Image denoising method based on double robust principal component analysis of graph

Abstract

The invention discloses a dual robust principal component analysis image denoising method based on a graph, and belongs to the technical field of image processing. The method comprises the following steps: flattening an image to be processed to construct a data matrix in a column vector form; constructing a graph structure by using a K nearest neighbor method, and generating a graph Laplacian matrix based on the graph structure; jointly considering an image reconstruction error, a sparse noise item, a linear mapping error item and a graph structure regular item, and constructing an optimization model; and carrying out variable alternating optimization by adopting an augmented Lagrange multiplier method and an alternating direction solution method, and obtaining an image denoising result according to the low-rank principal component. According to the method, image structure information and a double constraint mechanism are introduced into a robust principal component analysis framework, so that the detail retention capability and the structure consistency of the image are effectively enhanced, the robustness and the visual quality of image denoising are improved, and the method is suitable for application scenes such as image processing, video monitoring and target detection under complex backgrounds.

Description

Technical Field

[0001] The present invention relates to image processing and computer vision technologies, and particularly to a graph-based double robust principal component analysis image denoising method. Background Art

[0002] Images are often disturbed by various noises during actual acquisition and transmission. Especially in application scenarios such as video surveillance, image communication, and medical imaging, high-quality image restoration is of great significance for subsequent object detection, recognition, and analysis. Current image denoising technologies mainly include spatial domain filtering methods, transform domain denoising methods, statistical modeling methods, and low-rank decomposition methods, etc.

[0003] Among them, Robust Principal Component Analysis (RPCA) has become a research hotspot in recent years because it can represent an image as the superposition of a low-rank component and sparse noise. The RPCA model realizes the effective restoration of the image background under certain noise interference by constructing a convex optimization problem. However, the standard RPCA method does not fully utilize the structural information between image frames. Especially when dealing with image sequences with complex spatial distributions and strong structural coherence, problems such as edge blurring and detail loss exist in the restoration results.

[0004] To enhance the ability to preserve image structural information, some studies have begun to introduce the idea of graph regularization, using the graph Laplacian matrix to restrict the principal component direction, so that similar image frames or pixels have more consistent low-rank representations. However, most existing graph-based RPCA methods only impose graph constraints in the principal component restoration stage of the image, and fail to synchronously consider graph structure constraints in residual modeling and transformation structures, resulting in limited model expression ability and insufficient denoising accuracy.

[0005] In addition, many existing methods cannot suppress global noise while preserving image details, or have defects such as high computational complexity, slow convergence speed, and strong sensitivity to parameters. Therefore, there is an urgent need to design a new type of image denoising method that can simultaneously utilize graph structure information and double low-rank sparse constraints to improve image restoration quality and model robustness and meet the image processing requirements in more actual complex scenarios. Summary of the Invention

[0006] Object of the Invention: Aiming at the above problems, the object of the present invention is to provide a graph-based double robust principal component analysis image denoising method. By introducing a graph regularization term into the low-rank term, the structural similarity and local manifold features in the image data are fully exploited, so as to achieve a more refined image denoising effect.

[0007] Technical Solution: The graph-based double robust principal component analysis image denoising method of the present invention includes the following steps: 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; Use the K-nearest neighbor method to construct a graph structure, and generate a graph Laplacian matrix based on this graph structure; Jointly consider the image reconstruction error, sparse noise term, linear mapping error term and graph structure regularization term, and construct an optimization model; For the optimization model, use the augmented Lagrangian multiplier method and the alternating direction solution method to perform variable alternating optimization, gradually solve the low-rank principal components, and obtain the image denoising result according to the low-rank principal components.

[0008] Further, the steps of flattening the image to be processed to construct a data matrix in the form of a column vector include: Uniformly adjust each frame of the image to be processed to a fixed resolution, and flatten it into a vector according to the column-major order rule; Stitch all the images in the column direction to construct an image data matrix , where d is the pixel dimension after flattening a single-frame image, that is, the product of the image height and width, n is the number of image frames, represents the column vector formed after flattening the nth frame image in the sequence. This representation method can convert the original image sequence into a matrix form, which is convenient for subsequent matrix decomposition and optimization processing.

[0009] Further, the steps of using the K-nearest neighbor method to construct a graph structure and generating a graph Laplacian matrix based on this graph structure include: Step 201, based on the image data matrix , taking the column vector as a unit, calculate the Euclidean distance between any two samples , and the formula is: ; Step 202, take each sample as a node in the graph structure, and for each sample x i select the K samples with the smallest Euclidean distance from it as its neighbors to construct a neighbor set , forming an asymmetric K-nearest neighbor relationship; Step 203, define the weighted adjacency matrix of the graph , and its elements are defined as: , where σ is the kernel function bandwidth, 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, and the formula is: , Construct all the node degree values into a degree matrix ; Step 205: Generate a graph Laplacian matrix based on the weighted adjacency matrix and degree matrix as , where Φ represents the change intensity and relative distribution among samples in the graph structure.

[0010] Furthermore, the expression of the optimized 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 the sparse noise, residual penalty, nuclear norm of the transformation matrix, and graph structure regularization term respectively. tr is the trace operation, is the nuclear norm, is norm, is the transpose operation of the matrix; represents the constraint condition.

[0011] Furthermore, for the optimized model, the augmented Lagrangian multiplier method and the alternating direction solution method are used for variable alternating optimization. The steps for gradually solving the low-rank principal component include: Introduce two auxiliary variables N and M to replace A and H - AX in the optimized model. Then the optimized model is expressed as: , , where M is the reconstruction residual matrix and N is the auxiliary variable of A; Use the Lagrangian multiplier method to transform the constrained optimization problem of the optimized model into an unconstrained optimization problem, which is expressed as: , where Y, Q, and R are Lagrangian multipliers, η is the graph regularization term coefficient, and μ is the adjustment parameter, represents the Frobenius norm term.

[0012] Furthermore, for the optimized model, the augmented Lagrangian multiplier method and the alternating direction solution method are used for variable alternating optimization. The steps for gradually solving the low-rank principal component also include: Update the low-rank principal component H, fix other variables, extract the part related to the low-rank principal component H, and solve the sub-problem of H, which is expressed as: , where the parameter ; is the singular value shrinkage operator; The objective function of this sub-problem consists of three parts: the nuclear norm term , the Frobenius norm term , the graph regularization term ; The proximal gradient method is used to solve the sub - problem of H, including: First, calculate the gradient of the smooth part, and the formula is: , Then, perform a gradient descent step update, and the formula is: , In the formula, is the current value of H at the k - th iteration, is the step - size parameter, which controls the step - size of the gradient descent; Perform singular - value soft - thresholding on through the singular - value shrinkage operator , and the formula is: , Let be the singular - value decomposition of , then there is: , where the soft - threshold operator is defined as: , In the formula, x represents the input data; represents the threshold parameter; sgn(x) represents the sign function, which takes the value of 1 when x > 0, - 1 when x < 0, and 0 when x = 0; represents taking the non - negative maximum value; Finally, after the proximal gradient method and singular - value soft - thresholding, the solution of H is expressed as: .

[0013] Furthermore, for the optimization model, the augmented Lagrangian multiplier method and the alternating - direction solution method are used for variable - alternating optimization. The steps to gradually solve the low - rank principal component also include: Solve the sub - problem of S, and the solution of S is expressed as: , where ; Solve the sub - problem of N, and the solution of N is expressed as: , where is the intermediate matrix obtained through the previous variable update; Solve the sub - problem of M, and the solution of M is expressed as: , where is a correction term defined by variable substitution; Solve the sub-problem of A, expressed as: , The sub-problem of A is a standard linear least squares problem, and the solution is in closed form, expressed as: where ; Update the Lagrange multiplier: , where , the constraints are and , is the growth factor, is the maximum allowable step size.

[0014] Furthermore, according to the low-rank principal component H, the image denoising result is obtained, including: Based on the low-rank principal component obtained from the optimization iteration process, determine the final low-rank image denoising component H, with ; Inverse-transform the low-rank image denoising component H in the form of column vectors to restore the corresponding image frames, form a low-rank image sequence, and use the low-rank image sequence as the image denoising result.

[0015] Beneficial effects: Compared with the prior art, the significant advantages of the present invention are: 1. The present invention proposes a graph-based double 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; 2. The present invention effectively enhances the consistency constraint of the low-rank component on the image adjacency structure by introducing a graph-structure-based regularization term into the low-rank term, enabling the restored low-rank background image to better retain edge contours and detailed texture information while maintaining overall smoothness, improving the image quality and visual effect; 3. The present invention further effectively realizes the dynamic fitting between the low-rank term and the input data by jointly introducing constraint terms and regularization terms, improving the low-rank modeling ability, enabling the low-rank term to not only retain the graph structure information but also adaptively adjust the fitting ability through the mapping matrix A, enhancing the flexibility and robustness of the model in complex image scenarios; 4. The present invention does not rely on a fixed singular value truncation threshold. Instead, through the combined action of the graph regularization term and the dynamic mapping constraint, it automatically adjusts the rank structure of the low-rank term, can better adapt to the real features of multi-scale and multi-structure in the image, and can avoid the over-smoothing phenomenon while maintaining global consistency. 5. The present invention can achieve high-quality recovery of the low-rank background during image denoising, has strong ability to retain details, natural edge transition, excellent noise suppression effect, and 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. Description of the Drawings

[0016] Figure 1 is the flowchart of the graph-based double robust principal component analysis image denoising method; Figure 2 is the image denoising result. Detailed Embodiments

[0017] In order to make the objectives, technical solutions and advantages of the present application clearer, the present application will be further described in detail below with reference to the drawings and embodiments.

[0018] The graph-based double robust principal component analysis image denoising method of the present invention effectively enhances the ability to retain image details and structural consistency, improves the robustness and visual quality of image denoising by introducing graph structure information and double constraint mechanism in the robust principal component analysis framework, and is applicable to application scenarios such as image processing, video surveillance and target detection under complex backgrounds. Combining Figure 1 as shown, the method specifically includes the following steps: 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.

[0019] Further, the step of flattening the image to be processed to construct a data matrix in the form of a column vector includes: uniformly adjust each frame of the image to be processed to a fixed resolution, and flatten it into a vector according to the column-major order rule; concatenate all the images in the column direction to construct an image data matrix , where d is the pixel dimension after flattening a single-frame image, that is, the product of the height and width of the image, n is the number of image frames, represents the column vector formed after flattening the nth frame image in the sequence.

[0020] Perform unified size adjustment and vectorization processing on the original image sequence, flatten multiple frames of images into vectors according to the column-major order rule, and concatenate them in the column direction to construct an image data matrix X ∈ , this process can effectively standardize the structure of the input image, unify the spatial dimensions of each frame of the image, and facilitate subsequent matrix decomposition and modeling. In this way, the temporal consistency and spatial redundancy characteristics in the sequence images can be fully utilized, which helps to improve the effects of low-rank modeling and anomaly detection. At the same time, the data expression in matrix form is also convenient for algorithm implementation and accelerated calculation, which is particularly important for the processing and analysis of large-scale image sequences, especially in application scenarios that require real-time processing or efficient compression.

[0021] Step 2: Use the K-nearest neighbor method to construct a graph structure and generate a graph Laplacian matrix based on this graph structure.

[0022] After completing the construction of the image data matrix, to explore the structural relationships between samples, a K-nearest neighbor graph structure is further constructed between the column vectors of the matrix to introduce the graph Laplacian constraint for subsequent modeling.

[0023] Furthermore, the steps of using the K-nearest neighbor method to construct a graph structure and generate a graph Laplacian matrix based on this graph structure include: Step 201: Based on the image data matrix , taking the column vectors as units, regarding each column vector as a sample, calculate the Euclidean distance between any two samples , and the formula is: ; Step 202: Take each sample as a node in the graph structure. For each sample , select the K samples with the smallest Euclidean distance to it as its neighbors, and construct a neighbor set to form an asymmetric K-nearest neighbor relationship; Step 203: Define the weighted adjacency matrix of the graph, and its elements are defined as: , where σ is the kernel function bandwidth, 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, and the formula is: , Construct all the node degree values into a degree matrix ; Step 205: According to the weighted adjacency matrix and the degree matrix, generate the graph Laplacian matrix as , and use Φ 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.

[0024] Through the above process, the local similarity and global manifold structure between image frames can be effectively mined, and the geometric prior of the data can be strengthened. The adjacency weight information contained in the graph Laplacian matrix helps to constrain the smoothness and consistency of each frame image in the local neighborhood during the low-rank modeling process, thereby improving the denoising or background modeling effect. It is particularly valuable when dealing with video sequences with complex dynamic backgrounds or large changes between image frames. This process also lays a good structural foundation for the subsequent introduction of graph regularization terms.

[0025] Step 3: Jointly consider the image reconstruction error, sparse noise term, linear mapping error term, and graph structure regularization term, and construct an optimization model.

[0026] Furthermore, 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 the sparse noise, residual penalty, nuclear norm of the transformation matrix, and graph structure regularization term respectively. tr is the trace operation, is the nuclear norm, is norm, is the transpose operation of the matrix; represents the constraint condition.

[0027] Step 4: For the optimization model, use the augmented Lagrangian multiplier method and the alternating direction solution method to perform variable alternating optimization, and gradually solve the low-rank principal component. Obtain the image denoising result according to the low-rank principal component.

[0028] Furthermore, the steps of using the augmented Lagrangian multiplier method and the alternating direction solution method to perform variable alternating optimization and gradually solve the low-rank principal component for the optimization model include: Introduce two auxiliary variables N and M to replace A and H - AX in the optimization model, then the optimization model is expressed as: , , where M is the reconstruction residual matrix and N is the auxiliary variable of A; Use the Lagrangian multiplier method to transform the constrained optimization problem of the optimization model into an unconstrained optimization problem, which is expressed as: , where Y, Q, and R are Lagrangian multipliers, η is the graph regularization term coefficient, and μ is the adjustment parameter.

[0029] The optimized model is decomposed into six sub-problems for step-by-step solution, enabling the optimized model to more effectively distinguish foreground from background, enhance the consistency of the low-rank background among time frames, and at the same time have good suppression ability for complex noise. The solution process of these six sub-problems includes: Update the low-rank principal component H, fix other variables, extract the part related to the low-rank principal component H, and solve the sub-problem of H, which is expressed as: , In the formula, the parameter ; is the singular value shrinkage operator; The objective function of this sub-problem consists of three parts: the nuclear norm term , which makes H have low rank and extracts the global latent structure in the data; the Frobenius norm term makes H converge to the intermediate matrix ; the graph regularization term , which makes H keep smoothness on the graph structure and enhances local consistency; If there is no graph regularization term, then this formula is a classical SVT problem, and solving is enough; but due to the addition of the graph regularization term, the proximal gradient method is used to solve it here; using the proximal gradient method to solve the sub-problem of H includes: First, calculate the gradient of the smooth part, and the formula is: , Then, perform the gradient descent step update, and the formula is: , In the formula, is the current value of H at the k-th iteration, is the step size parameter, which controls the step size of the gradient descent; Perform singular value soft thresholding (SVT) processing on through the singular value shrinkage operator , and the formula is: , Let be the singular value decomposition (SVD) of , then there is: , where the soft threshold operator is defined as: , In the formula, x represents the input data; denotes the threshold parameter; sgn(x) represents the sign function, where sgn(x) takes the value of 1 when x > 0, -1 when x < 0, and 0 when x = 0; denotes taking the non - negative maximum; Finally, through the proximal gradient method and singular value soft - thresholding processing, the solution of H is expressed as: , By splitting the optimization problem into gradient descent for the smooth term and low - rank prior regularization for the non - smooth term, the gradient descent process maintains stable convergence. The singular value soft - thresholding operation promotes the solution to automatically suppress unimportant singular values in each iteration, thereby enhancing the low - rank property of the solution and effectively improving the solution quality and computational efficiency of the low - rank constraint term in the optimization process.

[0030] Solving the sub - problem of S, the solution of S is expressed as: , where ; By introducing the Lagrange multiplier and variable substitution, it is transformed into a standard soft - threshold problem, and the soft - threshold operator is used for updating; it can effectively ensure the gradual convergence and sparsity of the sparse matrix S; the soft - threshold operator naturally has the property of enhancing the sparsity of the input signal, and can adaptively suppress low - amplitude noise components in each iteration, only retaining significant foreground change information. Therefore, this step plays a key role in achieving efficient decoupling of the background and sparse noise terms in complex scenarios.

[0031] Fixing other variables, solving the sub - problem of N, the solution of N is expressed as: , where , is the intermediate matrix obtained through the previous variable update; this sub - problem is an optimization problem containing the nuclear norm (i.e., the sum of singular values, promoting low - rankness). Since the nuclear norm has the property of promoting low - rankness and the problem structure corresponds to the singular value soft - thresholding problem, it can be efficiently solved by performing a singular value decomposition on Z and applying the soft - threshold operation to the singular values. This operation can effectively compress non - dominant singular values, making the updated N matrix tend to a low - rank structure, which helps to extract the principal components or global structure information from the data.

[0032] Fixing other variables, solving the sub - problem of M, the solution of M is expressed as: , where , is a correction term defined by variable substitution; this problem, like the S sub-problem, can be efficiently solved by applying the soft-threshold operator: through this step, the updated result of M will exhibit sparsity, which can effectively capture the specific sparse structure or outliers modeled by the AX term in the input matrix, further enhancing the model's robustness to complex noise or local outliers; By introducing variable substitution, the optimization objective is transformed into a soft-threshold optimization problem with a regularization term, and the soft-threshold operator can be directly used for

[0033] efficient solution, keeping the update step simple and efficient. This step naturally makes the updated result of M exhibit sparsity, which can effectively capture the specific sparse structure or outliers in the AX term of the input matrix, thereby enhancing the model's robustness to complex noise or local outliers. Through this process, the M term plays a compensatory role in the decomposition for the part that cannot be fully modeled in the AX product term, which helps 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 result. , The sub-problem of A is a standard linear least squares problem, and the solution is in closed form, expressed as: , where .

[0034] In the sub-problem of A, by constructing a standard linear least squares optimization model, the closed-form solution can be efficiently solved, greatly improving the 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 the low-rank component and the linear mapping component in the data. Through the iterative optimization of A, it helps to improve the overall model's ability to model global structure changes, linear trends, and temporal consistency in the input image sequence, while providing a more reasonable decomposition basis for the residual term in the subsequent sub-problems, enhancing the overall expressiveness and generalization ability of the model.

[0035] After completing the update of each sub-variable of S, N, M, A, and H, enter the Lagrange multiplier update stage. Update the Lagrange multiplier: , where, , the constraint is and .

[0036] In this stage, by updating the multipliers Y, Q, and R, the satisfaction degree of the original constraints X = H + S, A = N, and M = AX + H is gradually enhanced, thereby promoting the convergence of the overall model solution and the constraint consistency. At the same time, the algorithm adopts a dynamic step-size growth strategy to adaptively adjust the Lagrangian penalty factor μ, with the formula , where is the growth factor, is the upper limit value. By gradually increasing μ round by round, the penalty intensity for each constraint residual can be effectively enhanced, thereby accelerating the model convergence speed, suppressing the oscillation phenomenon, and improving the algorithm stability and robustness. Overall, the Lagrangian multiplier update stage plays a "correcting deviation" role by dynamically adjusting the penalty term, which helps to balance the structural constraints of the model and the coordination consistency between variables, ensuring that the final solution result has good constraint consistency and global convergence.

[0037] So far, the algorithm has formed a complete alternating optimization solution framework by alternately updating the variables S, N, M, A, and H and supplemented by the dynamic update of the Lagrangian multiplier. Through the above steps, the algorithm can effectively balance low-rankness, sparsity, and structural constraints, and has good modeling ability and practical effects in tasks such as image processing, video analysis, and anomaly detection.

[0038] Furthermore, the image denoising result is obtained according to the low-rank principal component H, including: According to the low-rank principal component obtained from the optimization iteration process , the final low-rank image denoising component H is determined, with ; The low-rank image denoising component H is inverse-transformed and restored to the corresponding image frame in the form of column vectors to form a low-rank image sequence, and the low-rank image sequence is used as the image denoising result.

[0039] To further illustrate the effectiveness and denoising superiority of the graph-based double robust principal component analysis image denoising method described in the present invention, the following examples are used for illustration.

[0040] 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 effect is verified through the CAVIAR1 (Context Aware Vision using Image-based Active Recognition) dataset, and comparisons are made with traditional RPCA, graph-based RPCAG, and PSSV (Partial Sum of Singular Values).

[0041] The CAVIAR1 dataset was released by INRIA and the EC-funded CAVIAR project, mainly for researching behavior analysis in video surveillance scenarios. The CAVIAR1 dataset demonstrates diverse behavior patterns and complex dynamic background conditions in indoor surveillance scenarios. The video sequences included in this dataset widely cover typical surveillance scenarios such as pedestrians moving alone or interactively, background object changes, occlusions, and lighting changes, which can effectively reflect the dynamic complexity and challenges in real video surveillance. This dataset is often used to evaluate the effects of low-rank sparse decomposition, video background modeling, and image denoising algorithms in aspects such as moving object detection, background restoration, and video quality enhancement. The representative video sequences selected in the CAVIAR1 dataset are uniformly processed into sets of consecutive frames, with obvious foreground objects (such as pedestrians) and stable or gradually changing background information within the frames, making it suitable as standard test data for low-rank + sparse modeling tasks. Different video sequences cover diverse surveillance scenario changes, such as changes in the number of people, differences in interactive behaviors, the appearance and disappearance of occlusions, and fluctuations in lighting conditions. Each set of data can reflect the robustness and adaptability of video processing algorithms in dealing with dynamic environments and complex background changes.

[0042] The experiments were evaluated using three metrics: mean squared error (MSE), peak signal-to-noise ratio (PSNR), and structural similarity (SSIM). A lower MSE value indicates a smaller pixel difference between the reconstructed image and the reference image, and a higher image restoration quality; a higher PSNR value indicates a better noise suppression effect in the reconstructed image and a better overall visual quality of the image; the SSIM metric is used to evaluate the consistency of image structure, texture, and contrast. A higher SSIM value indicates that the quality of the restored image is closer to the reference image in terms of visual perception.

[0043] Table 1

[0044] As shown in Table 1, for the mean values of PSNR, MSE, and SSIM of each method in the CAVIAR1 dataset, in the comparative tests of different image denoising methods, the method described in the present invention performs better than other methods both in terms of image restoration quality and structural similarity metrics. In particular, the best results are achieved in metrics such as PSNR, MSE, and SSIM. The DRPCAG method demonstrates extremely high detail retention ability and image quality restoration effect in the visual effect of the image, such as Figure 2 shown in the result graphs of denoising images using different methods. It can be seen from Figure 2 that the present invention can effectively restore a large amount of detailed information in images severely polluted by noise. Even in complex backgrounds and high noise levels, high-definition image reconstruction can still be achieved.

[0045] Compared with the traditional RPCA, PSSV, and RPCAG methods, the image denoising method of the present invention has a significant improvement in the average PSNR and SSIM metrics, and the MSE is the lowest, which fully verifies the superior performance and strong generalization ability of the proposed method in the scenarios of image denoising and low-rank sparse modeling.

[0046] The present invention provides an image denoising method based on graph-regularized double robust principal component analysis (DRPCAG). There are various methods and ways to implement the technical solution. The above is only the preferred embodiment of the present invention. It should be noted that for those of ordinary skill in the art, without departing from the principle and core idea of the present invention, various improvements and optimizations can also be made according to specific application scenarios and actual needs, and these improvements and optimizations should also be regarded as the protection scope of the present invention. Each component not clearly described in this embodiment can be realized by the prior art.

Claims

1. A graph-based double robust principal component analysis image denoising method, characterized in that It includes the following steps: 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; Use the K-nearest neighbor method to construct a graph structure, and generate a graph Laplacian matrix based on this graph structure; Jointly consider the image reconstruction error, sparse noise term, linear mapping error term, and graph structure regularization term to construct an optimization model; For the optimization model, use the augmented Lagrangian multiplier method and the alternating direction solution method to perform variable alternating optimization, gradually solve the low-rank principal component, and obtain the image denoising result according to the low-rank principal component.

2. The graph-based double robust principal component analysis image denoising method according to claim 1, wherein The step of flattening the image to be processed to construct a data matrix in the form of a column vector includes: Uniformly adjust each frame of the image to be processed to a fixed resolution, and flatten it into a vector by column-major order; Stitch all the images in the column direction to construct an image data matrix , where d is the pixel dimension after flattening a single-frame image, that is, the product of the height and width of the image, and n is the number of image frames. represents the column vector formed after flattening the nth frame image in the sequence.

3. The graph-based dual robust principal component analysis image denoising method according to claim 2, wherein The step of using the K-nearest neighbor method to construct a graph structure and generating a graph Laplacian matrix based on this graph structure includes: Step 201, based on the image data matrix , taking column vectors as units, calculate the Euclidean distance between any two samples . The formula is as follows: ； Step 202: Take each sample as a node in the graph structure. For each sample x i select the K samples with the smallest Euclidean distance from it as its neighbors, and construct a neighbor set , forming an asymmetric K-nearest neighbor relationship; Step 203, define the weighted adjacency matrix of the graph , and its elements are defined as: ， In the formula, σ is the kernel function bandwidth, which is used to adjust the sensitivity of the adjacency strength to distance; Step 204, calculate the node degree value corresponding to each node according to the weighted adjacency matrix W, and the formula is: ， Construct a degree matrix from all node degree values ; Step 205: Generate a graph Laplacian matrix based on the weighted adjacency matrix and the degree matrix as , where Φ represents the change intensity and relative distribution among samples in the graph structure.

4. The graph-based double robust principal component analysis image denoising method according to claim 3, wherein The expression of the optimization model is: ， ， Wherein, 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, η are balance parameters, which are respectively used to control the weights of the sparse noise, residual penalty, nuclear norm of the transformation matrix, and graph structure regularization term, tr is the trace operation, is the nuclear norm, is the norm, is the transpose operation of the matrix; represents the constraint condition.

5. The graph-based double robust principal component analysis image denoising method according to claim 4, characterized in that For the optimization model, the steps of using the augmented Lagrangian multiplier method and the alternating direction solution method to perform variable alternating optimization and gradually solve the low-rank principal component include: Introduce two auxiliary variables N and M to replace A and H - AX in the optimization model, then the optimization model is expressed as: ， ， In the formula, M is the reconstruction residual matrix, and N is the auxiliary variable of A; Use the Lagrangian multiplier method to transform the constrained optimization problem of the optimization model into an unconstrained optimization problem, which is expressed as: ， where Y, Q, and R are Lagrange multipliers, η is the coefficient of the graph regularization term, and μ is the adjustment parameter, denotes the Frobenius norm term.

6. The graph-based double robust principal component analysis image denoising method according to claim 5, characterized in that, For the optimization model, the steps of using the augmented Lagrangian multiplier method and the alternating direction solution method to perform variable alternating optimization and gradually solve the low-rank principal component also include: Update the low-rank principal component H, fix other variables, take out the part related to the low-rank principal component H, and solve the sub-problem of H, which is expressed as: ， In the formula, the parameter ; is the singular value shrinkage operator; The objective function of this sub-problem consists of three parts: the nuclear norm term , the Frobenius norm term , and the graph regularization term ; Use the proximal gradient method to solve the sub-problem of H, including: First, calculate the gradient of the smooth part, and the formula is: ， Then perform the gradient descent step update, and the formula is: ， wherein, is the current value of H at the k-th iteration, is the step size parameter that controls the step size of the gradient descent; By means of the singular value shrinkage operator for perform singular value soft thresholding processing, and the formula is: ， Let be the singular value decomposition of, then we have: ， Among them, the soft threshold operator is defined as: ， Where x represents the input data; represents the threshold parameter; sgn(x) represents the sign function, where sgn(x) takes the value of 1 when x > 0, -1 when x < 0, and 0 when x = 0; represents taking the non - negative maximum value; Finally, after the proximal gradient method and singular value soft thresholding processing, the solution of H is expressed as: 。 7. The graph-based double robust principal component analysis image denoising method according to claim 6, wherein For the optimization model, the steps of using the augmented Lagrangian multiplier method and the alternating direction solution method to perform variable alternating optimization and gradually solve the low-rank principal component also include: Solve the sub-problem of S, and the solution of S is expressed as: ， Among them ; Solve the sub-problem of N, and the solution of N is expressed as: ， Among them is the intermediate matrix obtained by updating the previous-step variables; Solve the sub-problem of M, and the solution of M is expressed as: ， wherein is a correction term defined by variable substitution; Solve the sub-problem of A, which is expressed as: ， The sub-problem of A is a standard linear least squares problem, and the solution is a closed form, which is expressed as: , where ; Update the Lagrangian multiplier: ， Among them, , the constraint is and , is the growth factor, is the maximum allowable step size.

8. The graph-based double robust principal component analysis image denoising method according to claim 7, characterized in that, Obtain the image denoising result according to the low-rank principal component H, including: Low-rank principal components obtained according to the optimization iteration process , the final low-rank image denoising component H is determined as ; the low-rank image denoising component H is inversely transformed and restored to the corresponding image frame in the form of column vectors to form a low-rank image sequence, and the low-rank image sequence is used as the image denoising result.

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