weights to the above minimization model can improve the low-rank and sparsity of the obtained results. Therefore, a reweighting strategy is introduced in (6), and the following equation is obtained:

mintr(W_CοΣ)+λ||W_αοα||₁ mintr(W _C οΣ)+λ||W _α οα|| ₁

约束条件 $C = S (I_{n} &CircleTimes; (y - D α)), α &Element; Ω - - - (7)$ Restrictions $C = S (I_{no} &CircleTimes; (the y - D. α)), α &Element; Ω - - - (7)$

其中,tr(·)表示矩阵的迹，是一个由矩阵C的奇异值构成的对角线矩阵，diag(·)表示对角线矩阵，是矩阵的奇异值，W_C和W_α分别是低秩项和稀疏项对应的权重矩阵，符号ο表示两个矩阵的点乘运算； Among them, tr( ) represents the trace of the matrix, is a diagonal matrix composed of the singular values of the matrix C, diag(·) represents the diagonal matrix, is the singular value of the matrix, W _C and W _α are the weight matrix corresponding to the low-rank item and the sparse item respectively, and the symbol ο represents the point multiplication operation of the two matrices;

5)利用增广拉格朗日方法对方程进行最终求解，经过若干次迭代达到收敛，最终得到需要的低秩矩阵C和摩尔纹图案m在DCT字典下稀疏表示的系数向量α，从而得到织物纹理x与摩尔纹图案m： 5) Use the augmented Lagrangian method to finally solve the equation, and reach convergence after several iterations, and finally obtain the required low-rank matrix C and the coefficient vector α sparsely represented by the moiré pattern m under the DCT dictionary, so as to obtain the fabric Texture x and moiré pattern m:

$x x = = {(({Σ Σ}_{i i = = 11}^{n no} {s the s}_{i i}^{T T} {s the s}_{i i}))}^{- - 11} {Σ Σ}_{i i = = 11}^{n no} {s the s}_{i i}^{T T} {c c}_{i i} - - - - - - ((1919))$

m＝Dα(20) m=Dα(20)

其中，是求和公式，(·)^-1表示括号内矩阵的逆； in, is the summation formula, (·) ^-1 represents the inverse of the matrix in the brackets;

6)对带摩尔纹图案的图像区域所有图像块做上述的处理，最终得到不含摩尔纹图案的织物纹理图像I和分离出的摩尔纹成分图像M。 6) Perform the above-mentioned processing on all image blocks in the image area with moiré pattern, and finally obtain the fabric texture image I without moiré pattern and the separated moiré component image M.

构建稀疏约束项步骤进一步具体是，21)某些情况下，彩色图像的红、蓝两通道有摩尔纹图案而绿色通道不包含任何摩尔纹成分，所以选取红、蓝通道的带摩尔纹的图像块若干及绿色通道对应位置的不带摩尔纹的图像块若干，分别得到带摩尔纹的图像块与不带摩尔纹的图像块对应的DCT系数的统计分布图，得出结论：摩尔纹图案在频率域能量集中分布在中低频位置，因此将α对应高频位置的系数全部置零，即Ω为DCT域的中低频位置； The step of constructing the sparse constraint item is further specifically, 21) In some cases, the red and blue channels of the color image have moiré patterns and the green channel does not contain any moiré components, so the images with moiré patterns in the red and blue channels are selected A number of blocks and a number of image blocks without moiré in the corresponding position of the green channel, and the statistical distribution diagrams of the DCT coefficients corresponding to the image block with moiré and the image block without moiré are respectively obtained, and the conclusion is drawn: the moiré pattern is The energy in the frequency domain is concentrated in the middle and low frequency positions, so the coefficients of α corresponding to the high frequency position are all set to zero, that is, Ω is the middle and low frequency position of the DCT domain;

22)以防错误地把织物图像中的褶皱、边缘、阴影等位置划分为为摩尔纹成分，利用范数梯度最小化方法平滑图像块y′，获得包含褶皱、边缘、阴影等却不含摩尔纹与织物纹理的图像块，根据其在DCT域的能量分布，进一步约束系数α的有效位置，即对Ω进行限定。 22) In case of mistakenly dividing the folds, edges, shadows and other positions in the fabric image into moiré components, use The norm gradient minimization method smoothes the image block y′, and obtains an image block that contains wrinkles, edges, shadows, etc. but does not contain moiré patterns and fabric textures. According to its energy distribution in the DCT domain, the effective position of the coefficient α is further constrained, namely Ω is defined.

5)利用增广拉格朗日方法对方程进行最终求解，具体步骤如下： 5) Using the augmented Lagrangian method to finally solve the equation, the specific steps are as follows:

51)将方程(7)转换成拉格朗日函数形式： 51) Transform equation (7) into Lagrange function form:

L(C,α,Q)即为增广拉格朗日函数，其中，μ是惩罚因子，Q为拉格朗日乘子矩阵，<·，·>表示两个矩阵的内积，||·||_F表示矩阵的弗罗贝尼乌斯范数； L(C,α,Q) is the augmented Lagrangian function, where μ is the penalty factor, Q is the Lagrange multiplier matrix, <·,·> represents the inner product of two matrices, || ·|| _F represents the Frobenius norm of the matrix;

52)(8)式的迭代求解方程如下： 52) The iterative solution equation of (8) is as follows:

$\{\begin{matrix} {C C}^{k k + + 11,, τ τ} = = {argmin argmin}_{C C} {{L L ((C C,, {α α}^{k k,, τ τ},, {Q Q}^{k k,, τ τ}))}} \\ {α α}^{k k + + 11,, τ τ} = = {argmin argmin}_{α α} {{L L (({C C}^{k k + + 11,, τ τ},, α α,, {Q Q}^{k k,, τ τ}))}} \\ {Q Q}^{k k + + 11,, τ τ} = = {Q Q}^{k k,, τ τ} + + {μ μ}^{k k,, τ τ} (({C C}^{k k + + 11,, τ τ} - - S S (({I I}_{n no} &CircleTimes; &CircleTimes; ((y the y - - {Dα Dα}^{k k + + 11,, τ τ})))))) \\ {μ μ}^{k k + + 11,, τ τ} = = {ρμ ρμ}^{k k,, τ τ} \\ {W W}_{C C,, g g}^{τ τ + + 11} = = \frac{11}{{σ σ}_{g g}^{τ τ} + + ξ ξ},, {W W}_{α α,, g g}^{τ τ + + 11} = = \frac{11}{| | {α α}_{g g}^{τ τ} | | + + ξ ξ} \end{matrix} - - - - - - ((99))$

上式中argmin_X｛·｝表示使目标函数最小值时变量X的值，ρ为倍数因子，上标k,k+1以及τ,τ+1是迭代次数，ξ为任意小的正实数，下标g表示矩阵元素的位置坐标； In the above formula, argmin _X { } represents the value of variable X when the objective function is minimized, ρ is the multiple factor, superscripts k, k+1 and τ, τ+1 are the number of iterations, ξ is any small positive real number, The subscript g represents the position coordinates of the matrix elements;

53)求解C^k+1，τ,去掉式子(9)中求解C^k+1,τ的目标函数里与C无关的项得到: 53) Solve C ^{k+1, τ} , remove the C ^{k+1, τ in the objective function of solving C k+1,} the term that has nothing to do with C in the objective function of formula (9):

使用配方法将上式改写成: Using the formula method, the above formula can be rewritten as:

其中： in:

${P P}^{k k,, τ τ} = = S S (({I I}_{n no} &CircleTimes; &CircleTimes; ((y the y - - {Dα Dα}^{k k,, τ τ})))) - - \frac{{Q Q}^{k k,, τ τ}}{{μ μ}^{k k,, τ τ}}$

对于式(10)使用奇异值阈值分解法解得： For equation (10), use the singular value threshold decomposition method to solve:

${C C}^{k k + + 11,, τ τ} = = {U u}^{k k,, τ τ} s the s o o f f t t (({Σ Σ}^{k k,, τ τ},, \frac{{W W}_{C C}^{τ τ}}{{μ μ}^{k k,, τ τ}})) {(({V V}^{k k,, τ τ}))}^{T T} - - - - - - ((1111))$

其中U^k,τ，V^k,τ分别是P^k,τ的左奇异矩阵和右奇异矩阵，soft(·，·)为收缩算子； Among them, U ^{k, τ} , V ^{k, τ} are the left singular matrix and right singular matrix of P ^k, τ respectively, and soft(·,·) is the contraction operator;

54)使用加速近邻梯度算法求解α^k+1,τ，去掉式子(9)中求解α^k+1，τ的目标函数里与α无关的项得到如下方程： 54) Use the accelerated neighbor gradient algorithm to solve α ^k+1,τ , and remove the item in the objective function of solving α ^k+1,τ in equation (9) that has nothing to do with α to get the following equation:

使用泰勒展开的方法，构造出一个二阶函数来逼近上式，然后针对这个二阶的函数来求解原方程： Using the method of Taylor expansion, construct a second-order function to approximate the above formula, and then solve the original equation for this second-order function:

令 $f (α) = < Q^{k, τ}, C^{k + 1, τ} - S (I_{n} &CircleTimes; (y - {Dα}^{τ})) > + \frac{μ^{k, τ}}{2} {| | C^{k + 1, τ} - S (I_{n} &CircleTimes; (y - {Dα}^{τ})) | |}_{F}^{2},$ 再引入变量β，定义如下函数: make $f (α) = < Q^{k, τ}, C^{k + 1, τ} - S (I_{no} &CircleTimes; (the y - {Dα}^{τ})) > + \frac{μ^{k, τ}}{2} {| | C^{k + 1, τ} - S (I_{no} &CircleTimes; (the y - {Dα}^{τ})) | |}_{f}^{2},$ Then introduce the variable β and define the following function:

其中，为f(β)的梯度，L_f是一个常数值为用来保证对所有的β都有F(β)≤T(α,β)； in, is the gradient of f(β), L _f is a constant value Used to ensure that F(β)≤T(α,β) for all β;

经过上步转化，式子(12)转化成求解T(α,β_j)的最小值问题，通过配方得到如下形式： After the transformation in the previous step, the formula (12) is transformed into the problem of solving the minimum value of T(α,β _j ), and the following form is obtained through the formula:

其中，变量β_j的更新规则如下： in, The update rule of variable β _j is as follows:

$\{\begin{matrix} {β β}_{j j + + 11} = = {α α}_{j j + + 11}^{k k,, τ τ} + + \frac{{t t}_{j j + + 11} - - 11}{{t t}_{j j + + 11}} (({α α}_{j j + + 11}^{k k,, τ τ} - - {α α}_{j j}^{k k,, τ τ})) \\ {t t}_{j j + + 11} = = ((11 + + \sqrt{44 {t t}_{j j}^{22} + + 11})) / / 22 \end{matrix} - - - - - - ((1515))$

t_j是一组常数序列，j和j+1是迭代次数，由此解得： t _j is a set of constant sequences, and j and j+1 are the number of iterations, so the solution is:

${α α}^{k k + + 11,, τ τ} = = {α α}_{j j + + 11}^{k k,, τ τ} = = s the s o o f f t t (({U u}_{j j + + 11},, \frac{λ λ}{{L L}_{f f}} {W W}_{α α}^{τ τ})) - - - - - - ((1616))$

55)更新惩罚因子μ及迭代计数参数k： 55) Update penalty factor μ and iteration count parameter k:

μ^k+1，τ＝ρμ^k，τ,k＝k+1(17) μ ^k+1,τ =ρμ ^k,τ ,k=k+1(17)

56)当k达到最大迭代次数即时，更新权重因子及迭代计数参数τ： 56) When k reaches the maximum number of iterations, update the weight factor and iteration count parameter τ:

${W W}_{C C,, g g}^{τ τ + + 11} = = \frac{11}{{σ σ}_{g g}^{τ τ} + + ξ ξ},, {W W}_{α α,, g g}^{τ τ + + 11} = = \frac{11}{| | {α α}_{g g}^{τ τ} | | + + ξ ξ},, τ τ = = τ τ + + 11 - - - - - - ((1818))$

57)当τ达到重加权矩阵的最大迭代次数时，可以得到我们期望的低秩矩阵C和摩尔纹图案m在DCT字典下稀疏表示的系数向量α，从而得到织物纹理x与摩尔纹图案m： 57) When τ reaches the maximum number of iterations of the reweighting matrix, we can obtain the coefficient vector α of the sparse representation of our expected low-rank matrix C and moiré pattern m under the DCT dictionary, so as to obtain the fabric texture x and moiré pattern m:

m＝Dα(20)。 m=Dα(20).

本发明的技术特点及效果： Technical characteristics and effects of the present invention:

本发明方法避免了对成像设备硬件上的改变,采用后处理方法,通过用低秩稀疏矩阵分解模型,将图像中织物纹理与摩尔纹图案分离,即实现了织物图像摩尔纹的消除。具有以下特点： The method of the invention avoids changes to the hardware of the imaging device, adopts a post-processing method, and uses a low-rank sparse matrix decomposition model to separate the fabric texture from the moiré pattern in the image, thereby realizing the elimination of the moiré pattern in the fabric image. Has the following characteristics:

1、程序简单，易于实现。 1. The program is simple and easy to implement.

2、实现了织物图像中摩尔纹的消除。 2. Realized the elimination of moiré pattern in the fabric image.

3、采用低秩系数矩阵分解模型对织物图像摩尔纹进行去除：将织物图像中摩尔纹的消除问题归结于图像分割问题，利用织物纹理的局部自相似性以及摩尔纹在频率域的能量集中分布特性，构建低秩稀疏矩阵分解模型，将织物纹理与摩尔纹图案分离。这样可以很好地利用织物纹理与摩尔纹在空间域和频率域的不同点，更好地区分织物纹理成分与摩尔纹成分。 3. Use the low-rank coefficient matrix decomposition model to remove the moiré pattern of the fabric image: attribute the elimination of the moiré pattern in the fabric image to the image segmentation problem, and use the local self-similarity of the fabric texture and the concentrated energy distribution of the moiré pattern in the frequency domain features, building a low-rank sparse matrix factorization model to separate fabric texture from moiré patterns. In this way, the difference between the fabric texture and the moiré pattern in the space domain and the frequency domain can be well utilized, and the fabric texture component and the moiré pattern component can be better distinguished.

4、通过对摩尔纹图案在DCT字典下稀疏表示的系数进行有效位置约束，提高了织物纹理与摩尔纹图案的分离性能。这样可以保证不会错误地把织物图像中的褶皱、边缘、阴影等位置误分为摩尔纹成分。 4. By constraining the effective positions of the coefficients of the moiré pattern sparsely represented under the DCT dictionary, the separation performance of the fabric texture and the moiré pattern is improved. This ensures that the folds, edges, shadows, etc. in the fabric image will not be mistakenly classified as moiré components.

附图说明 Description of drawings

本发明上述的优点从下面结合附图对实施例的描述中将变得明显和容易理解，其中： The above-mentioned advantages of the present invention will become obvious and easy to understand from the following description of the embodiments in conjunction with the accompanying drawings, wherein:

图1是本发明方法的流程图； Fig. 1 is a flow chart of the inventive method;

图2是带摩尔纹的图像块与不带摩尔纹的图像块的DCT系数的统计分布对比图：1)是带摩尔纹的图像块的DCT系数的统计分布图；2)是不含摩尔纹的图像块的DCT系数统计分布图；3)两者的差异图； Fig. 2 is a comparison chart of the statistical distribution of DCT coefficients of image blocks with moiré and image blocks without moiré: 1) is a statistical distribution map of DCT coefficients of image blocks with moiré; 2) is without moiré The DCT coefficient statistical distribution diagram of the image block; 3) the difference diagram of the two;

图3是带摩尔纹的织物图像； Figure 3 is a fabric image with moiré patterns;

图4是织物图像摩尔纹消除结果对比图：1)是Photoshop软件对图3的处理结果；2)是本发明方法对图3的处理结果。 Fig. 4 is the comparison chart of the moiré elimination results of fabric images: 1) is the processing result of Fig. 3 by Photoshop software; 2) is the processing result of Fig. 3 by the method of the present invention.

具体实施方式 detailed description

下面结合实施例和附图对本发明基于低秩稀疏矩阵分解的织物图像摩尔纹消除方法做出详细说明。 The method for eliminating moiré in fabric images based on low-rank sparse matrix decomposition of the present invention will be described in detail below in conjunction with the embodiments and accompanying drawings.

本发明采用低秩稀疏矩阵分解模型对织物图像摩尔纹进行去除：将织物图像中摩尔纹的消除问题归结于图像分割问题，利用织物纹理的局部自相似性以及摩尔纹在频率域的能量集中分布特性，构建低秩稀疏矩阵分解模型，将织物纹理与摩尔纹图案分离；通过对摩尔纹在频率域的分布进行约束，提高了织物纹理与摩尔纹图案的分离性能避免将织物图像中的褶皱、边缘、阴影等位置误判为摩尔纹图案。具体方法包括以下步骤： The invention adopts a low-rank sparse matrix decomposition model to remove the moiré pattern of the fabric image: the problem of eliminating the moiré pattern in the fabric image is attributed to the image segmentation problem, and the local self-similarity of the fabric texture and the energy concentration distribution of the moiré pattern in the frequency domain are used characteristics, build a low-rank sparse matrix factorization model, and separate the fabric texture from the moiré pattern; by constraining the distribution of the moiré pattern in the frequency domain, the separation performance of the fabric texture and the moiré pattern is improved to avoid the folds and moiré patterns in the fabric image. Edges, shadows, etc. are misjudged as moiré patterns. The specific method includes the following steps:

1)构造初始数据： 1) Construct initial data:

11)采用从谷歌上下载的带有摩尔纹的织物图像作为实验数据； 11) Adopt the fabric image with moiré patterns downloaded from Google as the experimental data;

12)以u/2为步进间隔有重叠地将图像I_M划分成u×u大小的图像块，然后每个图像块y′进行处理，以实现织物纹理x′与摩尔纹图案m′的分离，即： 12) Divide the image I _M into image blocks of u×u size with u/2 as the step interval with overlapping, and then process each image block y' to realize the texture x' of the fabric and the moiré pattern m' separate, that is:

y′＝x′+m′(1) y'=x'+m'(1)

13)考虑到离散余弦变换(DCT)域比离散傅立叶变换(DFT)域有更高的分辨率，因此构建N×p大小的过完备DCT字典来稀疏表示摩尔纹图案。 13) Considering that the discrete cosine transform (DCT) domain has a higher resolution than the discrete Fourier transform (DFT) domain, an over-complete DCT dictionary of size N×p is constructed to sparsely represent moiré patterns.

2)构建稀疏约束项：利用摩尔纹图案在频率域的能量集中分布特性，假设摩尔纹图案在DCT字典下是可以稀疏表示的，公式如下： 2) Construct the sparse constraint item: Utilize the energy concentration and distribution characteristics of the moiré pattern in the frequency domain, assuming that the moiré pattern can be sparsely represented under the DCT dictionary, the formula is as follows:

其中，min表示求对应的最小值，||·||₁表示矩阵的范数，是构建的DCT字典，表示实数空间，N,p表示空间维度的大小，表示大小N×p的二维实数空间且N＝u²,p>>N，符号∈表示属于，m是待分离的摩尔纹图像块m′的向量形式，α是摩尔纹m在字典下稀疏表示的系数矩阵，Ω是对系数α有效位置的约束，即将非有效位置处的系数α的值全部置零； Among them, min means to seek the corresponding minimum value, ||·|| ₁ means the matrix norm, is the constructed DCT dictionary, Represents the real number space, N, p represents the size of the space dimension, Represents a two-dimensional real number space of size N×p and N=u ² , p>>N, the symbol ∈ represents belonging to, m is the vector form of the moiré image block m′ to be separated, α is the moiré pattern m is sparse under the dictionary The coefficient matrix represented by Ω is the constraint on the effective position of the coefficient α, that is, the value of the coefficient α at the non-effective position is all set to zero;

21)某些情况下，彩色图像的红、蓝两通道有摩尔纹图案而绿色通道不包含任何摩尔纹成分。所以选取红、蓝通道的带摩尔纹的图像块若干及绿色通道对应位置的不带摩尔纹的图像块若干，分别得到带摩尔纹的图像块与不带摩尔纹的图像块对应的DCT系数的统计分布图，得出结论：摩尔纹图案在频率域能量集中分布在中低频位置。因此将α对应高频位置的系数全部置零，即Ω为DCT域的中低频位置； 21) In some cases, the red and blue channels of a color image have moiré patterns and the green channel does not contain any moiré components. Therefore, select a number of image blocks with moiré patterns in the red and blue channels and a number of image blocks without moiré patterns in the corresponding positions of the green channel, and obtain the corresponding DCT coefficients of the image blocks with moiré patterns and the image blocks without moiré patterns Statistical distribution diagram, it is concluded that the energy of moiré patterns in the frequency domain is concentrated in the middle and low frequency positions. Therefore, all the coefficients of α corresponding to the high frequency position are set to zero, that is, Ω is the middle and low frequency position of the DCT domain;

3)构建低秩约束项：将织物纹理图像块x′分割成k×k的子块并转换成对应的向量形式，将这些子块向量排成矩阵C，利用织物纹理在空间域的局部自相似特性，添加低秩约束如下： 3) Construct a low-rank constraint item: divide the fabric texture image block x′ into k×k sub-blocks and convert them into corresponding vector forms, arrange these sub-block vectors into a matrix C, and use the local automaticity of the fabric texture in the spatial domain Similar properties, adding low-rank constraints as follows:

min||C||_*(3) min||C|| _* (3)

其中，||·||_*表示矩阵的核范数，C＝[c₁，c₂,…,c_n]，其中是第i个子块的向量形式，n是子块的个数，k²表示空间维度的大小，表示大小为k²的一维实数空间(即向量)。矩阵C的构建方程如下： Among them, ||·|| _* represents the nuclear norm of the matrix, C=[c ₁ , c ₂ ,…,c _n ], where is the vector form of the i-th sub-block, n is the number of sub-blocks, k ² represents the size of the spatial dimension, Represents a one ^- dimensional space of real numbers (ie, a vector) of size k2. The construction equation of matrix C is as follows:

其中，符号是克罗内克算子，I_n是一个n×n的对角线矩阵，x织物纹理图像块x′向量形式，S＝[s₁，s₂,…,s_n]是用来提取图像块子块的矩阵，满足c_i＝s_ix,i＝1,2,…,n。 Among them, the symbol is the Kronecker operator, I _n is an n×n diagonal matrix, x fabric texture image block x′ vector form, S=[s ₁ , s ₂ ,…,s _n ] is used to extract the image The matrix of blocks and sub-blocks satisfies c _i =s _i x,i=1,2,...,n.

min||C||_*+λ||α||₁ min||C|| _* +λ||α|| ₁

其中，λ是加权因子，平衡两项之间的权重。为了降低多个约束条件导致的收敛结果的误差，本发明将(5)式中的约束条件合并，得到如下的分解模型： Among them, λ is a weighting factor, which balances the weight between the two terms. In order to reduce the error of the convergence result caused by multiple constraints, the present invention combines the constraints in formula (5) to obtain the following decomposition model:

min||C||_*+λ||α||₁ min||C|| _* +λ||α|| ₁

最近有关低秩矩阵恢复和稀疏表示的研究成果表明，给上述最小化模型添加权重能够提高所得结果的低秩性和稀疏性。因此，本发明在(6)式中引入了重加权策略，获得如下方程： Recent work on low-rank matrix recovery and sparse representation shows that adding weights to the minimization model described above can improve the low-rank and sparsity of the resulting results. Therefore, the present invention has introduced reweighting strategy in (6) formula, obtains following equation:

mintr(W_CοΣ)+λ||W_αοα||₁ mintr(W _C οΣ)+λ||W _α οα|| ₁

其中,tr(·)表示矩阵的迹，是一个由矩阵C的奇异值构成的对角线矩阵，diag(·)表示对角线矩阵，是矩阵的奇异值，W_C和W_α分别是低秩项和稀疏项对应的权重矩阵，符号ο表示两个矩阵的点乘运算。 Among them, tr( ) represents the trace of the matrix, is a diagonal matrix composed of the singular values of the matrix C, diag(·) represents the diagonal matrix, is the singular value of the matrix, W _C and W _α are the weight matrices corresponding to the low-rank items and sparse items respectively, and the symbol ο indicates the point multiplication operation of the two matrices.

L(C,α,Q)即为增广拉格朗日函数。其中，μ是惩罚因子，Q为拉格朗日乘子矩阵，<·，·>表示两个矩阵的内积，||·||_F表示矩阵的弗罗贝尼乌斯范数； L(C,α,Q) is the augmented Lagrangian function. Among them, μ is the penalty factor, Q is the Lagrange multiplier matrix, <·,·> represents the inner product of two matrices, ||·|| _F represents the Frobenius norm of the matrix;

上式中argmin_X{·}表示使目标函数最小值时变量X的值，ρ为倍数因子，上标k,k+1以及τ,τ+1是迭代次数，ξ为任意小的正实数，下标g表示矩阵元素的位置坐标； In the above formula, argmin _X { } represents the value of variable X when the objective function is minimized, ρ is the multiple factor, superscript k, k+1 and τ, τ+1 are the number of iterations, ξ is any small positive real number, The subscript g represents the position coordinates of the matrix elements;

53)求解C^k+1,τ,去掉式子(9)中求解C^k+1,τ的目标函数里与C无关的项得到: 53) To solve for C ^k+1,τ , remove the item that has nothing to do with C in the objective function of solving C ^k+1,τ in formula (9):

其中： in:

对于式(10)使用奇异值阈值分解法解得： For formula (10), use the singular value threshold decomposition method to solve:

其中U^k,τ，V^k,τ分别是P^k,τ的左奇异矩阵和右奇异矩阵，soft(·，·)为收缩算子。 Among them, U ^{k, τ} , V ^{k, τ} are the left singular matrix and right singular matrix of P ^k, τ respectively, and soft(·,·) is the contraction operator.

使用泰勒展开的方法，构造出一个二阶函数来逼近上式，然后针对这个二阶的函数来求解原方程。 Using the method of Taylor expansion, a second-order function is constructed to approximate the above formula, and then the original equation is solved for this second-order function.

其中，为f(β)的梯度，L_f是一个常数值为用来保证对所有的β都有F(β)≤T(α,β)。 in, is the gradient of f(β), L _f is a constant value It is used to ensure that F(β)≤T(α,β) for all β.

t_j是一组常数序列，j和j+1是迭代次数。由此解得： t _j is a set of constant sequences, and j and j+1 are the number of iterations. From this we get:

μ^k+1,τ＝ρμ^k,τ,k＝k+1(17) μ ^k+1,τ =ρμ ^k,τ ,k=k+1(17)

57)当τ达到重加权矩阵的最大迭代次数时，可以得到我们期望的低秩矩阵C和摩尔纹图案m在DCT字典下稀疏表示的系数矩阵α。从而可以得到织物纹理x与摩尔纹图案m： 57) When τ reaches the maximum number of iterations of the reweighting matrix, we can get our expected low-rank matrix C and the coefficient matrix α sparsely represented by the moiré pattern m under the DCT dictionary. Thus, the fabric texture x and the moiré pattern m can be obtained:

m＝Dα(20) m=Dα(20)

其中，是求和公式，(·)^-1表示括号内矩阵的逆。 in, is the summation formula, and (·) ^-1 represents the inverse of the matrix within the brackets.

本发明提出了一种低秩稀疏矩阵分解的图像摩尔纹消除方法(如图1的流程所示)，结合附图及实施例详细说明如下： The present invention proposes a kind of image moiré elimination method of low-rank sparse matrix decomposition (as shown in the flow process of Fig. 1), detailed description is as follows in conjunction with accompanying drawing and embodiment:

1)构造初始数据： 1) Construct initial data:

12)以32为步进间隔有重叠地将图像I_M(如图3所示)划分成64×64大小的图像块，然后每个图像块y′进行处理，以实现织物纹理x′与摩尔纹图案m′的分离，即： 12) Divide the image I _M (as shown in Figure 3) into image blocks of 64×64 size with a step interval of 32, and then process each image block y′ to realize the fabric texture x′ and Moore The separation of grain pattern m′, namely:

y′＝x′+m′(1) y'=x'+m'(1)

13)考虑到离散余弦变换(DCT)域比离散傅立叶变换(DFT)域有更高的分辨率，因此构建4096×6400大小的过完备DCT字典来稀疏表示摩尔纹图案； 13) Considering that the discrete cosine transform (DCT) domain has a higher resolution than the discrete Fourier transform (DFT) domain, an over-complete DCT dictionary with a size of 4096×6400 is constructed to sparsely represent moiré patterns;

其中，min表示求对应的最小值，||·||₁表示矩阵的范数，是构建的DCT字典，表示实数空间，4096,6400是空间维度的大小，表示大小4096×6400的二维实数空间，符号∈表示属于，m是待分离的摩尔纹图像块m′的向量形式，α是摩尔纹m在字典下稀疏表示的系数矩阵，Ω是对系数α有效位置的约束，即将非有效位置处的系数α的值全部置零； Among them, min means to seek the corresponding minimum value, ||·|| ₁ means the matrix norm, is the constructed DCT dictionary, Indicates the real number space, 4096,6400 is the size of the space dimension, Represents a two-dimensional real number space with a size of 4096×6400, the symbol ∈ represents belonging to, m is the vector form of the moiré image block m′ to be separated, α is the coefficient matrix sparsely represented by the moiré m in the dictionary, and Ω is the pair coefficient α Constraints on effective positions, that is, all values of coefficient α at non-effective positions are set to zero;

21)选取红、蓝通道的带摩尔纹的图像块若干及绿色通道对应位置的不带摩尔纹的图像块若干，分别得到带摩尔纹的图像块与不带摩尔纹的图像块的DCT系数的统计分布图(如图2所示)，得出结论：摩尔纹图案在频率域能量集中分布在中低频位置。因此将α对应高频位置系数全部置零，即Ω为DCT域的中低频位置； 21) Select a number of image blocks with moiré patterns in the red and blue channels and a number of image blocks without moiré patterns in the corresponding positions of the green channel, and obtain the DCT coefficients of the image blocks with moiré patterns and the image blocks without moiré patterns respectively According to the statistical distribution diagram (as shown in Figure 2), it is concluded that the energy of the moiré pattern in the frequency domain is concentrated in the middle and low frequency positions. Therefore, all the coefficients corresponding to the high frequency position of α are set to zero, that is, Ω is the middle and low frequency position of the DCT domain;

22)以防错误地把织物图像中的褶皱、边缘、阴影等位置划分为为摩尔纹成分，利用范数梯度最小化方法平滑图像块y′，获得包含褶皱、边缘、阴影等却不含摩尔纹与织物纹理的图像块，根据其在DCT域的能量分布，将其能量分布位置处的系数α的取值全部置零,进一步约束系数α的有效位置，即对Ω进行限定； 22) In case of mistakenly dividing the folds, edges, shadows and other positions in the fabric image into moiré components, use The norm gradient minimization method smoothes the image block y′, and obtains an image block that contains wrinkles, edges, shadows, etc. but does not contain moiré patterns and fabric textures. According to its energy distribution in the DCT domain, the coefficient α at the position of its energy distribution is calculated as The values of are all set to zero, further restricting the effective position of the coefficient α, that is, limiting Ω;

3)构建低秩约束项：将织物纹理图像块x′分割成8×8的子块并转换成对应的向量形式，将这些子块向量排成矩阵C，利用织物纹理在空间域的局部自相似特性，添加低秩约束如下： 3) Construct a low-rank constraint item: divide the fabric texture image block x′ into 8×8 sub-blocks and convert them into corresponding vector forms, arrange these sub-block vectors into a matrix C, and use the local self-regulation of the fabric texture in the space domain Similar properties, adding low-rank constraints as follows:

min||C||_*(3) min||C|| _* (3)

其中，||·||_*表示矩阵的核范数，C＝[c_1，c₂,…,c_n]，其中是第i个子块的向量形式，n＝225是子块的个数，64表示空间维度的大小，表示大小为64的一维实数空间(即向量)。矩阵C的构建方程如下： Among them, ||·|| _* represents the nuclear norm of the matrix, C=[c _1, c ₂ ,…,c _n ], where is the vector form of the i-th sub-block, n=225 is the number of sub-blocks, 64 represents the size of the spatial dimension, Represents a one-dimensional real number space (ie, vector) of size 64. The construction equation of matrix C is as follows:

其中，符号是克罗内克算子，I_n是一个225×225的对角线矩阵，x织物纹理图像块x′向量形式，S＝[s_1，s₂,…,s_n]是用来提取图像块子块的矩阵，满足c_i＝s_ix,i＝1,2,…,225。 Among them, the symbol is the Kronecker operator, I _n is a 225×225 diagonal matrix, x fabric texture image block x′ vector form, S=[s _1, s ₂ ,…,s _n ] is used to extract the image The matrix of blocks and sub-blocks satisfies c _i =s _i x, i=1, 2, . . . , 225.

min||C||_*+λ||α||₁ min||C|| _* +λ||α|| ₁

其中，λ是加权因子,取值为0.015，用作平衡两项之间的权重。为了降低多个约束条件导致的收敛结果的误差，本发明将(5)式中的约束条件合并，得到如下的分解模型： Among them, 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 multiple constraints, the present invention combines the constraints in formula (5) to obtain the following decomposition model:

min||C||_*+λ||α||₁ min||C|| _* +λ||α|| ₁

mintr(W_CοΣ)+λ||W_αοα||₁ mintr(W _C οΣ)+λ||W _α οα|| ₁

其中,tr(·)表示矩阵的迹，Σ＝diag([σ₁,σ₂,…,σ₆₄])是一个由矩阵C的奇异值构成的对角线矩阵，diag(·)表示对角线矩阵，σ₁,σ₂,…,σ₆₄是矩阵的奇异值，W_C和W_α分别是低秩项和稀疏项对应的权重矩阵，符号ο表示个两矩阵的点乘运算。 Among them, tr(·) represents the trace of the matrix, Σ=diag([σ ₁ ,σ ₂ ,…,σ ₆₄ ]) is a diagonal matrix composed of the singular values of matrix C, diag(·) represents the diagonal Line matrix, σ ₁ , σ ₂ ,..., σ ₆₄ are the singular values of the matrix, W _C and W _α are the weight matrices corresponding to the low-rank items and sparse items, respectively, and the symbol ο represents the point product operation of two matrices.

L(C,α,Q)即为增广拉格朗日函数。其中，μ是惩罚因子，Q为拉格朗日乘子矩阵，<·，·>表示两个矩阵的内积，||·||_F表示矩阵的弗罗贝尼乌斯范数。 L(C,α,Q) is the augmented Lagrangian function. Among them, μ is the penalty factor, Q is the Lagrange multiplier matrix, <·,·> represents the inner product of two matrices, and ||·|| _F represents the Frobenius norm of the matrix.

上式中argmin_X{·}表示使目标函数最小值时变量X的值，ρ为倍数因子，上标k,k+1以及τ,τ+1是迭代次数，ξ为任意小的正实数，下标g表示矩阵元素的位置坐标。实验中设定初值为： $k = 1, τ = 1, ρ = 1.2, μ = 1.4, C^{1} = Q^{1} = 0, α^{1} = 0, W_{C}^{1} = 1, W_{α}^{1} = 1, ξ = 10^{- 5} .$ In the above formula, argmin _X { } represents the value of variable X when the objective function is minimized, ρ is the multiple factor, superscript k, k+1 and τ, τ+1 are the number of iterations, ξ is any small positive real number, The subscript g represents the position coordinates of the matrix elements. The initial value set in the experiment is: $k = 1, τ = 1, ρ = 1.2, μ = 1.4, C^{1} = Q^{1} = 0, α^{1} = 0, W_{C}^{1} = 1, W_{α}^{1} = 1, ξ = 10^{- 5} .$

53)求解C^k+1,τ,去掉式子(9)中C^k+1,τ的目标函数里与C无关的项得到: 53) Solving for C ^k+1,τ , and removing the items that have nothing to do with C in the objective function of C ^k+1,τ in formula (9):

将上式配方改写成: The above formula formula is rewritten as:

其中： in:

54)使用加速近邻梯度算法求解α^k+1,τ，去掉式子(9)中α^k+1,τ的目标函数里与α无关的项得到如下方程： 54) Use the accelerated neighbor gradient algorithm to solve α ^k+1,τ , and remove the items that have nothing to do with α in the objective function of α ^k+1,τ in formula (9) to get the following equation:

其中，为f(β)的梯度，L_f是一个常数值为，用来保证对所有的β都有F(β)≤T(α,β)。 in, is the gradient of f(β), L _f is a constant value , which is used to ensure that F(β)≤T(α,β) for all β.

t_j是一组常数序列，j和j+1是迭代次数。设定初始值为：j＝1,t₁＝1,β₁＝1。 t _j is a set of constant sequences, and j and j+1 are the number of iterations. The initial values are set as follows: j=1, t ₁ =1, β ₁ =1.

收敛时可以解得： When it converges, it can be solved:

μ^k+1，τ＝ρμ^k,τ,k＝k+1(17) μ ^k+1,τ =ρμ ^k,τ ,k=k+1(17)

56)迭代次数设置为40，达到最大迭代次数即k＝40时，更新权重矩阵及迭代计数参数τ： 56) The number of iterations is set to 40, when the maximum number of iterations is reached, i.e. k=40, update the weight matrix and iteration count parameter τ:

57)重加权矩阵的迭代次数设置为4，即τ＝4时，可以得到我们期望的低秩矩阵C和摩尔纹图案m在DCT字典下稀疏表示的系数向量α。从而可以得到织物纹理x与摩尔纹图案m： 57) The number of iterations of the reweighting matrix is set to 4, that is, when τ=4, we can obtain the coefficient vector α sparsely represented by our expected low-rank matrix C and moiré pattern m under the DCT dictionary. Thus, the fabric texture x and the moiré pattern m can be obtained:

m＝Dα(20) m=Dα(20)

6)对带摩尔纹图案的图像区域所有图像块做上述的处理，最终得到不含摩尔纹图案的织物纹理图像I(如图4所示)和分离出的摩尔纹成分图像M。 6) Perform the above-mentioned processing on all image blocks in the image area with moiré patterns, and finally obtain the fabric texture image I without moiré patterns (as shown in FIG. 4 ) and the separated moiré pattern component image M.

Claims

1. A fabric image moiré elimination method based on low-rank sparse matrix decomposition, is characterized in that, utilizes the local self-similarity of fabric texture and the energy concentration distribution characteristic of moiré pattern in the frequency domain to construct a low-rank sparse matrix decomposition model, Solve the problem of eliminating moiré in fabric images by separating the fabric texture from the moiré pattern.

1) Divide the image I _M into image blocks of u×u size with u/2 as the step interval with overlapping, and then process each image block y' to realize the texture x' of the fabric and the moiré pattern m' separate, that is:

y'=x'+m'(1)

2) Constructing sparse constraint items: Utilizing the energy concentration and distribution characteristics of the moiré pattern in the frequency domain, assuming that the moiré pattern is sparsely represented by a discrete cosine transform (DCT) dictionary, the formula is as follows:

_min‖α‖1 constraint m=Dα,α∈Ω(2)

Among them, min means to find the corresponding minimum value, ‖·‖ ₁ means the l1 norm of the matrix, is the constructed DCT dictionary, Represents the real number space, N, p represents the size of the space dimension, Indicates a two-dimensional real number space with a size of N×p and N=u ² , p>>N, the symbol ∈ indicates belonging, m is the vector form of the moiré image block m′ to be separated, and α is the moiré m under the dictionary Sparse representation of the coefficient matrix, Ω is a constraint on the effective position of the coefficient α, that is, the value of the coefficient α at the non-effective position is all set to zero;

3) Construct a low-rank constraint item: convert the fabric texture image block x′ into k×k sub-blocks and convert them into corresponding vector forms, arrange these sub-block vectors into a matrix C, and use the local self-similarity of the fabric texture in the spatial domain feature, adding the low-rank constraint as follows:

min‖C‖ _* (3)

Among them, ‖·‖ _* represents the kernel norm of the matrix, C=[c ₁ ,c ₂ ,…,c _n ], where is the vector form of the i-th sub-block, n is the number of sub-blocks, k ² represents the size of the spatial dimension, Represents a one-dimensional real number space of size k ² , that is, a vector; the construction equation of matrix C is as follows:

Among them, the symbol is the Kronecker operator, I _n is an n×n diagonal matrix, x fabric texture image block x′ vector form, S=[s ₁ ,s ₂ ,…,s _n ] is used to extract the image The matrix of blocks and sub-blocks satisfies c _i =s _i x,i=1,2,...,n;

4) Combine the sparse constraint item of the moiré pattern and the low-rank constraint item of the fabric texture similar block matrix with a factor λ to construct the final low-rank sparse matrix decomposition equation, the specific formula is as follows:

min‖C‖ _* +λ‖α‖ ₁

Restrictions

Among them, λ is a weighting factor, which balances the weight between the two items. In order to reduce the error of the convergence result caused by multiple constraints, the present invention combines the constraints in formula (5) to obtain the following decomposition model:

min‖C‖ _* +λ‖α‖ ₁

Restrictions

Recent research results on low-rank matrix recovery and sparse representation show that adding weights to the above minimization model can improve the low-rank and sparsity of the obtained results. Therefore, a reweighting strategy is introduced in (6), and the following equation is obtained:

mintr(W _C оΣ)+λ‖W _α оα‖ ₁

Restrictions

Among them, tr( ) represents the trace of the matrix, is a diagonal matrix composed of the singular values of the matrix C, diag(·) represents the diagonal matrix, is the singular value of the matrix, W _C and W _α are the weight matrices corresponding to low-rank items and sparse items respectively, and the symbol о indicates the point multiplication operation of two matrices;

5) Use the augmented Lagrangian method to finally solve the equation, and reach convergence after several iterations, and finally obtain the required low-rank matrix C and the coefficient vector α sparsely represented by the moiré pattern m under the DCT dictionary, so as to obtain the fabric Texture x and moiré pattern m:

m=Dα(20)

in, is the summation formula, (·) ^-1 represents the inverse of the matrix in the brackets;

6) Perform the above-mentioned processing on all image blocks in the image area with moiré pattern, and finally obtain the fabric texture image I without moiré pattern and the separated moiré component image M.

2. the fabric image moiré elimination method based on low rank sparse matrix decomposition as claimed in claim 1, is characterized in that, concrete method comprises the following steps:

The step of constructing the sparse constraint item is further specifically, 21) In some cases, the red and blue channels of the color image have moiré patterns and the green channel does not contain any moiré components, so the images with moiré patterns in the red and blue channels are selected A number of blocks and a number of image blocks without moiré in the corresponding position of the green channel, and the statistical distribution diagrams of the DCT coefficients corresponding to the image block with moiré and the image block without moiré are respectively obtained, and the conclusion is drawn: the moiré pattern is The energy in the frequency domain is concentrated in the middle and low frequency positions, so the coefficients of α corresponding to the high frequency position are all set to zero, that is, Ω is the middle and low frequency position of the DCT domain;

22) In order to prevent the folds, edges, shadows, etc. in the fabric image from being mistakenly divided into moiré components, use the l0 norm gradient minimization method to smooth the image block y′, and obtain the folds, edges, shadows, etc. The image blocks of moiré pattern and fabric texture, according to their energy distribution in the DCT domain, further constrain the effective position of the coefficient α, that is, limit Ω.

3. the fabric image moiré elimination method based on low-rank sparse matrix decomposition as claimed in claim 1, is characterized in that, 5) utilizes augmented Lagrange method to carry out final solution to equation, and concrete steps are as follows:

51) Transform equation (7) into Lagrange function form:

L(C, α, Q) is the augmented Lagrangian function, where μ is the penalty factor, Q is the Lagrange multiplier matrix, <·,·> represents the inner product of two matrices, || ·|| _F represents the Frobenius norm of the matrix;

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

In the above formula, argmin _X { } represents the value of variable X when the objective function is minimized, ρ is the multiple factor, superscript k, k+1 and τ, τ+1 is the number of iterations, ξ is any small positive real number, The subscript g represents the position coordinates of the matrix elements;

53) Solve for C ^{k+1, τ} , remove C k+1 in formula (9), and get in the objective function of ^τ that has nothing to do with C:

Using the formula method, the above formula can be rewritten as:

in:

For formula (10), use the singular value threshold decomposition method to solve:

Where U ^{k, τ} , V ^{k, τ} are the left singular matrix and right singular matrix of P ^k, τ respectively, and soft( , ) is the contraction operator;

54) Use the accelerated neighbor gradient algorithm to solve α ^{k+1, τ} , and remove the item in the objective function for solving α ^{k+1, τ} in equation (9) that has nothing to do with α to get the following equation:

Using the method of Taylor expansion, construct a second-order function to approximate the above formula, and then solve the original equation for this second-order function:

make Then introduce the variable β and define the following function:

in, is the gradient of f(β), L _f is a constant value It is used to ensure that F(β)≤T(α,β) for all β;

After the transformation in the previous step, the formula (12) is transformed into the problem of solving the minimum value of T(α, β _j ), and the following form is obtained through the formula:

in, The update rule of variable β _j is as follows:

t _j is a set of constant sequences, j and j+1 are the number of iterations, thus the solution is:

55) Update penalty factor μ and iteration count parameter k:

μ ^k+1,τ =ρμ ^k,τ ,k=k+1(17)

56) When k reaches the maximum number of iterations, update the weight factor and iteration count parameter τ:

57) When τ reaches the maximum number of iterations of the reweighting matrix, we can obtain the coefficient vector α of the sparse representation of our expected low-rank matrix C and moiré pattern m under the DCT dictionary, so as to obtain the fabric texture x and moiré pattern m:

m=Dα(20).