CN115965556A - Binary image restoration method - Google Patents

Abstract

The invention relates to a binary image restoration method, which comprises the following steps: collecting an original binary image with noise; inputting an original binary image; constructing a model selection target function based on the original binary image; linearizing a nonlinear constraint condition of the model selection objective function; and (4) selecting a target function for the model, and carrying out iterative gradient solving and linear programming solving until the function value is converged, and finally restoring a binary image. Compared with the prior art, the polynomial kernel image learning coefficient is optimized by utilizing the model to select the image learning objective function, and the noisy binary image is restored by the linearized MSGL constraint condition and the Frank-Wolfe iterative linear programming algorithm.

Description

A binary image restoration method

Technical Field

The present invention relates to the technical field of image processing, and in particular to a binary image restoration method.

Background Art

Graph learning has been extensively studied for more than a decade, and in recent years, there has been a surge in cross-research between statistical machine learning, such as Graph Signal Processing (GSP) and its applications in data classification: social network management, regression-based image quality assessment, image enhancement, and clustering. GSP focuses on processing signals defined on a graph, which consists of pairwise similarities of vertices supporting data and weighted edges representing observations. The key to machine learning-based graph signal processing is to design a signal that captures the relationship between the processed data observations and formulate a signal prior, such as a graph signal smoothness prior, to effectively filter the graph signal, for example, to facilitate data classification through signal extrapolation or label propagation.

Graph structures can be learned from two aspects: 1) collected raw data (such as time series); 2) constructed features from data samples. Because successful graph filtering depends on the underlying graph model being the structure of the collected data, graph learning is crucial. Graph learning methods can be divided into two types: 1) data-driven graph learning, which learns graphs based on collected raw data; 2) feature-enabled graph learning that uses constructed feature learning graphs.

For data-driven graph learning, the focus is on assigning modeling relationships to nodes between multiple spatial and/or temporal observations (graph signals) of a graph. Existing graph learning methods mostly update each graph node by solving a system of linear equations or learning pairwise distance metrics, where the number of optimization variables increases dramatically with the increase in the number of data samples or feature dimensions.

How to determine the number of optimized variables in graph learning methods so as to achieve high-accuracy restoration of noisy binary images is a problem that needs to be solved urgently.

Summary of the invention

The purpose of the present invention is to provide a binary image restoration method to improve the processing performance of binary image pixel restoration in view of the shortcomings of the existing technology in the field of binary image restoration.

The purpose of the present invention can be achieved by the following technical solutions:

A binary image restoration method comprises the following steps:

Collect the original binary image with noise;

Input the original binary image;

Based on the original binary image, construct a model selection objective function;

Linearizing the nonlinear constraints of the model selection objective function;

The objective function of the model is selected to iteratively calculate the gradient and linear programming solution until the function value converges, and finally the binary image is restored.

Furthermore, inputting the original binary image includes inputting multiple features of pixels of the original binary image, and the multiple features include binary image pixel values, pixel horizontal coordinate values, and pixel vertical coordinate values.

Furthermore, the constructing of the model selection objective function comprises the following steps:

Construct a multi-feature adjacency matrix;

Based on the multiple feature adjacency matrix, construct a regularized graph Laplacian matrix;

Based on the regularized graph Laplace matrix, construct a polynomial graph kernel function;

Based on the polynomial kernel function, a model is constructed for pixels in the original binary image to select an objective function.

Furthermore, the multi-feature adjacency matrix is recorded as A, and the expression of the elements in the multi-feature adjacency matrix A is:

Where _fi represents the multiple features of the original binary image pixels, and M is the measurement matrix between the multiple features of the original binary image pixels.

Furthermore, the regularized graph Laplacian matrix is denoted as L, and its expression is:

L＝D ^-1/2 (DA)D ^-1/2

Where D is the degree matrix, and the element expression is:

其表达式为：Furthermore, the polynomial graph kernel function is recorded as

Its expression is:

Where U is the eigenvector matrix of the regularized graph Laplace matrix L, U is an N-order matrix, λ _i is the eigenvalue corresponding to the eigenvector, and β _i represents the polynomial kernel coefficient.

Furthermore, the expression of the model selection objective function is:

Furthermore, the nonlinear constraints of the model selection objective function are linearized, that is, the positive constraints in the model selection objective function are split into inequality constraints, and the model selection objective function expression obtained by linearizing the constraints is:

正定的容许量。Among them, a is the value range limit, μ is a very small value to ensure

Positive tolerance.

Furthermore, the gradient of the given point is calculated for the model selection objective function of the linear transformation of the constraint condition, and the gradient is used as the objective function to obtain the following linear programming problem:

Iterative gradient calculation and linear programming problem solving are then performed to obtain the polynomial kernel coefficients β _i .

的函数值，进而计算求得复原的二值图像像素值

为：Further, based on the obtained polynomial kernel coefficient β _i , the polynomial graph kernel function is obtained:

The function value of , and then calculate the restored binary image pixel value

for:

表示基准二值图像像素的像素值，

表示待复原二值图像像素与基准二值图像像素的正则化图拉普拉斯副矩阵，

表示待复原二值图像像素的正则化图拉普拉斯副矩阵的逆矩阵。in

represents the pixel value of the base binary image pixel,

Represents the regularized graph Laplacian submatrix of the binary image pixels to be restored and the reference binary image pixels,

The inverse matrix of the regularized graph Laplacian matrix representing the binary image pixels to be restored.

Compared with the prior art, the present invention has the following beneficial effects:

1. The present invention optimizes the polynomial kernel graph learning coefficients by using the model selection graph learning objective function, and restores the noisy binary image by linearizing the MSGL constraint conditions and the Frank-Wolfe iterative linear programming algorithm. It has the advantages of high solution efficiency and high accuracy of the restored binary image.

2. The present invention utilizes binary image pixel values, pixel ordinate values, and pixel abscissa values to form a multiple feature set of binary image pixels, and then describes the correlation between binary image pixel points through an adjacency matrix, thereby improving the accuracy of mathematical description of the correlation between pixel points of the binary image acquired under noisy conditions, and ensuring the accuracy of subsequent binary image restoration.

3. The present invention utilizes a degree matrix to regularize the diagonal elements of the graph Laplacian matrix to 1, and at the same time, performs corresponding transformation on the non-diagonal elements of the graph Laplacian matrix, thereby improving the accuracy of pixel restoration of the overall binary image.

4. The present invention avoids the learning of the feature metric relationship in the graph Laplacian matrix by defining the graph Laplacian matrices of neighboring pixel points of binary image pixels at different levels for linear combination, so that the number of optimization variables is the same as the number of neighbor points defined by the user, thereby greatly improving the efficiency of binary image pixel restoration.

5. The present invention constructs a minimization optimization problem, defines data matching terms and complexity penalty terms, and adds positive constraints. Without introducing additional parameters, a convex optimization problem is constructed to solve the coefficients of the polynomial kernel function of binary image pixels, so that the combination coefficients can be solved quickly later.

6. The present invention linearizes the positive definite nonlinear constraints in the binary image pixel model selection objective function MSGL by converting the positive definite nonlinear constraints into a series of linear constraints, each of which is related to the eigenvalues of the Laplacian matrix of the binary image pixel graph, and then can solve MSGL through a linear solver, thereby greatly improving the solution efficiency.

7. The present invention calculates the gradient of a given point on the binary image pixel objective function MSGL transformed by linearization of constraint conditions, and uses the gradient as the objective function to obtain a specific linear programming problem, so that the binary image pixel restoration function avoids nonlinear problems in the solution process and improves the solution efficiency.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of the process of the present invention.

DETAILED DESCRIPTION

The present invention is described in detail below in conjunction with the accompanying drawings and specific embodiments. This embodiment is implemented based on the technical solution of the present invention, and provides a detailed implementation method and specific operation process, but the protection scope of the present invention is not limited to the following embodiments.

The present invention provides a binary image restoration method, which is implemented by a graph learning method based on linear constraint optimization. The method optimizes the polynomial kernel graph learning coefficients by using a model selection graph learning (MSGL) objective function, and restores a noisy binary image by linearizing the constraints of the MSGL objective function and a Frank-Wolfe iterative linear programming algorithm.

The Frank-Wolfe iterative linear programming algorithm is an algorithm for solving nonlinear programming problems under linear constraints. The innovation of the present invention is to combine the linear constraint model selection graph learning objective function MSGL of the Frank-Wolfe iterative linear programming algorithm to automatically restore the noisy binary image.

The purpose of the present invention is to address the deficiencies of the existing technology in the field of binary image restoration and to provide a graph learning method and a binary image restoration method based on linear constraint optimization, so as to improve the restoration accuracy of workers in the field of binary image processing when designing graph learning methods and binary image restoration methods based on linear constraint optimization, and effectively improve the processing performance of binary image pixel restoration.

The present invention belongs to the field of semi-supervised learning. Semi-supervised learning (SSL) is a key issue in the field of pattern recognition and machine learning. It is a learning method that combines supervised learning with unsupervised learning. Semi-supervised learning uses a large amount of unlabeled data and labeled data to perform pattern recognition. When using semi-supervised learning, as few people as possible are required to do the work, while at the same time, it can bring relatively high accuracy.

As shown in Figure 1, the present invention comprises the following steps:

Collect the original binary image with noise;

Input the original binary image;

Based on the original binary image, construct a model selection objective function;

Linearizing the nonlinear constraints of the model selection objective function;

The objective function of the model is selected to iteratively calculate the gradient and linear programming solution until the function value converges, and finally the binary image is restored.

The specific technical solution steps include:

Step 1: Input the initial binary image with noise and construct the multi-feature adjacency matrix A:

Wherein _fi represents the multiple features of the input binary image pixels, including the binary image pixel value, the pixel horizontal coordinate value, and the pixel vertical coordinate value, M is the measurement matrix between the multiple features of the binary image pixels, and _Aij represents the elements of the multiple feature adjacency matrix A.

Step 2: Construct the binary image pixel regularization graph Laplacian matrix:

L＝D ^-1/2 (DA)D ^-1/2

Where D is the degree matrix, and its element size is:

Step 3: Construct binary image pixel polynomial kernel function:

Where U is the eigenvector matrix of the regularized Laplacian matrix L of the binary image pixel, λ _i is the eigenvalue corresponding to the eigenvector, and β _i represents the polynomial kernel coefficient of the binary image pixel.

Step 4: Construct binary image pixel model and select target function MSGL:

Step 5: Binary image pixel objective function MSGL constraint linearization:

可转化为：From step 4, we can see that the second term of g(β)

can be transformed into:

Taking the derivative of the binary image pixel objective function g(β) and setting it to zero yields:

Multiply both sides of the equation by the binary image pixel polynomial kernel coefficient β _i :

From the above derivation, we can further deduce that:

From this, we can derive the following constraint linear transformation binary image pixel model selection objective function MSGL:

正定的容许量。Among them, a is the value range limit, μ is a very small value to ensure

Positive tolerance.

Step 6: Frank-Wolfe iterative solution of binary image pixel objective function:

For the binary image pixel model transformed by linearization of constraints in step 5, select the objective function MSGL to calculate the gradient of a given point, and use the gradient as the objective function, and the following linear programming problem can be obtained:

Iterative gradient calculation and linear programming problem solving are then performed to obtain the binary image pixel polynomial kernel coefficient β _i .

的函数值，进而计算求得复原的二值图像像素值y：Step 7: Obtain the binary image pixel polynomial kernel function using the binary image pixel polynomial kernel coefficient β _i obtained in step 6

The function value of , and then calculate the restored binary image pixel value y:

表示基准二值图像像素的像素值，

表示对应待复原的二值图像像素的正则化图拉普拉斯副矩阵的逆矩阵，

表示对应待复原与基准二值图像像素的正则化图拉普拉斯副矩阵，

表示待复原二值图像像素的像素值。in

represents the pixel value of the base binary image pixel,

represents the inverse matrix of the regularized graph Laplacian submatrix corresponding to the binary image pixels to be restored,

represents the regularized graph Laplacian submatrix corresponding to the pixels of the to-be-restored and reference binary images,

Represents the pixel value of the binary image to be restored.

As shown in Table 1, the accuracy of the binary image pixel restoration in the Matlab image library using the method proposed in the present invention is verified. It can be seen that the method proposed in the present invention has a high accuracy rate in restoring the binary image pixels.

Table 1: Accuracy verification of binary image pixel restoration using MSGL in Matlab image library

The multiple feature adjacency matrix in the step of Figure 1 realizes the use of binary image pixel values, pixel ordinate values, and pixel abscissa values to form multiple feature sets of binary image pixels, and then describes the correlation between binary image pixels through the adjacency matrix, improves the accuracy of mathematical description of the correlation between pixels of the binary image obtained under noise conditions, and ensures the accuracy of subsequent binary image restoration. The specific technical features of this implementation are: based on the binary image pixel values, pixel ordinate values, and characteristic distances between pixel abscissa values of binary image pixels, a feature measurement matrix is constructed, and then the characteristic distance between binary image pixels is defined, thereby constructing a multiple feature adjacency matrix. The innovation of this implementation is to simultaneously use the pixel values of binary image pixels and the spatial positions of the pixels, combine the features in different mode states such as pixel value attributes and spatial attributes, and obtain highly robust multiple features. The technical effect of this implementation is to improve the accuracy of mathematical description of the correlation between pixels of the binary image obtained under noise conditions, and ensure the accuracy of subsequent binary image restoration.

The step of regularizing the graph Laplace matrix in Figure 1 realizes the use of the degree matrix to regularize the diagonal elements of the graph Laplace matrix to 1, and at the same time, the non-diagonal elements of the graph Laplace matrix are transformed accordingly, thereby improving the accuracy of the overall binary image pixel restoration. The specific technical features of this implementation are: using the degree matrix to regularize the diagonal elements of the graph Laplace matrix to 1, and at the same time, the non-diagonal elements of the graph Laplace matrix are transformed accordingly. The innovation of this implementation is to ensure that the diagonal elements of the graph Laplace matrix are all 1, to ensure the uniformity of subsequent binary image pixel restoration, and the technical effect of this implementation is that the diagonal elements of the graph Laplace matrix are regularized to 1, and at the same time, the non-diagonal elements of the graph Laplace matrix are also transformed accordingly.

The step polynomial graph kernel function in Figure 1 realizes the simultaneous consideration of the graph structure of neighboring pixels of binary image pixels at different levels, thereby improving the accuracy of binary image pixel restoration. The specific technical features of this implementation are: linearly combining the graph Laplacian matrices of neighboring pixels of binary image pixels at different levels, and defining the combination coefficient as the optimization variable. The innovation of this implementation is to define the linear combination of the graph Laplacian matrices of neighboring pixels of binary image pixels at different levels, thereby avoiding the learning of the feature metric relationship in the graph Laplacian matrix, so that the number of optimization variables is the same as the number of neighbors defined by the user, greatly improving the efficiency of binary image pixel restoration. The technical effect of this implementation is to linearly combine the graph Laplacian matrices of neighboring pixels of binary image pixels at different levels, define the combination coefficient as the optimization variable, and then quickly solve the combination coefficient subsequently.

The step model selection objective function MSGL (i.e., model selection graph learning function, English name: Model-Selection Graph learning, abbreviated as MSGL) in Figure 1 realizes solving the coefficients of the polynomial graph kernel function of binary image pixels by constructing an optimization problem. The specific technical features of this implementation are: constructing a minimization optimization problem, defining data matching terms and complexity penalty terms, and adding positive definite constraints. The innovation of this implementation is to construct a convex optimization problem for solving the coefficients of the polynomial graph kernel function of binary image pixels without introducing additional parameters, so that the combination coefficients can be solved quickly subsequently. The technical effect of this implementation is that the combination coefficients can be solved quickly subsequently.

The step objective function MSGL constraint linearization in Figure 1 realizes the linearization of the positive definite nonlinear constraint conditions in the binary image pixel model selection objective function MSGL, and then the MSGL can be solved by a linear solver, which greatly improves the solution efficiency. The specific technical features of this implementation are: converting the positive definite nonlinear constraint conditions into a series of linear constraints, each of which is related to the eigenvalues of the Laplacian matrix of the binary image pixel graph. The innovation of this implementation is to linearize the positive definite nonlinear constraint conditions in the binary image pixel model selection objective function MSGL, and then the MSGL can be solved by a linear solver, which greatly improves the solution efficiency. The technical effect of this implementation is that MSGL can be solved by a linear solver, which greatly improves the solution efficiency.

The Frank-Wolfe iterative solution in step 1 of FIG1 realizes the use of linear programming to solve the binary image pixel model selection objective function MSGL with linear constraints, which significantly improves the solution efficiency. The specific technical features of this implementation are: the gradient of the binary image pixel objective function MSGL transformed by the linear transformation of the constraint condition is calculated at a given point, and the gradient is used as the objective function to obtain a specific linear programming problem. The innovation of this implementation is to iteratively calculate the gradient of the binary image pixel objective function MSGL transformed by the linear transformation of the constraint condition at a given point, and use the gradient as the objective function to obtain a specific linear programming problem, so that the binary image pixel restoration function avoids nonlinear problems in the solution process, thereby improving the solution efficiency. The technical effect of this implementation is to iteratively calculate the gradient of the binary image pixel objective function MSGL transformed by the linear transformation of the constraint condition at a given point, and use the gradient as the objective function to obtain a specific linear programming problem, thereby improving the solution efficiency.

The preferred specific embodiments of the present invention are described in detail above. It should be understood that a person skilled in the art can make many modifications and changes based on the concept of the present invention without creative work. Therefore, any technical solution that can be obtained by a person skilled in the art through logical analysis, reasoning or limited experiments based on the concept of the present invention on the basis of the prior art should be within the scope of protection determined by the claims.

Claims (10)

1. A binary image restoration method, characterized in that it comprises the following steps: Collect the original binary image with noise; Input the original binary image; Based on the original binary image, construct a model selection objective function; Linearizing the nonlinear constraints of the model selection objective function; The objective function of the model is selected to iteratively calculate the gradient and linear programming solution until the function value converges, and finally the binary image is restored. 2. A binary image restoration method according to claim 1, characterized in that inputting the original binary image includes inputting multiple features of the original binary image pixels, and the multiple features include binary image pixel values, pixel horizontal coordinate values and pixel vertical coordinate values. 3. A binary image restoration method according to claim 2, characterized in that the step of constructing a model to select an objective function comprises the following steps: Construct a multi-feature adjacency matrix; Based on the multiple feature adjacency matrix, construct a regularized graph Laplacian matrix; Based on the regularized graph Laplace matrix, construct a polynomial graph kernel function; Based on the polynomial kernel function, a model is constructed for pixels in the original binary image to select an objective function. 4. A binary image restoration method according to claim 3, characterized in that the multiple feature adjacency matrix is recorded as A, and the expression of the elements in the multiple feature adjacency matrix A is:

Where _fi represents the multiple features of the original binary image pixels, and M is the measurement matrix between the multiple features of the original binary image pixels. 5. A binary image restoration method according to claim 4, characterized in that the regularized graph Laplacian matrix is denoted as L, and its expression is: L＝D ^-1/2 (DA)D ^-1/2 Where D is the degree matrix, and the element expression is:

6. A binary image restoration method according to claim 5, characterized in that the polynomial kernel function is expressed as

Its expression is:

Where U is the eigenvector matrix of the regularized graph Laplace matrix L, U is an N-order matrix, λ _i is the eigenvalue corresponding to the eigenvector, and β _i represents the polynomial kernel coefficient. 7. A binary image restoration method according to claim 6, characterized in that the expression of the model selection objective function is:

8. A binary image restoration method according to claim 7, characterized in that the nonlinear constraint condition of the model selection objective function is linearized, that is, the positive constraint condition in the model selection objective function is split into inequality constraints, and the model selection objective function expression obtained by linearizing the constraint condition is:

Among them, a is the value range limit, μ is to ensure

Positive tolerance. 9. A binary image restoration method according to claim 8, characterized in that the gradient of the model selection objective function of the linear transformation of the constraint condition is calculated at a given point, and the gradient is used as the objective function to obtain the following linear programming problem:

Iterative gradient calculation and linear programming problem solving are then performed to obtain the polynomial kernel coefficients β _i . 10. A binary image restoration method according to claim 9, characterized in that the polynomial kernel function is obtained based on the obtained polynomial kernel coefficient β _i

The function value of , and then calculate the restored binary image pixel value

for:

in

represents the pixel value of the base binary image pixel,

Represents the regularized graph Laplacian submatrix of the binary image pixels to be restored and the reference binary image pixels,

The inverse matrix of the regularized graph Laplacian matrix representing the binary image pixels to be restored.

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