A Parametric Two-Dimensional Fusion LASSO Model Solution Method

Abstract

This invention discloses a parameterless method for solving a two-dimensional fused LASSO model. First, a two-dimensional signal model is constructed and columnarized. Then, the total variation norm and sparse norm are fused, and constraints are added to the rows and columns of the two-dimensional data to construct a two-dimensional fused LASSO model. Next, a normalized weighted matrix is ​​calculated based on the covariance fitting criterion to determine the optimal regularization parameter in the model, thus deriving the parameterless two-dimensional fused LASSO model. Finally, the model is solved using the CVX toolbox. The reconstruction results of this invention are superior to existing two-dimensional fused LASSO model methods. By establishing a parameterless two-dimensional fused LASSO model, it solves the problem of optimal selection of regularization parameters in existing two-dimensional fused LASSO models, eliminating the need for regularization parameter selection during the calculation process and expanding the application scope of this invention.

Description

Technical Field

[0001] This invention belongs to the field of signal processing technology, specifically relating to a method for solving a parameterless two-dimensional fused LASSO model. Background Technology

[0002] In the field of signal processing, signal reconstruction is a hot research topic. The Least Absolute Shrinkage and Selection Operator (LASSO) is widely used in image processing due to its excellent sparse reconstruction capabilities.

[0003] The paper "Zou J, Zhu M, Li X, et al. Norm regularization method and its application in radar azimuth super-resolution. IEEE International Conference of IEEE Region 10 (TENCON 2013). IEEE, 2013: 1-4" proposes a radar angular super-resolution method based on the LASSO model. It enhances the sparsity of the reconstructed target by introducing a sparse norm, but the reconstruction performance of a single constraint term is limited. The paper "Zhang Q, Zhang Y, Huang Y, et al. TV-sparse super-resolution method for radar forward-looking imaging. IEEE Transactions on Geoscience and Remote Sensing, 2020, 58(9): 6534-6549" proposes a fusion LASSO model solution method based on a combination of total variation norm and sparse norm. Introducing the total variation norm into the LASSO model helps to better recover the target's contour information; however, this method can only process one-dimensional signals and is not suitable for two-dimensional image processing. The paper "Q. Zhang, C. Liu and G. He, 'An Improved Wiener Filter Based on Adaptive SNR MRI Image Denoising Algorithm, 2022 International Conference on Computing, Communication, Perception and Quantum Technology (CCPQT), Xiamen, China, 2022, pp. 164-168'" proposes an improved Wiener filter image reconstruction method; however, this method depends on the noise distribution and has limited reconstruction performance. The paper "Wahid A, Lee HJ. Image denoising method based on directional total variation filtering. International Conference on Information and Communication Technology Convergence (ICTC). IEEE, 2017: 798-802" proposes a two-dimensional signal reconstruction method based on total variation filtering and applies it to the field of image denoising.This method can effectively preserve the edge information of the image, but it relies on several manually adjusted regularization parameters. Summary of the Invention

[0004] To address the aforementioned technical problems, this invention provides a parameterless two-dimensional fusion LASSO model solution method, solving the challenge of optimal selection of regularization parameters.

[0005] The technical solution adopted in this invention is: a parameter-free two-dimensional fusion LASSO model solution method, the specific steps of which are as follows:

[0006] Step 1: Construct a two-dimensional signal model;

[0007] The two-dimensional signal model in image processing is represented as:

[0008] Y = AX + E (1)

[0009] in, Represents a two-dimensional measurement data matrix. Represents the original two-dimensional data matrix. Represents a dictionary matrix. Represents the additive noise matrix; The vector space dimension is represented by M and N, which represent the row and column dimensions of the two-dimensional data matrix, respectively.

[0010] Step 2: Model columnization processing;

[0011] The model is columnarized, and three matrices F1, F2, and F3 are assumed to exist. According to matrix columnarization theory, we can obtain:

[0012]

[0013] Where vec(·) represents the matrix columnization operation, This represents the Kronecker product, (·). T This represents the matrix transpose operation.

[0014] Equation (1) can be transformed into a linear model:

[0015] y = Hx + e (3)

[0016] Where y = vec(Y), x = vec(X), e = vec(E), Denotes the identity matrix, and

[0017] Step 3: Construct a two-dimensional fused LASSO model;

[0018] By simultaneously adding sparsity constraints and total variation constraints to the rows and columns of the model in equation (1), a two-dimensional fused LASSO model is obtained, which can be specifically expressed by the linear model in equation (3):

[0019]

[0020] Where, x′=vec(X T ), where λ represents the regularization parameter, and D represents the total variation operator, represented by the difference matrix:

[0021]

[0022] Where ||Dx||1 represents the fusion constraint on the column data of the two-dimensional matrix, and ||Dx′||1 represents the fusion constraint on the row data of the two-dimensional matrix.

[0023] Step 4: Calculate the normalized weighted matrix;

[0024] The normalization parameter is removed by constructing a normalized weighted matrix to obtain the optimal solution.

[0025] First, let's rephrase equation (4) as follows:

[0026]

[0027] Next, let B = HD -1 B = [b1, b2, ..., b MN According to the covariance fitting criterion, the normalized weighting matrix W x It can be represented as:

[0028] W x =diag([w1,w2,...,w MN (7)

[0029]

[0030] Here, diag(·) represents converting a vector into a diagonal matrix.

[0031] Step 5: Derive the parameter-free two-dimensional fusion LASSO model;

[0032] Assuming uniform noise, according to the covariance fitting criterion, substituting the normalized weighting matrix into equation (4) yields a parameter-free two-dimensional fusion LASSO model:

[0033]

[0034] Step 6: Solve the model;

[0035] The model obtained in step five is a convex optimization problem, which can be solved quickly using the CVX toolbox in Matlab. The CVX solution result is as follows:

[0036]

[0037] in, This indicates the target reconstruction result, and CVX(·) indicates the solution process of the CVX toolbox.

[0038] Finally, by solving the parameterless two-dimensional fusion LASSO model, two-dimensional target reconstruction was achieved, solving the problem of optimal selection of regularization parameters.

[0039] The beneficial effects of this invention are as follows: The method of this invention first constructs a two-dimensional signal model, then performs columnar processing on the two-dimensional signal model. Next, it fuses the total variation norm and the sparse norm, adding constraints to the row and column data of the two-dimensional data respectively, constructing a two-dimensional fused LASSO model. Then, it calculates the normalized weighted matrix according to the covariance fitting criterion, calculates the optimal regularization parameter in the model, and derives a parameter-free two-dimensional fused LASSO model. Finally, it solves the problem using the CVX toolbox. The reconstruction effect of this invention is superior to existing two-dimensional fused LASSO model methods. By establishing a parameter-free two-dimensional fused LASSO model, it solves the problem of optimal selection of regularization parameters in existing two-dimensional fused LASSO models. Since the selection of regularization parameters is not required during the calculation process, the application scope of this invention is expanded. Attached Figure Description

[0040] Figure 1 This is a flowchart of a parameter-free two-dimensional fusion LASSO model solution method according to the present invention.

[0041] Figure 2 This is the original diagram used in the embodiments of the present invention.

[0042] Figure 3 This is a noise map after noise has been added in an embodiment of the present invention.

[0043] Figure 4 The diagram shows the solution results of the existing two-dimensional fusion LASSO model in an embodiment of the present invention.

[0044] Figure 5 The figure shows the solution result of the parameterless two-dimensional fusion LASSO model in an embodiment of the present invention. Detailed Implementation

[0045] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0046] This embodiment uses simulation based on measured data to demonstrate the effectiveness of the proposed method. All steps and conclusions of this invention are verified on the Matlab2020b simulation platform.

[0047] like Figure 1 The flowchart of a parameter-free two-dimensional fusion LASSO model solution method of the present invention is shown below, and the specific steps are as follows:

[0048] Step 1: Construct a two-dimensional signal model;

[0049] The two-dimensional signal model in image processing is represented as follows:

[0050] Y = AX + E (1)

[0051] in, Represents a two-dimensional measurement data matrix. Represents the original two-dimensional data matrix. Represents a dictionary matrix. Represents the additive noise matrix; The vector space dimension is represented by M and N, which represent the row and column dimensions of the two-dimensional data matrix, respectively.

[0052] In this embodiment, the two-dimensional signal model is applied to the field of optical image denoising to denoise images with added Gaussian noise.

[0053] Where M = 90, N = 90, A is the identity matrix, and Y is an image with additive Gaussian noise added, which has a mean of 0 and a variance of 0.01.

[0054] Step 2: Model columnization processing;

[0055] To solve the two-dimensional signal model using optimization methods, the model needs to be columnarized. Assuming there are three matrices F1, F2, and F3, according to matrix columnarization theory, we can obtain:

[0056]

[0057] Where vec(·) represents the matrix columnization operation, This represents the Kronecker product, (·). T This represents the matrix transpose operation.

[0058] Equation (1) can be transformed into a linear model:

[0059] y = Hx + e (3)

[0060] Where y = vec(Y), x = vec(X), e = vec(E), Denotes the identity matrix, and

[0061] A linear model is obtained by columnarization, and this model can be solved using optimization methods.

[0062] Step 3: Construct a two-dimensional fused LASSO model;

[0063] By simultaneously adding sparsity constraints and total variation constraints to the rows and columns of the model in equation (1), a two-dimensional fused LASSO model is obtained, which improves the contour constraint capability of the model and achieves better reconstruction results. In two-dimensional space, the fused LASSO model should not only consider the difference between a feature and the differences to the left and right of that feature, but also the differences above and below that feature. This can be achieved by adding a constraint term to each row and each column of the feature. This method can be specifically expressed by the linear model in equation (3) above:

[0064]

[0065] Where, x′=vec(X T ), where λ represents the regularization parameter used to balance the constraint terms, and D represents the total variation operator, expressed as a difference matrix:

[0066]

[0067] In equation (4), ||Dx||1 represents the fusion constraint on the column data of the two-dimensional matrix, and ||Dx′||1 represents the fusion constraint on the row data of the two-dimensional matrix, thus achieving the two-dimensional constraint effect. Directly solving this model requires selecting a suitable regularization parameter λ, which is very difficult in practical applications.

[0068] In this embodiment, the regularization parameter λ = 0.0239 was selected using the generalized cross-validation method.

[0069] Step 4: Calculate the normalized weighted matrix;

[0070] To avoid the problem of finding the optimal regularization parameter λ in equation (4), a normalized weighted matrix is ​​constructed to remove the regularization parameter and obtain the optimal solution.

[0071] First, equation (4) can be reformulated as

[0072]

[0073] Next, let B = HD -1 B = [b1, b2, ..., b MN According to the covariance fitting criterion, the normalized weighting matrix W x It can be represented as:

[0074] W x =diag([w1,w2,...,w MN (7)

[0075]

[0076] Here, diag(·) represents converting a vector into a diagonal matrix.

[0077] Step 5: Derive the parameter-free two-dimensional fusion LASSO model;

[0078] Assuming uniform noise, according to the covariance fitting criterion, substituting the normalized weighting matrix into equation (4) yields a parameter-free two-dimensional fusion LASSO model:

[0079]

[0080] This model can sparsely reconstruct two-dimensional signals and has good contour constraint capabilities. At the same time, it can avoid the selection of regularization parameters, which greatly simplifies the calculation process.

[0081] Step 6: Solve the model;

[0082] The model obtained in step five is a convex optimization problem, which can be solved quickly using the CVX toolbox in Matlab. The CVX solution result is as follows:

[0083]

[0084] in, This indicates the target reconstruction result, and CVX(·) indicates the solution process of the CVX toolbox.

[0085] Ultimately, by solving the parameterless two-dimensional fusion LASSO model, the selection of regularization parameters can be avoided, thus achieving two-dimensional reconstruction.

[0086] In this embodiment, to demonstrate the effectiveness of the method of the present invention, the parameterless two-dimensional fusion LASSO model solution method proposed in this invention is applied to the field of optical image denoising. The experiment is carried out on a 64-bit Matlab2020b platform, and the solution method uses the CVX toolbox of MATLAB.

[0087] The original images used in the experiment are as follows Figure 2 As shown, its dimensions are 90×90. Additive Gaussian noise with a mean of 0 and a variance of 0.01 is added to the image, resulting in a noisy image as shown. Figure 3 As shown. Figure 4 The solution results of the existing two-dimensional fusion LASSO model are presented, in which the parameter λ = 0.0239 is selected using the cross-validation method. Figure 5 The results are obtained from the parametric two-dimensional fusion LASSO model. To quantitatively evaluate the denoising effect, the Peak Signal-to-Noise Ratio (PSNR) of each method compared to the original image is used. A higher PSNR indicates that the image is closer to the original image, and the better the denoising effect. The PSNR of each method is shown in Table 1.

[0088] Table 1

[0089]

[0090] As can be seen from the table, the denoising effect of the method of the present invention is slightly better than the existing two-dimensional fusion LASSO model method. Furthermore, the method of the present invention does not require the selection of regularization parameters during the calculation process, thus solving the problem of optimal selection of regularization parameters, which greatly expands the application scope of the method of the present invention.

[0091] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Various modifications and variations can be made to the invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the invention should be included within the scope of the claims of the invention.

Claims

1. A parameter-free two-dimensional fusion LASSO model solution method, the specific steps of which are as follows: Step 1: Construct a two-dimensional signal model; The two-dimensional signal model in image processing is represented as: Y = AX + E (1) in, Represents a two-dimensional measurement data matrix. Represents the original two-dimensional data matrix. Represents a dictionary matrix. Represents the additive noise matrix; Let M and N represent the dimensions of the vector space, respectively, and M and N represent the row and column dimensions of the two-dimensional data matrix. This two-dimensional signal model is applied to the field of optical image denoising, where Y is an image with additive Gaussian noise added. Step 2: Model columnization processing; The model is columnarized, and three matrices F1, F2, and F3 are assumed to exist. According to matrix columnarization theory, we can obtain: Where vec(·) represents the matrix columnization operation, This represents the Kronecker product, (·). T This represents the matrix transpose operation; Equation (1) can be transformed into a linear model: y = Hx + e (3) Where y = vec(Y), x = vec(X), e = vec(E), Denotes the identity matrix, and Step 3: Construct a two-dimensional fused LASSO model; By simultaneously adding sparsity constraints and total variation constraints to the rows and columns of the model in equation (1), a two-dimensional fused LASSO model is obtained, which can be specifically expressed by the linear model in equation (3): Where, x′=vec(X T ), where λ represents the regularization parameter, and D represents the total variation operator, represented by the difference matrix: Where ||Dx||1 represents the fusion constraint on the column data of the two-dimensional matrix, and ||Dx′||1 represents the fusion constraint on the row data of the two-dimensional matrix; Step 4: Calculate the normalized weighted matrix; The optimal solution is obtained by constructing a normalized weighted matrix to remove the regularization parameter. First, let's rephrase equation (4) as follows: Next, let B = HD -1 B = [b1, b2, ..., b MN According to the covariance fitting criterion, the normalized weighting matrix W x It can be represented as: IN x =diag([w1,w2,...,w MN ]) (7) Where, diag(·) represents converting a vector into a diagonal matrix; Step 5: Derive the parameter-free two-dimensional fusion LASSO model; Assuming uniform noise, according to the covariance fitting criterion, substituting the normalized weighting matrix into equation (4) yields a parameter-free two-dimensional fusion LASSO model: Step 6: Solve the model; The model obtained in step five is a convex optimization problem, which can be solved quickly using the CVX toolbox in Matlab. The CVX solution result is as follows: in, The result represents the target reconstruction, and CVX(·) represents the solution process of the CVX toolbox. Finally, by solving the parameterless two-dimensional fusion LASSO model, two-dimensional target reconstruction was achieved, solving the problem of optimal selection of regularization parameters.

Images