Graph-based dual robust principal component analysis image denoising method

Abstract

The present invention discloses a graph-based dual robust principal component analysis image denoising method, which belongs to the field of image processing technology. The method includes: flattening the image to be processed to construct a data matrix in the form of a column vector; constructing a graph structure using the K-nearest neighbor method, and generating a graph Laplacian matrix based on the graph structure; jointly considering the image reconstruction error, sparse noise term, linear mapping error term and graph structure regularization term to construct an optimization model; using the augmented Lagrange multiplier method and the alternating direction solution method to perform variable alternating optimization, and obtaining the image denoising result based on the low-rank principal component. The present invention effectively enhances the image detail retention ability and structural consistency by introducing graph structure information and a dual constraint mechanism into the robust principal component analysis framework, thereby improving the robustness and visual quality of image denoising, and is suitable for application scenarios such as image processing, video surveillance and target detection under complex backgrounds.

Description

Technical Field

[0001] The present invention relates to image processing and computer vision technology, and in particular to a graph-based dual robust principal component analysis image denoising method. Background Art

[0002] Images are often subject to various noise interferences during actual acquisition and transmission. This is particularly true in applications such as video surveillance, image communication, and medical imaging. High-quality image restoration is crucial for subsequent target detection, recognition, and analysis. Current image denoising techniques primarily include spatial domain filtering, transform domain denoising, statistical modeling, and low-rank decomposition.

[0003] Robust principal component analysis (RPCA) has become a hot topic of research in recent years due to its ability to represent images as a superposition of low-rank components and sparse noise. The RPCA model effectively recovers image backgrounds in the presence of certain noise interference by formulating a convex optimization problem. However, standard RPCA methods do not fully utilize the structural information between image frames. In particular, when processing image sequences with complex spatial distribution and strong structural coherence, the restored results suffer from blurred edges and loss of detail.

[0004] To enhance the ability to preserve image structural information, some research has begun to introduce graph regularization, using the graph Laplacian matrix to constrain the direction of the principal components, resulting in more consistent low-rank representations for similar image frames or pixels. However, most existing graph-based RPCA methods only impose graph constraints during the principal component recovery phase, failing to simultaneously consider graph structural constraints in residual modeling and transformation structures. This results in limited model expressiveness and insufficient denoising accuracy.

[0005] Furthermore, many existing methods fail to suppress global noise while preserving image details, or suffer from drawbacks such as high computational complexity, slow convergence, and strong parameter sensitivity. Therefore, there is an urgent need to design a new image denoising method that can simultaneously leverage graph structural information and dual low-rank sparsity constraints to improve image restoration quality and model robustness, and adapt to image processing needs in more complex real-world scenarios. Summary of the Invention

[0006] Purpose of the invention: In response to the above problems, the purpose of the present invention is to provide a graph-based dual robust principal component analysis image denoising method. By introducing graph regularization terms in low-rank terms, the structural similarity and local manifold features in image data are fully exploited, thereby achieving a more refined image denoising effect.

[0007] Technical solution: The graph-based dual robust principal component analysis image denoising method of the present invention comprises the following steps:

[0008] Obtain an image to be processed, and flatten the image to be processed to construct a data matrix in the form of a column vector;

[0009] Use the K-nearest neighbor method to construct a graph structure and generate a graph Laplacian matrix based on the graph structure;

[0010] The optimization model is constructed by jointly considering the image reconstruction error, sparse noise term, linear mapping error term and graph structure regularization term.

[0011] For the optimization model, the augmented Lagrange multiplier method and the alternating direction solution method are used to perform variable alternating optimization, gradually solve the low-rank principal components, and obtain the image denoising results based on the low-rank principal components.

[0012] Furthermore, the step of flattening the image to be processed into a data matrix in the form of a column vector includes:

[0013] Each frame of the image to be processed is uniformly adjusted to a fixed resolution and flattened into a vector using the column-major order rule;

[0014] Construct an image data matrix by splicing all images in the column direction , where d is the pixel dimension of a single frame image after flattening, that is, the product of image height and width, and n is the number of image frames. Represents the column vector formed by flattening the n-th frame image in the sequence. This representation can convert the original image sequence into a matrix form, which is convenient for subsequent matrix decomposition and optimization processing.

[0015] Furthermore, the steps of constructing a graph structure using the K-nearest neighbor method and generating a graph Laplacian matrix based on the graph structure include:

[0016] Step 201, based on the image data matrix , in units of column vectors, calculate any two samples The Euclidean distance between them is:

[0017] ;

[0018] Step 202: Each sample is treated as a node in the graph structure. x i Select the K samples with the smallest Euclidean distance as its nearest neighbors and build a nearest neighbor set , forming an asymmetric K nearest neighbor relationship;

[0019] Step 203: Define the weighted adjacency matrix of the graph , whose elements are defined as:

[0020] ,

[0021] Where σ is the bandwidth of the kernel function, which is used to adjust the sensitivity of the adjacency strength to the distance;

[0022] Step 204: Calculate the node degree value corresponding to each node according to the weighted adjacency matrix W. The formula is:

[0023] ,

[0024] Construct all node degree values ​​into a degree matrix ;

[0025] Step 205: Generate the graph Laplacian matrix according to the weighted adjacency matrix and the degree matrix: , using Φ to characterize the change intensity and relative distribution between samples in the graph structure.

[0026] Furthermore, the expression of the optimization model is:

[0027] ,

[0028] ,

[0029] Where X is the input observation matrix, H is the low-rank principal component image representation, S is the sparse noise term, A is the low-rank transformation matrix, and AX is the reconstructed enhanced clean matrix; α, β, γ, and η are balance parameters, which are used to control the weights of sparse noise, residual penalty, transformation matrix nuclear norm, and graph structure regularization term, respectively. tr is the trace operation. is the nuclear norm, yes norm, is the transpose operation of the matrix; Indicates constraints.

[0030] Furthermore, for the optimization model, the augmented Lagrange multiplier method and the alternating direction solution method are used to perform variable alternating optimization. The steps of gradually solving the low-rank principal components include:

[0031] Introducing two auxiliary variables N and M to replace A and H-AX in the optimization model, the optimization model is expressed as:

[0032] ,

[0033] ,

[0034] Where M is the reconstructed residual matrix, and N is the auxiliary variable of A;

[0035] The Lagrange multiplier method is used to transform the constrained optimization problem of the optimization model into an unconstrained optimization problem, which can be expressed as:

[0036] ,

[0037] Where Y, Q and R are Lagrange multipliers, η is the graph regularization coefficient, μ is the adjustment parameter, represents the Frobenius norm term.

[0038] Furthermore, for the optimization model, the augmented Lagrange multiplier method and the alternating direction solution method are used to perform variable alternating optimization, and the steps of gradually solving the low-rank principal components also include:

[0039] Update the low-rank principal component H, fix other variables, extract the parts related to the low-rank principal component H, and solve the sub-problem of H, which can be expressed as:

[0040] ,

[0041] In the formula, the parameters ; is the singular value contraction operator;

[0042] The objective function of this subproblem consists of three parts: the nuclear norm term , the Frobenius norm term , graph regularization term ;

[0043] The proximal gradient method is used to solve the subproblems of H, including:

[0044] First calculate the gradient of the smooth part, the formula is:

[0045] ,

[0046] Then perform the gradient descent step update, the formula is:

[0047] ,

[0048] Where, is the current value of H at the kth iteration, Is the step size parameter, which controls the step size of gradient descent;

[0049] Through the singular value contraction operator right Perform singular value soft thresholding processing, the formula is:

[0050] ,

[0051] make for The singular value decomposition of , then:

[0052] ,

[0053] The soft threshold operator is defined as:

[0054] ,

[0055] Where x represents the input data; represents the threshold parameter; sgn(x) represents the sign function, when x>0, sgn(x) takes the value of 1, when x<0, sgn(x) takes the value of -1, and when x=0, sgn(x) takes the value of 0; Indicates taking the non-negative maximum value;

[0056] Finally, after the proximal gradient method and singular value soft thresholding, the solution of H is expressed as:

[0057] .

[0058] Furthermore, for the optimization model, the augmented Lagrange multiplier method and the alternating direction solution method are used to perform variable alternating optimization, and the steps of gradually solving the low-rank principal components also include:

[0059] Solve the subproblem of S. The solution of S is expressed as:

[0060] ,

[0061] in ;

[0062] Solve the subproblem of N. The solution of N is expressed as:

[0063] ,

[0064] in It is the intermediate matrix obtained by updating the variables in the previous step;

[0065] Solve the subproblem of M. The solution of M is expressed as:

[0066] ,

[0067] in is the correction term defined by variable substitution;

[0068] Solve the subproblem of A, expressed as:

[0069] ,

[0070] The subproblem of A is a standard linear least squares problem, and the solution is closed form, expressed as:

[0071] ,in ;

[0072] Update the Lagrange multipliers:

[0073] ,

[0074] in, , constrained to and , is the growth factor, is the maximum allowed step size.

[0075] Furthermore, the image denoising result is obtained based on the low-rank principal component H, including:

[0076] The low-rank principal components obtained by the optimization iterative process , determine the final low-rank image denoising component H, there is ; The low-rank image denoising component H is inversely transformed into the corresponding image frame in the form of a column vector to form a low-rank image sequence, and the low-rank image sequence is used as the image denoising result.

[0077] Beneficial effects: Compared with the prior art, the present invention has the following significant advantages:

[0078] 1. This paper proposes a graph-based dual robust principal component analysis image denoising method. By introducing a graph regularization term into the low-rank term, it fully exploits the structural similarity and local manifold features in the image data, thereby achieving a more refined image denoising effect.

[0079] 2. This invention introduces a regularization term based on graph structure into the low-rank term , effectively enhances the consistency constraint of low-rank components on the image adjacency structure, so that the restored low-rank background image can better preserve edge contours and detailed texture information while maintaining overall smoothness, thereby improving image quality and visual effects;

[0080] 3. The present invention further introduces Constraints and The regularization term effectively realizes the dynamic fitting between the low-rank term and the input data, improving the low-rank modeling capability. The low-rank term not only preserves the graph structure information, but also adaptively adjusts the fitting capability through the mapping matrix A, enhancing the flexibility and robustness of the model in complex image scenarios.

[0081] 4. The present invention does not rely on a fixed singular value cutoff threshold. Instead, it automatically adjusts the rank structure of low-rank terms through the combined effect of graph regularization terms and dynamic mapping constraints. This can better adapt to the multi-scale and multi-structured real features of the image, while maintaining global consistency and avoiding over-smoothing.

[0082] 5. The present invention can achieve high-quality restoration of low-rank background in the image denoising process, with strong detail retention ability, natural edge transition, and excellent noise suppression effect. The restored image is superior to the existing RPCA, RPCAG and PSSV methods in terms of objective indicators such as MSE, PSNR, SSIM and subjective visual effects, and has broad application value and promotion prospects. BRIEF DESCRIPTION OF THE DRAWINGS

[0083] Figure 1 This is a flowchart of the graph-based dual robust principal component analysis image denoising method;

[0084] Figure 2 is the image denoising result. DETAILED DESCRIPTION

[0085] In order to make the purpose, technical solutions and advantages of this application more clear, this application is further described in detail below with reference to the accompanying drawings and embodiments.

[0086] The graph-based dual robust principal component analysis image denoising method of the present invention effectively enhances the image detail retention capability and structural consistency by introducing graph structure information and dual constraint mechanism into the robust principal component analysis framework, improves the robustness and visual quality of image denoising, and is suitable for application scenarios such as image processing, video surveillance and target detection in complex backgrounds. Figure 1 As shown, the method specifically includes the following steps:

[0087] Step 1: Obtain the image to be processed and flatten the image to be processed to construct a data matrix in the form of a column vector.

[0088] Furthermore, the step of flattening the image to be processed into a data matrix in the form of a column vector includes:

[0089] Each frame of the image to be processed is uniformly adjusted to a fixed resolution and flattened into a vector using the column-major order rule;

[0090] Construct an image data matrix by splicing all images in the column direction , where d is the pixel dimension of a single frame image after flattening, that is, the product of image height and width, and n is the number of image frames. Represents the column vector formed by flattening the n-th frame image in the sequence.

[0091] The original image sequence is uniformly resized and vectorized, and multiple frames of images are flattened into vectors according to the column-major order rule, and then spliced ​​in the column direction to construct the image data matrix X∈ This process effectively normalizes the structure of the input image and unifies the spatial dimensions of each frame, facilitating subsequent matrix decomposition and modeling. This approach fully exploits the temporal consistency and spatial redundancy of sequential images, helping to improve low-rank modeling and anomaly detection. Furthermore, matrix-based data representation facilitates algorithm implementation and accelerates computation, making it particularly important for processing and analyzing large-scale image sequences, especially in applications requiring real-time processing or efficient compression.

[0092] Step 2: Use the K-nearest neighbor method to construct a graph structure and generate a graph Laplacian matrix based on the graph structure.

[0093] After completing the construction of the image data matrix, in order to explore the structural relationship between samples, a K-nearest neighbor graph structure is further constructed between the matrix column vectors to introduce graph Laplace constraints for subsequent modeling.

[0094] Furthermore, the steps of constructing a graph structure using the K-nearest neighbor method and generating a graph Laplacian matrix based on the graph structure include:

[0095] Step 201, based on the image data matrix , taking column vector as unit, taking each column vector as a sample, and calculating any two samples The Euclidean distance between them is:

[0096] ;

[0097] Step 202: Each sample is treated as a node in the graph structure. Select the K samples with the smallest Euclidean distance as its nearest neighbors and build a nearest neighbor set , forming an asymmetric K nearest neighbor relationship;

[0098] Step 203: Define the weighted adjacency matrix of the graph , whose elements are defined as:

[0099] ,

[0100] Where σ is the bandwidth of the kernel function, which is used to adjust the sensitivity of the adjacency strength to the distance;

[0101] Step 204: Calculate the node degree value corresponding to each node according to the weighted adjacency matrix W. The formula is:

[0102] ,

[0103] Construct all node degree values ​​into a degree matrix ;

[0104] Step 205: Generate the graph Laplacian matrix according to the weighted adjacency matrix and the degree matrix: , Φ is used to characterize the change intensity and relative distribution between samples in the graph structure, which is used to constrain the low-rank principal components in the model to have graph structure consistency.

[0105] This process effectively exploits the local similarities and global manifold structure between image frames, strengthening the geometric priors of the data. The adjacency weight information contained in the graph Laplacian matrix helps constrain the smoothness and consistency of each image frame within its local neighborhood during low-rank modeling, thereby improving denoising and background modeling. This is particularly valuable when processing video sequences with complex dynamic backgrounds or large inter-frame variations. This process also lays a good structural foundation for the subsequent introduction of graph regularization.

[0106] Step 3: Jointly consider the image reconstruction error, sparse noise term, linear mapping error term and graph structure regularization term to construct an optimization model.

[0107] Furthermore, the expression of the optimization model is:

[0108] ,

[0109] ,

[0110] Where X is the input observation matrix, H is the low-rank principal component image representation, S is the sparse noise term, A is the low-rank transformation matrix, and AX is the reconstructed enhanced clean matrix; α, β, γ, and η are balance parameters, which are used to control the weights of sparse noise, residual penalty, transformation matrix nuclear norm, and graph structure regularization term, respectively. tr is the trace operation. is the nuclear norm, yes norm, is the transpose operation of the matrix; Indicates constraints.

[0111] Step 4: For the optimization model, the augmented Lagrange multiplier method and the alternating direction solution method are used to perform variable alternating optimization, gradually solve the low-rank principal components, and obtain the image denoising result based on the low-rank principal components.

[0112] Furthermore, for the optimization model, the augmented Lagrange multiplier method and the alternating direction solution method are used to perform variable alternating optimization. The steps of gradually solving the low-rank principal components include:

[0113] Introducing two auxiliary variables N and M to replace A and H-AX in the optimization model, the optimization model is expressed as:

[0114] ,

[0115] ,

[0116] Where M is the reconstructed residual matrix, and N is the auxiliary variable of A;

[0117] The Lagrange multiplier method is used to transform the constrained optimization problem of the optimization model into an unconstrained optimization problem, which can be expressed as:

[0118] ,

[0119] Where Y, Q, and R are Lagrange multipliers, η is the graph regularization coefficient, and μ is the tuning parameter.

[0120] The optimization model is decomposed into six sub-problems and solved step by step, so that the optimization model can more effectively distinguish the foreground and background, enhance the consistency of low-rank background between time frames, and have good suppression ability for complex noise. The solution process of the six sub-problems includes:

[0121] Update the low-rank principal component H, fix other variables, extract the parts related to the low-rank principal component H, and solve the sub-problem of H, which can be expressed as:

[0122] ,

[0123] In the formula, the parameters ; is the singular value contraction operator;

[0124] The objective function of this subproblem consists of three parts: the nuclear norm term , making H low-rank and extracting the global potential structure in the data; Frobenius norm term Make H the intermediate matrix Convergence; graph regularization term , so that H maintains smoothness in the graph structure and enhances local consistency;

[0125] If there is no graph regularization term, then this formula is a classic SVT problem. However, due to the addition of the graph regularization term, the Proximal Gradient Method is used here to solve the problem. The Proximal Gradient Method is used to solve the subproblems of H, including:

[0126] First calculate the gradient of the smooth part, the formula is:

[0127] ,

[0128] Then perform the gradient descent step update, the formula is:

[0129] ,

[0130] Where, is the current value of H at the kth iteration, Is the step size parameter, which controls the step size of gradient descent;

[0131] Through the singular value contraction operator right Perform singular value soft thresholding (SVT) processing, the formula is:

[0132] ,

[0133] make for The singular value decomposition (SVD) of , then:

[0134] ,

[0135] The soft threshold operator is defined as:

[0136] ,

[0137] Where x represents the input data; represents the threshold parameter; sgn(x) represents the sign function, when x>0, sgn(x) takes the value of 1, when x<0, sgn(x) takes the value of -1, and when x=0, sgn(x) takes the value of 0; Indicates taking the non-negative maximum value;

[0138] Finally, after the proximal gradient method and singular value soft thresholding, the solution of H is expressed as:

[0139] ,

[0140] By splitting the optimization problem into smooth term gradient descent and non-smooth term low-rank prior regularization, the gradient descent process maintains smooth convergence, and the singular value soft thresholding operation prompts the solution to automatically suppress unimportant singular values ​​in each iteration, thereby enhancing the low-rank characteristics of the solution and effectively improving the solution quality and computational efficiency of low-rank constraints in the optimization process.

[0141] Solve the subproblem of S. The solution of S is expressed as:

[0142] ,

[0143] in ;

[0144] By introducing Lagrange multipliers and variable substitution, it is transformed into a standard soft threshold problem and the soft threshold operator is used. The update is performed; it effectively ensures the gradual convergence and sparsity of the sparse matrix S. The soft threshold operator naturally enhances the sparsity of the input signal, adaptively suppressing low-amplitude noise components during each iteration, retaining only significant foreground change information. Therefore, this step plays a key role in achieving efficient decoupling of background and sparse noise terms in complex scenes.

[0145] Fix other variables and solve the subproblem of N. The solution of N is expressed as:

[0146] ,

[0147] in , is the intermediate matrix obtained by updating the variables in the previous step; this subproblem is a matrix containing the nuclear norm Because the nuclear norm promotes low rank, and the problem structure corresponds to the singular value soft thresholding problem, it can be efficiently solved by performing a singular value decomposition (SWD) on Z and applying a soft thresholding operation to the singular values. This operation effectively compresses non-dominant singular values, making the updated N matrix tend to have a low-rank structure, which helps to extract principal components or global structural information from the data.

[0148] Fix other variables and solve the subproblem of M. The solution of M is expressed as:

[0149] ,

[0150] in , is a correction term defined by variable substitution; this problem, like the S-subproblem, can be efficiently solved by applying a soft threshold operator: through this step, the update result of M will be sparse, which can effectively capture the specific sparse structure or abnormal terms modeled by the AX term in the input matrix, further enhancing the model's robustness to complex noise or local anomalies;

[0151] By introducing variable substitution, the optimization objective is transformed into a The soft threshold optimization problem of the regularization term can be directly solved by using the soft threshold operator Efficient solution, keeping the update steps simple and efficient. This step makes the updated results of M naturally sparse, which can effectively capture specific sparse structures or abnormal terms in the input matrix AX, thereby enhancing the model's robustness to complex noise or local anomalies. Through this process, the M term in the decomposition compensates for the under-modeled parts of the AX product term, helping to improve the overall model's adaptability to complex structures such as dynamic targets and local perturbations in the image, while ensuring the stability of the global low-rank background structure and the overall consistency of the decomposition results.

[0152] Fix other variables and solve the subproblem of A, which can be expressed as:

[0153] ,

[0154] The subproblem of A is a standard linear least squares problem, and the solution is closed form, expressed as:

[0155] ,in .

[0156] In subproblem A, by constructing a standard linear least-squares optimization model, a closed-form solution can be efficiently obtained, greatly improving computational efficiency and stability. This process enables the updated A to more accurately fit the mapping relationship between H and AX, thereby capturing the coupling characteristics between low-rank components and linear mapping components in the data. Through iterative optimization of A, the overall model's ability to model global structural changes, linear trends, and temporal consistency in the input image sequence is improved. At the same time, it provides a more reasonable decomposition basis for the residual terms in subsequent subproblems, enhancing the model's overall expressiveness and generalization capabilities.

[0157] After completing the update of the sub-variables S, N, M, A and H, the Lagrange multiplier update phase begins. Update the Lagrange multiplier:

[0158] ,

[0159] in, , constrained to and .

[0160] In this stage, the original constraints X=H+S, A=N, and M=AX+H are gradually enhanced by updating the multipliers Y, Q, and R, thereby promoting the convergence of the overall model solution and the constraint consistency. At the same time, the algorithm uses a dynamic step-size growth strategy to adaptively adjust the Lagrangian penalty factor μ, which is: ,in is the growth factor, is the upper limit. By increasing μ over each round, the penalty for each constraint residual can be effectively increased, thereby accelerating model convergence, suppressing oscillation, and improving algorithm stability and robustness. Overall, the Lagrange multiplier update phase dynamically adjusts the penalty term, providing a "corrective" effect. This helps balance the consistency between the model's structural constraints and variables, ensuring good constraint consistency and global convergence in the final solution.

[0161] At this point, the algorithm forms a complete alternating optimization framework by alternatingly updating the variables S, N, M, A, and H, supplemented by dynamic updates of Lagrange multipliers. Through these steps, the algorithm effectively balances low-rank, sparsity, and structural constraints, demonstrating excellent modeling capabilities and practical results in tasks such as image processing, video analysis, and anomaly detection.

[0162] Furthermore, the image denoising result is obtained based on the low-rank principal component H, including:

[0163] The low-rank principal components obtained by the optimization iterative process , determine the final low-rank image denoising component H, there is ; The low-rank image denoising component H is inversely transformed into the corresponding image frame in the form of a column vector to form a low-rank image sequence, and the low-rank image sequence is used as the image denoising result.

[0164] To further illustrate the effectiveness and denoising excellence of the graph-based dual robust principal component analysis image denoising method described in the present invention, the following examples are provided for illustration.

[0165] The image denoising method disclosed in this embodiment is denoted as DRPCAG (Double Robust Principal Component Analysis with Graph). In the experiments of this embodiment, the experimental results are verified using the CAVIAR1 (Context Aware Vision using Image-based Active Recognition) dataset and compared with traditional RPCA, graph-based RPCAG, and PSSV (Partial Sum of Singular Values).

[0166] The CAVIAR1 dataset, released by INRIA and the EC-funded CAVIAR project, is primarily used for research on behavioral analysis in video surveillance scenarios. The CAVIAR1 dataset showcases diverse behavioral patterns and complex dynamic background conditions in indoor surveillance scenarios. The video sequences included in this dataset cover a wide range of typical surveillance scenarios, including individual and interactive pedestrian motion, changing background objects, occlusions, and varying illumination. This dataset effectively reflects the dynamic complexity and challenges inherent in real-world video surveillance. This dataset is commonly used to evaluate the effectiveness of low-rank sparse decomposition, video background modeling, and image denoising algorithms in areas such as moving object detection, background restoration, and video quality enhancement. Representative video sequences selected from the CAVIAR1 dataset are uniformly processed into a continuous set of frames, each containing distinct foreground objects (such as pedestrians) and stable or gradually changing background information, making it suitable as benchmark data for low-rank and sparse modeling tasks. The different video sequences cover a wide range of surveillance scene variations, such as changes in the number of people, differences in interactive behavior, the presence and absence of occlusions, and fluctuating illumination conditions. Each dataset demonstrates the robustness and adaptability of video processing algorithms in handling dynamic environments and complex background changes.

[0167] The experiment used three metrics for evaluation: mean square error (MSE), peak signal-to-noise ratio (PSNR), and structural similarity (SSIM). A lower MSE value indicates smaller pixel differences between the reconstructed image and the reference image, resulting in higher image restoration quality. A higher PSNR value indicates better noise suppression in the reconstructed image, resulting in better overall visual quality. The SSIM metric is used to assess the consistency of image structure, texture, and contrast; a higher SSIM value indicates that the restored image is closer to the reference image in terms of visual perception.

[0168] Table 1

[0169]

[0170] As shown in Table 1, the PSNR, MSE, and SSIM mean values ​​of each method in the CAVIAR1 dataset show that in the comparative test of different image denoising methods, the method of the present invention outperforms other methods in terms of both image restoration quality and structural similarity indicators, especially in terms of PSNR, MSE, SSIM and other indicators. The DRPCAG method shows extremely high detail preservation ability and image quality restoration effect in terms of image visual effects. Figure 2 The results of image denoising using different methods are shown. Figure 2 It can be seen that the present invention can effectively restore a large amount of detail information in images with severe noise pollution, and can still achieve high-definition image reconstruction even in complex backgrounds and high noise levels.

[0171] Compared with the traditional RPCA, PSSV, and RPCAG methods, the image denoising method proposed in the present invention has significant improvements in the average PSNR and SSIM indicators, and has the lowest MSE, which fully verifies the superior performance and strong generalization ability of the proposed method in image denoising and low-rank sparse modeling scenarios.

[0172] This invention provides an image denoising method based on graph regularized dual robust principal component analysis (DRPCAG). Various methods and approaches exist for implementing this technical solution; the above merely illustrates preferred embodiments of the invention. It should be noted that those skilled in the art can, without departing from the principles and core concepts of this invention, make various improvements and optimizations based on specific application scenarios and practical needs, and these improvements and optimizations should also be considered within the scope of this invention. Components not explicitly described in this embodiment can be implemented using existing technologies.

Claims

1. A graph-based dual robust principal component analysis image denoising method, characterized by: The following steps are involved: Obtain an image to be processed, and flatten the image to be processed to construct a data matrix in the form of a column vector; Use the K-nearest neighbor method to construct a graph structure and generate a graph Laplacian matrix based on the graph structure; The optimization model is constructed by jointly considering the image reconstruction error, sparse noise term, linear mapping error term and graph structure regularization term. For the optimization model, the augmented Lagrange multiplier method and the alternating direction solution method are used to perform variable alternating optimization, gradually solving the low-rank principal components, and obtaining the image denoising results based on the low-rank principal components; The expression of the optimization model is: ， ， Where X is the input observation matrix, H is the low-rank principal component image representation, S is the sparse noise term, A is the low-rank transformation matrix, and AX is the reconstructed enhanced clean matrix; α, β, γ, and η are balance parameters, which are used to control the weights of sparse noise, residual penalty, transformation matrix nuclear norm, and graph structure regularization term, respectively. tr is the trace operation. is the nuclear norm, yes norm, is the transpose operation of the matrix; represents the constraints, represents the graph Laplacian matrix; For the optimization model, the augmented Lagrange multiplier method and the alternating direction solution method are used to perform variable alternating optimization. The steps of gradually solving the low-rank principal components include: Introducing two auxiliary variables N and M to replace A and H-AX in the optimization model, the optimization model is expressed as: ， ， Where M is the reconstructed residual matrix, and N is the auxiliary variable of A; The Lagrange multiplier method is used to transform the constrained optimization problem of the optimization model into an unconstrained optimization problem, which can be expressed as: ， Where Y, Q and R are Lagrange multipliers, η is the graph regularization coefficient, μ is the adjustment parameter, represents the Frobenius norm term.

2. The graph-based dual robust principal component analysis image denoising method according to claim 1, characterized in that: The steps of flattening the image to be processed into a data matrix in the form of column vectors include: Each frame of the image to be processed is uniformly adjusted to a fixed resolution and flattened into a vector using the column-major order rule; Construct an image data matrix by splicing all images in the column direction , where d is the pixel dimension of a single frame image after flattening, that is, the product of image height and width, and n is the number of image frames. Represents the column vector formed by flattening the n-th frame image in the sequence.

3. The graph-based dual robust principal component analysis image denoising method according to claim 2, characterized in that: The steps of constructing a graph structure using the K-nearest neighbor method and generating a graph Laplacian matrix based on the graph structure include: Step 201, based on the image data matrix , in units of column vectors, calculate any two samples The Euclidean distance between them is: ； Step 202: Each sample is treated as a node in the graph structure. x i Select the K samples with the smallest Euclidean distance as its nearest neighbors and build a nearest neighbor set , forming an asymmetric K nearest neighbor relationship; Step 203: Define the weighted adjacency matrix of the graph , whose elements are defined as: ， Where σ is the bandwidth of the kernel function, which is used to adjust the sensitivity of the adjacency strength to the distance; Step 204: Calculate the node degree value corresponding to each node according to the weighted adjacency matrix W. The formula is: ， Construct all node degree values ​​into a degree matrix ; Step 205: Generate the graph Laplacian matrix according to the weighted adjacency matrix and the degree matrix: , using Φ to characterize the change intensity and relative distribution between samples in the graph structure.

4. The graph-based dual robust principal component analysis image denoising method according to claim 3, characterized in that: For the optimization model, the augmented Lagrange multiplier method and the alternating direction solution method are used to perform variable alternating optimization. The steps of gradually solving the low-rank principal components also include: Update the low-rank principal component H, fix other variables, extract the parts related to the low-rank principal component H, and solve the sub-problem of H, which can be expressed as: ， In the formula, the parameters ; is the singular value contraction operator; The objective function of this subproblem consists of three parts: the nuclear norm term , the Frobenius norm term , graph regularization term ; The proximal gradient method is used to solve the subproblems of H, including: First calculate the gradient of the smooth part, the formula is: ， Then perform the gradient descent step update, the formula is: ， Where, is the current value of H at the kth iteration, Is the step size parameter, which controls the step size of gradient descent; Through the singular value contraction operator right Perform singular value soft thresholding processing, the formula is: ， make for The singular value decomposition of , then: ， The soft threshold operator is defined as: ， Where x represents the input data; represents the threshold parameter; sgn(x) represents the sign function, when x>0, sgn(x) takes the value of 1, when x<0, sgn(x) takes the value of -1, and when x=0, sgn(x) takes the value of 0; Indicates taking the non-negative maximum value; Finally, after the proximal gradient method and singular value soft thresholding, the solution of H is expressed as: 。 5. The graph-based dual robust principal component analysis image denoising method according to claim 4, characterized in that: For the optimization model, the augmented Lagrange multiplier method and the alternating direction solution method are used to perform variable alternating optimization. The steps of gradually solving the low-rank principal components also include: Solve the subproblem of S. The solution of S is expressed as: ， in ; Solve the subproblem of N. The solution of N is expressed as: ， in It is the intermediate matrix obtained by updating the variables in the previous step; Solve the subproblem of M. The solution of M is expressed as: ， in is the correction term defined by variable substitution; Solve the subproblem of A, expressed as: ， The subproblem of A is a standard linear least squares problem, and the solution is closed form, expressed as: ,in ; Update the Lagrange multipliers: ， in, , constrained to and , is the growth factor, is the maximum allowed step size.

6. The graph-based dual robust principal component analysis image denoising method according to claim 5, characterized in that: The image denoising results are obtained based on the low-rank principal component H, including: The low-rank principal components obtained by the optimization iterative process , determine the final low-rank image denoising component H, there is ; The low-rank image denoising component H is inversely transformed into the corresponding image frame in the form of a column vector to form a low-rank image sequence, and the low-rank image sequence is used as the image denoising result.

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