A robust principal component analysis method based on truncated weighted kernel norm

Abstract

The present application relates to the technical field of computer vision, and especially relates to a robust principal component analysis method based on truncated weighted kernel norm. The present application discloses a robust principal component analysis method based on truncated weighted kernel norm, which uses a weighted truncated kernel norm to approximate a rank function, fully utilizes prior rank information to retain main data components, retains several large singular values without punishment, so that the model obtains a clear low-rank part in low-rank and sparse matrix decomposition applications. Secondly, the weight value of the corresponding component of the smaller singular value is given, so that the punishment of the large singular value is smaller, and the punishment of the small singular value is more, so as to improve the robustness of the model to noise or outliers. Finally, each sub-problem of the model is solved in the form of a closed solution, so that the model is simple and fast to solve.

Description

Technical Field

[0001] This invention relates to the field of computer vision technology, and in particular to a robust principal component analysis method based on truncated weighted nuclear norm. Background Technology

[0002] Due to the rapid development of computer technology, the dimensionality of real data is increasing, which greatly increases the difficulty of data processing. Robust Principal Component Analysis (RPCA), an effective tool that can accurately separate the desired low-rank and sparse components from high-dimensional measurements, has attracted great attention from scholars in recent years. This tool has been widely used in various machine learning and low-visual tasks, including face shadow removal, image processing, fabric defect detection, and video background modeling.

[0003] The goal of RPCA (Real-Time Conversion) is to decompose the original data matrix into the sum of two matrices: a low-rank matrix and a sparse matrix. The low-rank matrix contains the signal in the data, while the sparse matrix contains noise and interference. For example, in video background modeling, since video clips captured by surveillance cameras are usually taken in static scenes with a background, RPCA can decompose the video matrix into two matrices: a low-rank matrix representing background information and a sparse matrix representing foreground information, thus achieving video separation. In facial image acquisition under different lighting conditions, RPCA can decompose facial images into background and foreground images. The background image contains basic structural information of the face, while the foreground image contains texture and detail information, thus achieving accurate removal of facial shadows. Due to its efficiency and effectiveness, RPCA has been widely used in information processing, machine learning, computer vision, and other fields.

[0004] Currently, existing techniques often employ non-convex methods to solve the RPCA problem. To improve the performance of low-rank sparse matrix factorization, an RPCA model based on truncated kernel norms has been proposed. This is because, in reality, the largest singular values ​​often represent the main information of the data and do not affect the rank of the matrix. Therefore, the truncated kernel norm preserves the information of the first r largest singular values, minimizing only the sum of the singular values ​​from r+1 to min(m,n). The specific RPCA model is shown below:

[0005] stX=L+S

[0006] in, r is the preset rank. The advantage of truncating the nuclear norm is that it fully utilizes the prior information of the rank, thus accurately capturing the low-rank and sparse structure in a given matrix. Besides the methods mentioned above, the weighted nuclear norm minimization model (WNNM-RPCA) is also an effective method for solving computer vision problems. Its model is as follows:

[0007] stX=L+S

[0008] in, The weight can be set to Due to the reasonable weight settings, this method can accurately capture the low-rank and sparse structure of data in computer vision applications such as video background modeling. When the weights of the current r singular values ​​are 0 and the weights of the remaining singular values ​​are 1, the truncated nuclear norm can also be regarded as a special case of the weighted nuclear norm. Although the above two models can effectively improve the model performance, the flexibility and robustness of the model need to be improved because they do not simultaneously consider the rank prior information and the weight information of the rank components. Summary of the Invention

[0009] The problem to be solved by the present invention is to provide a robust principal component analysis method based on truncated weighted nuclear norm, which uses the weighted truncated nuclear norm to approximate the rank function, thereby improving the regularization term of the non-convex model and enhancing the model's flexibility and robustness.

[0010] To solve the above-mentioned technical problems and achieve the above-mentioned objectives, the present invention adopts the following technical solution: a robust principal component analysis method based on truncated weighted nuclear norm, characterized by comprising the following steps:

[0011] S1. Convert the video to be tested into several image data, and preprocess the image data;

[0012] S2. Input the preprocessed data into a robust component analysis model based on a weighted truncation nuclear norm, and output the L matrix and S matrix of the preprocessed data;

[0013] S3. Restore the L and S matrices according to the preset data format;

[0014] S4. Obtain the image data after restoring the L matrix and the image data after restoring the S matrix, and display them in video format.

[0015] Furthermore, step S1 also includes the following sub-steps:

[0016] S101. Convert several test images into matrix form;

[0017] S102. Expand several images that have been converted into matrix form column by column to obtain vector data about this image;

[0018] S103. Stack several vector data into a matrix to obtain a matrix composed of several vector data.

[0019] Furthermore, step S2 includes the following sub-steps:

[0020] S201. Establishing an RPCA model by weighted truncated nuclear norm;

[0021] S202. The RPCA model established in step S201 is optimized and solved using the ADMM algorithm.

[0022] Step S202 includes the following sub-steps:

[0023] S202-1. List the augmented Lagrangian function of the RPCA model:

[0024]

[0025] Where A is a Lagrange multiplier, and μ>0 is the penalty coefficient;

[0026] S202-2. Update S, L, A, and μ sequentially using an iterative alternation method. That is, by fixing other variables, minimize S and L sequentially, then update the Lagrange multiplier A and the penalty coefficient μ. The update strategy is as follows:

[0027]

[0028] Specifically, in the (k+1)th iteration, S is first updated. k+1 :

[0029]

[0030] Secondly, update L k+1 :

[0031]

[0032] Finally, update A sequentially. k+1 μ k+1 :Right now

[0033] A k+1 =A k +μ k (XL k+1 -S k+1 ),

[0034] and

[0035] μ k+1 =ρ*μ k (ρ>1).

[0036] S203-3. Determine the iteration condition ||XL k+1 -S k+1 || F / ||X|| F Check if <θ satisfies the condition. If it does, output matrices L and S. If not, return to S202-2 and execute the loop.

[0037] Furthermore, the expression for the RPCA model in step S201 is:

[0038]

[0039] in, w i Singular value σ i The weights of (L). To make model (5) more flexible, the weights are adjusted in the (k+1)th iteration. Set it to the following format:

[0040]

[0041] Where C is a constant term; r is the preset rank.

[0042] Furthermore, the solution method for equation (6) in step S202-2 is as follows:

[0043] Given X∈R m×n The optimization problem satisfies the following form:

[0044]

[0045] The global optimal solution X * =S τ [y], where y is an element of matrix Y. Where S τ [y] = sgn(y)max(|y|-τ,0)

[0046] Seeking

[0047] Furthermore, the solution method for equation (7) in step S202-2 is as follows:

[0048] Given X∈R m×n Assuming m ≥ n, let U∑ G V T Given the SVD decomposition of G, the following optimization problem is solved:

[0049]

[0050] The global optimal solution X * It can be represented by the RSVT operator as: X * =USw (∑ G V T S here w (∑ G ) ii =max((∑ G ) ii -τw i ,0) is the soft threshold operator for weight w.

[0051] make for The SVD decomposition yields the solution to equation (7) as follows:

[0052] The beneficial effects of this invention are as follows: By providing a robust principal component analysis method based on truncated weighted nuclear norms, the method approximates the rank function using weighted truncated nuclear norms, fully utilizes prior rank information to retain the main data components, and preserves several large singular values ​​without penalty, thus enabling the model to obtain a clear low-rank component in low-rank and sparse matrix factorization applications. Secondly, by assigning weights to the corresponding components of smaller singular values, the penalty for large singular values ​​is reduced, while the penalty for small singular values ​​is increased, thereby improving the model's robustness to noise or outliers. Finally, solving each subproblem of this model is done in the form of a closed-form solution, making the solution process simple and fast. Attached Figure Description

[0053] Figure 1 A flowchart of an embodiment of a robust principal component analysis method based on truncated weighted nuclear norm of the present invention.

[0054] Figure 2 The algorithm framework diagram of a robust principal component analysis method based on truncated weighted nuclear norm of the present invention is shown.

[0055] Figure 3 A comparison of the effects of an embodiment of the robust principal component analysis method based on truncated weighted nuclear norm of the present invention.

[0056] Figure 4 A visual representation of an embodiment of a robust principal component analysis method based on truncated weighted nuclear norm of the present invention. Detailed Implementation

[0057] To enable those skilled in the art to better understand the present invention and to make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be further described in detail below with reference to embodiments and accompanying drawings. The illustrative embodiments and descriptions of the present invention are for illustrative purposes only and are not intended to further limit the present invention.

[0058] This invention provides a robust principal component analysis method based on truncated weighted nuclear norm, the algorithm as follows: Figure 1 As shown.

[0059] This invention provides a robust principal component analysis method based on truncated weighted nuclear norm, the process of which is as follows: Figure 2 As shown, taking Curtain from the I2R dataset as an example, this subset of video data contains 200 consecutive images taken from a static camera, each frame being 128*160 pixels in size. In this dataset, we need to separate the teacher from the background, obtaining images containing only the teacher and background.

[0060] S1. The video to be tested is divided into frames, resulting in a total of 200 images.

[0061] S2. Convert the 200 images into a matrix. Specifically, convert each image into a 128*160 matrix, expand it column-wise to form a 20480*1 vector. The specific steps are as follows:

[0062] S201. Establishing an RPCA model by weighted truncated nuclear norm;

[0063] S202. The RPCA model established in step S201 is optimized and solved using the ADMM algorithm.

[0064] Step S202 includes the following sub-steps:

[0065] S202-1. List the augmented Lagrangian function of the RPCA model:

[0066]

[0067] Where A is a Lagrange multiplier, and μ>0 is the penalty coefficient;

[0068] S202-2. Update S, L, A, and μ sequentially using an iterative alternation method. That is, by fixing other variables, minimize S and L sequentially, then update the Lagrange multiplier A and the penalty coefficient μ. The update strategy is as follows:

[0069]

[0070] Specifically, in the (k+1)th iteration, S is first updated. k+1 :

[0071]

[0072]

[0073] Secondly, update Lk+1 :

[0074]

[0075] Finally, update A sequentially. k+1 μ k+1 :Right now

[0076] A k+1 =A k +μ k (XL k+1 -S k+1 ),

[0077] and

[0078] μ k+1 =ρ*μ k (ρ>1).

[0079] S203-3. Determine the iteration condition ||XL k+1 -S k+1 || F / ||X|| F Check if <θ satisfies the condition. If it does, output matrices L and S. If not, return to S202-2 and execute the loop.

[0080] S3. Stack the 200 vectors into a matrix to form a 20480*200 matrix X.

[0081] S4. Input matrix X into the model and obtain matrix L and matrix S by following the steps in Part 4.

[0082] S4. Matrix L has dimensions 20480*200. It is split column-wise into 200 vectors of size 20480*1. For each vector, it is restored to a 128*160 matrix, then converted to RGB image format, thus obtaining one frame of image. After performing column-wise operations on the 200 vectors, 200 consecutive background images are obtained. Matrix S can be obtained similarly, resulting in 200 consecutive images containing only the teacher.

[0083] The final goal is to separate the Curtain dataset into two consecutive sets of images containing only the teacher and background images, and then display them in video format.

[0084] To evaluate the model's performance in separating video background and foreground, we assess it from the perspectives of visual effects and two evaluation metrics, assuming S... re To obtain the restored binary image, and S as the baseline binary image, we first convert the separated image from RGB format to binary format using MRF technology. Then, we measure the similarity between the obtained image and the original image using the following two evaluation metrics:

[0085] (1) Intersection-over-Union (IoU) represents the overlap ratio between the reconstructed image and the reference image, i.e., the ratio of their intersection to their union. A larger IoU indicates that the two images being compared are closer. Ideally, the ratio is 1. The formula for calculating IoU is:

[0086] (2) The F-measure (F) represents the harmonic mean of precision and recall. A larger F-measure indicates higher model precision. The calculation formula is:

[0087] To verify the effectiveness of the model proposed in this patent, this embodiment selects the following seven methods for comparison:

[0088] (1) Inexact Augmented Lagrange Multiplier (IALM): This method uses inexact generalized Lagrange multiplication to solve convex optimization problems.

[0089] (2) The Non-convex RPCA method: This method recovers low-rank and sparse matrices by optimizing the non-convex γ norm minimization problem, where the γ norm provides a closer approximation than the kernel norm.

[0090] (3) Weighted Nuclear Norm Minimization (WNNM): The weighted nuclear norm is used to replace the nuclear norm in the RPCA model. The key to the weighted nuclear norm is to adaptively assign weights to different singular values.

[0091] (4) The Truncated Nuclear Norm Minimization (TNN) method: This method transforms the original RPCA problem into a truncated nuclear norm minimization problem and solves it through a two-step iterative scheme.

[0092] (5) The Generalized Nonconvex Regularization for RPCA (GNR-RPCA). This is a method that uses generalized nonconvex regularization to decompose low-order sparse components from a given matrix.

[0093] (6) Weighted Schatten-p Norm Minimization (WSNM): This method uses the Schatten-p norm instead of the rank function, which can accurately approximate the original RPCA problem.

[0094] (7) Iterated Robust CUR Decomposition Method (IRCUR): This method updates the low-rank components through CUR decomposition, avoiding complex calculations on the entire matrix and improving the computational efficiency of solving the RPCA problem.

[0095] From the perspective of evaluation metrics, the performance comparison results of different algorithms based on IU and F metrics are presented in... Figure 3 It can be seen that the method proposed in this invention is superior to other methods in most cases, except for the Shoppingmall data subset, which is because the foreground remains stationary for too long and is misclassified as background.

[0096] From a visual perspective, Figure 3 The image shows visual results of representative frames from the Curtain, Watersurface, and Video2 sequences. Figure 3 As can be seen in Video 2, IALM, Non-convex RPCA, and GNR-RPCA fail to completely separate the foreground and background, and also fail to detect camouflaged foreground objects. However, WSNM, IRCUR, and TRNN-RPCA outperform other RPCAs in handling textureless foreground and background objects.

[0097] This invention is described from the perspectives of its intended use, effectiveness, progress, and novelty. Its practical and progressive features meet the functional enhancement and use requirements emphasized by the Patent Law. The above description and drawings are merely preferred embodiments of this application and are not intended to limit this application. Therefore, all structures, devices, features, etc., that are similar to or identical to those of this application, i.e., all equivalent substitutions or modifications made in accordance with the scope of this patent application, shall fall within the scope of protection of this patent application.

[0098] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A robust principal component analysis method based on truncated weighted nuclear norm, characterized in that, Includes the following steps: S1. Convert the video to be tested into several image data, and preprocess the set of image data; S2. Input the preprocessed data into a robust component analysis model based on a weighted truncation nuclear norm, and output the L matrix and S matrix of the preprocessed data; S3. Restore the L and S matrices according to the preset data format; S4. Obtain the image data after restoring the L matrix and the image data after restoring the S matrix; Step S2 includes the following sub-steps: S201. Establish the RPCA model by approximating the rank function with a weighted truncated nuclear norm; S202. The RPCA model established in step S201 is optimized and solved using the ADMM algorithm; Step S202 includes the following sub-steps: S202-1. List the augmented Lagrangian function of the RPCA model: Where A is a Lagrange multiplier, >0 is the penalty coefficient; S202-2. Update S, L, A sequentially using an iterative alternation method. That is, by fixing other variables, minimizing S and L sequentially, and then updating the Lagrange multiplier A and the penalty coefficient. Its update strategy is as follows: Specifically, in the (k+1)th iteration, S is first updated. k+1 : （6） Secondly, update L k+1 : （7） Finally, update A sequentially. k+1 μ k+1 :Right now and S203-3. Determining Iteration Conditions If the condition is met, output matrices L and S; otherwise, return to S202-2 and execute the loop. The expression for the RPCA model in step S201 is: (5) in, , Singular values The weights are adjusted in the (k+1)th iteration. Set it to the following format: Where C is a constant term; r is the preset rank.

2. The robust principal component analysis method based on truncated weighted nuclear norm according to claim 1, characterized in that, Step S1 includes the following sub-steps: S101. Convert several test images into matrix form; S102. Expand several images that have been converted into matrix form column by column to obtain vector data about this image; S103. Stack several vector data into a matrix to obtain a matrix composed of several vector data.

3. The robust principal component analysis method based on truncated weighted nuclear norm according to claim 1, characterized in that, The solution method for equation (6) in step S202-2 is as follows: Given The optimization problem satisfies the following form: global optimal solution ,in Let Y be an element of matrix Y, where Seeking .

4. The robust principal component analysis method based on truncated weighted nuclear norm according to claim 1, characterized in that, The solution method for equation (7) in step S202-2 is as follows: Given Assuming ,make for Given the SVD decomposition, the following optimization problem is: global optimal solution It can be represented by the RSVT operator as: , here Weight The soft threshold operator, let for The SVD decomposition yields the solution to equation (7) as follows: .

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