Single image rain streak removal based on direction estimation

Abstract

The application discloses a single image rain removal method based on direction estimation of rain streaks, and belongs to the field of image rain removal.(1) The linear property and angle distribution of rain streaks are considered, and the direction gradient operation can effectively locate rain streaks in the rain streak layer, so that the rain streak layer image can be estimated more accurately, the single image oblique rain streak removal is realized, and the rain removal precision is improved.(2) A sparse prior is used to describe the direction smoothness of the rain image for the estimated rain streak layer, and the original sparse prior along the horizontal direction (x direction) for the background layer image and the sparse prior along the vertical direction for the rain streak layer image (y direction) are combined to form a final sparse model; the alternating direction multiplier method is used to solve the proposed sparse model, so that an optimal solution is obtained, which is helpful to better restore the rain-free image.(3) Because the l0 norm is non-convex, and the l1 norm is convex, the l1 norm is used to represent the sparsity of rain streaks, so that the bad features of the rain-free area are avoided.

Description

Technical Field

[0001] This invention presents a single-image rain removal method based on a direction-estimated rain ripple layer. This method estimates the rain ripple layer through directional gradient operations to better remove oblique rain ripples. A sparse prior is applied to the estimated rain ripple layer, combined with existing sparse priors for the rain image along the horizontal and vertical directions, to form a final sparse model. The Alternating Direction Method of Multipliers (ADMM) is used to solve this model to recover the rain-free image. This method can be applied to image rain removal, accurately removing oblique rain ripples and improving rain removal accuracy. Background Technology

[0002] In rainy weather, most outdoor vision systems capture images with rain streaks, leading to reduced visibility of the background image. Image deraining, as a low-level vision task, has become a necessary preprocessing procedure for subsequent tasks such as semantic segmentation, face recognition, and scene analysis. Therefore, to improve the reliability of outdoor vision systems, it is necessary to develop effective methods to solve the image degradation problem caused by rain streaks in images.

[0003] Currently, methods for processing rain images mainly fall into two categories: one is the traditional model-driven single-image rain removal method, which uses rainless images or relatively simple rain streak priors to remove rain. This limits the rain removal capability for complex scenes and cannot effectively remove oblique rain streaks. The other is the deep learning-based single-image rain removal method, which uses network training to remove rain streaks to better obtain estimates of rainless images. However, this method requires a large amount of data and time for network training, and because image patches are severely contaminated, it cannot handle oblique rain streaks well. In recent years, single-image rain removal methods based on global sparse models have been proposed. For example, the Global Sparse Model (UGSM) method proposed by Deng et al. (Liang-Jian Deng, Ting-Zhu Huang, Xi-LeZhao, and Tai-Xiang Jiang, “A directional global sparse model for single imagerain removal,” Applied Mathematical Modelling, vol.59, pp.662–679, 2018.) establishes a global sparse model for rain streak removal. In most cases, this method can effectively remove rain streaks and has a significant image restoration effect. However, in windy conditions, rain images are damaged by oblique rain streaks. This method requires rotating the image to obtain vertical rain streaks, which leads to image pixel resampling and causes image degradation.

[0004] To overcome the limitations of the UGSM method, this invention proposes a single-image rain removal method based on a direction-estimated rain ripple layer. Rain ripples are detected based on pixel intensity to obtain a binary mask, and a directional gradient operation is used to adapt to arbitrary directions to estimate the rain ripple layer. Then, sparsity constraints are added to the estimated rain ripple layer to improve the accuracy of removing oblique rain ripples. Summary of the Invention

[0005] This invention, based on a global sparse model, designs a single-image rain removal method using a direction estimation rain ripple layer. This improves rain removal accuracy and image clarity, reduces the number of rain ripples in the image to be restored, and allows the de-rained image to retain more detail. Since humans are more sensitive to changes in brightness, we convert RGB images to YUV space and apply the algorithm only to the luminance channel (Y) for oblique rain ripple removal. Specifically, since observed rain ripples typically have higher pixel intensity compared to other adjacent image structures, we first detect rain ripples based on pixel intensity, thereby obtaining a coarse binary mask M for the rain ripples. cThen, Principal Component Analysis (PCA) is applied to obtain the angular distribution. Using angular statistics and histograms, the distribution of the most frequently occurring angles can be obtained, thus yielding a refined binary mask M. r Finally, the directional gradient operation is used to adapt to arbitrary directions, thereby estimating the rain ripple layer. For the estimated rain ripple layer, a sparse prior is used to describe it, combining the original sparse priors for the background image along the horizontal direction (x-direction) and for the rain ripple image along the vertical direction (y-direction) to form the final sparse model. The proposed sparse model is solved using ADMM to obtain the optimal solution, which helps to better recover the rain-free image. Then, the desired rain-free image can be obtained by subtracting the estimated rain ripple image from the input rain image.

[0006] To achieve the above objectives, the present invention provides the following solution:

[0007] A single-image rain removal method based on direction estimation rain streak layer, the method comprising:

[0008] Step 1: Estimate the direction of the rain streaks;

[0009] Step 2: Sparsity constraints on the image;

[0010] Step 3: Sparse Model;

[0011] The method for estimating the direction of rain streaks specifically includes the following steps:

[0012] Step 1): Obtain the coarse binary mask M c Since the observed rain streaks typically have higher intensity compared to other adjacent image structures, we first detect rain streaks based on pixel intensity to obtain a coarse binary mask M for the first-stage rain streaks. c .

[0013] Step 2): Obtain the angle distribution. Next, we use the rain ripples as the coarse binary mask M obtained in the first stage. c The connection components in the second stage. For the i-th connection component, the pixel can be represented as... By applying PCA, the angle θ of the dominant rain ripple direction can be obtained. i Length along the direction of the rain streaks Length along the vertical direction of the rain streak By applying the same PCA procedure to all connected parts, the angle distribution can be obtained.

[0014] Step 3): Obtain the fine binary mask M rBased on the angle distribution obtained in the second stage, the most frequently occurring angle distribution can be obtained using angle statistics and statistical histograms. Masking is then improved using (1) to obtain a refined binary mask M. r .

[0015]

[0016] Where, Δθ i This represents the difference between the angle of the i-th connected component and the most common angle. Indicates the length along the direction of the rain streaks. T1 represents the length along the vertical direction of the rain streak, and T2 and T3 represent the threshold values, respectively.

[0017] Step 4): Calculate the final angle θ of the rain ripples. Since the angular variance of the rain ripples is small, we use the truncated angle distribution near the peak to obtain the final angle θ, as shown in Equation (2).

[0018]

[0019] Where, θ i W represents the angle of the i-th connected component. i Represents angle θ i The frequency of θ. In formula (2), the direction of rain streaks is different for different input rain images, therefore θ in all tested rain images i and W i The value of θ determines the final angle θ of the rain pattern.

[0020] Step 5): Oriented Gradient Operation. Since the orientation of rain streaks in the input rain image is complex, it is necessary to utilize traditional gradient operations to adapt to arbitrary orientations. The introduced oriented gradient operator is as follows:

[0021]

[0022] Here, w1(θ) and w2(θ) are functions of the final angle θ of the rain streaks, and their values ​​range from [0,1]. The directional gradient operator is designed for two reasons. First, most elements are constants, making it easy to compute. Second, w2(θ) represents the deviation of the rain streaks from the vertical direction. However, this operation can only be applied to rain streaks originating from the southwest. To the south (w1 = 1, θ = 0), we can transform any direction to this range through row and column arrangements or matrix mirroring operations. When When w1(θ) and w2(θ) are calculated, they are as follows:

[0023]

[0024] The sparsity constraint of the image specifically includes the following steps:

[0025] Step 1): Rain Model. The simplest and most popular model used in existing research is the additive composite model, which decomposes the input rain image into the following form:

[0026] O = B + S (5)

[0027] Where, O∈R m×n The input is a rain image, B∈R m×n It is a rainless background layer, S∈R m×n This is the rain pattern layer, where R represents the spatial region, and m and n represent the length and width of the input rain image, respectively. The goal of rain pattern removal is to estimate the rainless background layer B from a given input rain image O.

[0028] Step 2): Sparsity of rain streaks along the vertical direction. Assume that the rain streaks are sparse. The sparsity of rain streaks is usually described by the l0 norm, which represents the number of non-zero elements. Since the l0 norm is non-convex while the l1 norm is convex, this paper uses the l1 norm to represent the sparsity of rain streaks. The l1 norm can guarantee global optimality in the ADMM algorithm and avoid adverse effects on rainless areas. In real rainfall scenarios, rain streaks usually descend from top to bottom, which is theoretically called the vertical direction (y-direction) of the rain image. Due to the smoothness within the rain, the difference between adjacent image pixels is very small (approximately zero). Therefore, we also consider the sparse prior of rain streaks varying along the vertical direction. When the angle between the vertical direction and the real rainfall direction is small, the sparsity along the vertical direction remains unchanged. Equation (6) represents the sparsity regularization term varying along the vertical direction.

[0029]

[0030] Where ||·||1 represents the l1 norm, This indicates a difference operation along the vertical direction. express In vector form.

[0031] Step 3): Estimate the sparsity of the rain ripple layer. Because rain ripples sometimes do not fall strictly vertically in windy conditions, and the angle between the vertical direction and the actual rainfall direction is large, the vertical sparsity constraint is difficult to apply. To solve this problem, it is necessary to estimate the direction of the rain ripples to effectively remove oblique rain ripples. Equation (7) represents the sparsity regularization term for estimating the direction of the rain ripples.

[0032]

[0033] in, This represents the directional gradient operation, where θ represents the direction of the rain ripples. express In vector form.

[0034] Step 4): Sparsity of the rainless image along the horizontal direction. To obtain a robust rain removal effect, variational information of the rain stripe direction is used to determine the discontinuity of the rain stripes. In this case, the horizontal direction is considered to be the direction across the rain stripes. Furthermore, due to the relationship b = os, equation (8) represents the sparsity regularization term of the rainless image varying along the horizontal direction.

[0035]

[0036] in, This indicates a difference operation along the horizontal direction. express In vector form.

[0037] Step 5) Nonnegativity constraint. Typically, rain streaks s are nonnegative and usually have the brightest intensity in the rain image. Therefore, the input rain image o and the rain streak layer s have the constraint relationship shown in (9).

[0038] o≥s≥0 (9)

[0039] After studying the properties of rain streaks and introducing three sparsity regularization terms, combined with three sparsity constraints, we can find a sparse model for estimating the horizontal, vertical, and rain streak directions in the rain removal problem. The model can be expressed as follows:

[0040]

[0041] Here, λ1 and λ2 represent two positive normalization parameters. The nonnegativity constraint o≥s≥0 is achieved through a projection strategy, as detailed in formula (22).

[0042] The modeling and optimization process for the sparse model is as follows:

[0043] Step 1): Since the proposed model (10) is non-differentiable, we performed variable substitution and addressed the following issues:

[0044]

[0045] Step 2): The augmented Lagrangian function for problem (11) is expressed in the following form:

[0046]

[0047] Where, p i (i = 1, 2, 3) represents the Lagrange multiplier. <p i ,·>(i=1,2,3) represents a linear term, βi (i = 1, 2, 3) represents the normalization parameters. This represents a quadratic term.

[0048] Step 3): The sparse model is optimized using ADMM. The iterative optimization process of ADMM is as follows:

[0049] First, update u. To solve the subproblem concerning u, it can be transformed into solving the following optimization problem:

[0050]

[0051] The u-subproblem can be solved using a soft thresholding strategy and can be expressed as follows:

[0052]

[0053] Among them, s k Let s represent the input s for the k-th iteration. Let p represent the value of the input p in the k-th iteration of the subproblem u. k+1 Let represent the solution to u obtained in the k-th iteration.

[0054] Where Shrink(a,b) = sign(a)max(a|-b,0) and sign(a) is represented as in equation (15).

[0055]

[0056] Then, update v. Similar to updating u, to solve the subproblem with respect to v, we can transform it into solving the following optimization problem:

[0057]

[0058] The subproblem v can be solved using a soft thresholding strategy and can be expressed as follows:

[0059]

[0060] in, Let p represent the value of the input p in the k-th iteration of the subproblem v. k+1 Let v be the solution obtained in the k-th iteration. In model (11) It can be done Perform calculations; * indicates a convolution operation.

[0061] Secondly, update w. Similar to updating u and v, to solve the subproblem concerning w, we can transform it into solving the following optimization problem:

[0062]

[0063] The w subproblem can be solved using a soft thresholding strategy and can be expressed as follows:

[0064]

[0065] in, Let p represent the value of the input p in the k-th iteration of the subproblem w. k+1 Let w be the solution obtained in the k-th iteration.

[0066] Then, update s. To solve the subproblem concerning s, it can be transformed into solving the following optimization problem:

[0067]

[0068] Since the quadratic s subproblem (20) is differentiable, the solution in closed form can be solved by (21).

[0069]

[0070] in, express The transpose of , where I denotes the identity matrix. express The transpose of the matrix, Let represent the closed-form solution of s obtained in the k-th iteration. Equation (21) can be solved using the Fast Fourier Transform (FFT). Due to the nonnegativity constraint (9), s k+1 It can be estimated using equation (22).

[0071]

[0072] Among them, s k+1 Let represent the solution to s obtained in the k-th iteration.

[0073] Finally, update the Lagrange multipliers p1, p2, p3. The Lagrange multipliers are updated based on ADMM, and the update formula is shown in (23).

[0074]

[0075] in, This represents the value of p in the input of the k-th iteration. This represents the value of p obtained in the k-th iteration.

[0076] Therefore, the subproblems v, u, and w can be effectively solved using a soft thresholding strategy, while the subproblem s can be computed using the FFT algorithm. All four subproblems have their closed-form solutions, which can guarantee the speed and accuracy of the algorithm.

[0077] Beneficial effects:

[0078] This invention proposes a single-image deraining method based on orientation-based estimation of the rain ripple layer. First, the rain ripple layer is estimated using a proposed directional gradient operation. Second, a sparse prior is applied to the estimated rain ripple layer, combined with existing sparse priors for the background image along the horizontal direction (x-direction) and for the rain ripple layer image along the vertical direction (y-direction), forming a final sparse model. Finally, ADMM is used to solve this sparse model. This method can accurately estimate the rain ripple layer image, improving deraining accuracy and image clarity, reducing the amount of rain ripples to be restored, and allowing the derained image to retain more detail. Attached Figure Description

[0079] Figure 1 This is a flowchart of the single-image rain removal method based on direction estimation rain streak layer provided by the present invention;

[0080] Figure 2 This is a schematic diagram of the rain ripple layer estimation process of the single-image rain removal method based on direction estimation provided by the present invention;

[0081] Figure 3 This is a single-image rain removal result image based on the direction estimation rain ripple layer method provided by this invention. The left column is the rain image, and the right column is the image after rain removal. Detailed Implementation

[0082] The purpose of this invention is to provide a single-image rain removal method based on direction estimation of the rain ripple layer. This method estimates the rain ripple layer and applies a sparse prior to the estimated rain ripple layer. This sparse prior is then combined with existing sparse priors for the background image along the horizontal direction (x-direction) and for the rain ripple layer image along the vertical direction (y-direction) to form a final sparse model. Finally, ADMM is used to solve this sparse model. This single-image rain removal method based on direction estimation of the rain ripple layer not only improves rain removal accuracy and image clarity, reduces the amount of rain ripples to be restored, but also consumes less runtime.

[0083] The present invention will now be described in detail with reference to the accompanying drawings. It should be noted that the described embodiments are intended only to facilitate the understanding of the present invention and are not intended to limit it in any way.

[0084] Figure 1 This is a flowchart of the single-image rain removal method based on direction estimation rain streak layer provided by the present invention; Figure 2 This is a schematic diagram of the rain ripple layer estimation process of the single-image rain removal method based on direction estimation provided by the present invention; Figure 3 This is a single-image deraining result image of the single-image deraining method based on direction estimation rain ripple layer provided by the present invention, wherein the left column is the rain image and the right column is the image after deraining.

[0085] The rain removal method for a single image based on direction estimation rain streak layer provided by this invention specifically includes:

[0086] Step 1: Estimate the direction of the rain streaks;

[0087] Step 2: Sparsity constraints on the image;

[0088] Step 3: Sparse Model;

[0089] Step 1: Estimate the direction of the rain streaks;

[0090] In order to estimate oblique rain streaks under windy conditions, this invention uses directional gradient operations to estimate the rain streak layer. First, a coarse binary mask M needs to be obtained. c And the angle distribution, and then the fine binary mask M needs to be obtained. r Finally, the final angle θ of the rain ripples is calculated. Since the direction of the rain ripples in the input rain image is complex, it is necessary to utilize traditional gradient operations to adapt to arbitrary directions. This method can effectively estimate the angle of the rain ripples, thereby providing a better estimate of the rain ripple layer.

[0091] In this invention, a fine binary mask M is obtained. r The distribution of the most frequent angles can be obtained by using angle statistics and statistical histograms. The mask is then improved using formula (1) to obtain a refined binary mask M. r In formula (1), the threshold values ​​T1 are 5, T2 are 8°, and T3 are 3.

[0092] Step 2: Sparsity constraints on the image;

[0093] After estimating the direction of the rain streaks, this invention assumes that the rain streaks are sparse in order to better describe them, and therefore proposes a sparsity constraint in this step. The input rain image is decomposed into a rainless background layer image and a rain streak layer image, and then sparsity constraints are applied to the estimated rain streak layer image, the rainless background layer image along the horizontal direction, and the rain streak layer image along the vertical direction. Furthermore, due to the non-convexity of the l0 norm and the convexity of the l1 norm, this invention utilizes the l1 norm to represent the sparsity of the rain streaks. Simultaneously, since the rain streaks s are non-negative and have the brightest intensity in the rain image, the input rain image and the rain streak layer image have a non-negative constraint relationship. By adding the sparsity constraint to the image, feature selection is better achieved, and the model becomes more interpretable.

[0094] Step 3: Sparse Model;

[0095] After adding sparsity constraints, in order to obtain a rainless image, this invention needs to combine three sparsity constraints to find a sparse model in the rain removal problem along the horizontal direction, the estimated rain streak direction, and the vertical direction. Furthermore, since the model is non-differentiable, variable substitution is used, and the augmented Lagrangian function for solving the model is given.

[0096] In this embodiment of the invention, in order to better recover the rainless image and retain its detailed information, the normalization parameters λ1 are set to 0.95, λ2 to 0.08, and β1, β2, and β3 to 100; the positive tolerance value tol is 10. -3 Maximum number of iterations M iter The value is 400. After proposing the sparse model, the augmented Lagrangian function is solved using the ADMM iterative optimization process. Finally, the Lagrange multipliers p1, p2, and p3 are updated using ADMM. The initial values ​​of p1, p2, and p3 are 0. Optimizing the sparse model allows for a better estimation of rainless images.

[0097] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any modifications and substitutions that can be conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A single-image rain removal method based on direction estimation rain streak layer, characterized in that, Includes the following steps: Step 1: Estimate the direction of the rain streaks; first, a coarse binary mask M needs to be obtained. c And the angle distribution, and then the fine binary mask M needs to be obtained. r Finally, calculate the final angle of the rain streaks. And it uses gradient operations to adapt to arbitrary directions; Step 2: Sparsity constraint of the image; After estimating the direction of the rain streaks, a sparsity constraint is added to the estimated rain streak layer image; The specific steps for implementing the sparsity constraint of the image are as follows: First, the input rain image is decomposed into the following form: ； in, The input is a rain image. It is a rainless background layer. This is the rain pattern layer, where R represents the spatial region, and m and n represent the length and width of the input rain image, respectively. Then, rain streaks typically descend from top to bottom, theoretically referred to as the vertical direction (y-direction) of the rain image; therefore, sparse priors varying along the vertical direction of the rain image are also considered; equation Represents a sparsity regularization term that varies along the vertical direction; ； in, express Norm, This indicates a difference operation along the vertical direction. express The vector form; Secondly, in windy conditions, rain streaks sometimes do not fall strictly vertically, and the angle between the vertical direction and the actual rainfall direction is large. In this case, the sparsity constraint of the vertical direction is difficult to apply. To solve this problem, it is necessary to estimate the direction of the rain streaks; equation This represents the sparsity regularization term for estimating the direction of rain streaks; ； in, Indicates the directional gradient operation. Indicates the direction of the rain streaks. express The vector form; Then, to achieve a robust rain removal effect, variational information about the rain ripple direction is used to determine the discontinuity of the rain ripples; in this case, the horizontal direction is considered to be the direction spanning the rain ripples; furthermore, since Relationship, equation The sparsity regularization term represents the rainless image that varies along the horizontal direction; ； in, This indicates a difference operation along the horizontal direction. express The vector form; Finally, since the rain streaks s are non-negative and typically have the brightest intensity in a rain image, the input rain image o and the rain streak layer s have the following characteristics: The constraints shown; ； After studying the properties of rain streaks, and combining three sparsity regularization terms, a sparse model is found for the horizontal direction, the estimated rain streak direction, and the vertical direction in the rain removal problem; the model is expressed in the following form: ； in, Indicates two positive normalization parameters; set It is 0.

95. It is 0.08; its nonnegativity constraint This is achieved through a projection strategy; Step 3: Sparse Model; Combining the sparsity constraints on the estimated rain ripple layer in Step 2, and the original sparsity constraints on the rainless background layer along the horizontal direction and on the rain ripple layer along the vertical direction, the final sparse model is formed; After proposing the sparse model, the alternating direction multiplier method is used to solve the model.

2. The single-image rain removal method according to claim 1, characterized in that, The specific steps for estimating the rain streak direction described in step 1 are as follows: First, obtain the coarse binary mask M. c Since the observed rain streaks typically have higher intensity compared to other adjacent image structures, rain streaks are first detected based on pixel intensity to obtain a coarse binary mask M for the first-stage rain streaks. c ; Then, obtain the angular distribution; use the rain ripples as the coarse binary mask M obtained in the first stage. c The connection components in the second stage; for the i-th connection component, the pixel is represented as Principal component analysis was used to obtain the angle of the dominant direction of rain streaks. Length along the direction of the rain streaks Length along the vertical direction of the rain streaks The same PCA procedure was applied to all connecting parts to obtain the angle distribution. Secondly, obtain the fine binary mask M. r Based on the angle distribution obtained in the second stage, the distribution of the most frequently occurring angles is obtained using angle statistics and statistical histograms; By improving the mask, a refined binary mask M can be obtained. r ; ； in, This represents the difference between the angle of the i-th connected component and the most common angle. Indicates the length along the direction of the rain streaks. This represents the length along the vertical direction of the rain streak; T1, T2, and T3 represent the threshold values, respectively; the parameters T1, T2, and T3 are set to 5, 8°, and 3, respectively. Then, calculate the final angle of the rain streaks. Since the angular variance of rain streaks is small, the final angle is obtained by utilizing the truncated angular distribution near the peak. , such as equation As shown; ； in, This represents the angle of the i-th connected component. Indicates angle The frequency quantity; in the formula Among them, different input rain images have different directions of rain streaks, therefore different rain images and Different values ​​result in different final angles of rain streaks obtained from different input rain images. They are also different; Finally, since the orientation of rain streaks in the input rain image is complex, it is necessary to utilize traditional gradient operations to adapt to arbitrary orientations; the introduced directional gradient operator is as follows: ； in, and It's about the final angle of the rain streaks. The function, and The range of values ​​is , and The calculation is as follows: 。 3. The single-image rain removal method according to claim 1, characterized in that, The specific steps for modeling and optimizing the sparse model are as follows: First, due to the proposed model Since it is not differentiable, a variable substitution was performed, and the following problem was solved: ； Secondly, regarding the question The augmented Lagrange function can be expressed in the following form: ； in, Represents the Lagrange multipliers. Represents linear terms, Represents normalized parameters, Representing quadratic terms; setting normalization parameters. It is 0.

95. It is 0.

08. 100; positive tolerance value Maximum number of iterations M iter It is 400; The initial value is 0; Finally, ADMM is used to optimize and solve the sparse model. The specific steps of the ADMM iterative optimization process are as follows: Step 1): Update u; To solve the subproblem about u, we transform it into solving the following optimization problem: ； The u-subproblem is solved using a soft thresholding strategy and can be expressed as follows: ； in, Let s represent the input s for the k-th iteration. This represents the value of p as input to the k-th iteration of the subproblem u. Let represent the solution to u obtained in the k-th iteration; And sign(a) is represented as follows As shown; ； Step 2): Update v; similar to updating u, to solve the subproblem about v, we transform it into solving the following optimization problem: ； The subproblem v is solved using a soft thresholding strategy and can be expressed as follows: ； in, This represents the value of p as input to the k-th iteration of the subproblem v. This represents the solution to v obtained in the k-th iteration; in the model In pass Perform calculations. Indicates the convolution operation; Step 3): Update w; similar to updating u and v, to solve the subproblem with respect to w, we transform it into solving the following optimization problem: ； The w subproblem is solved using a soft thresholding strategy and can be expressed as follows: ； in, This represents the value of p as input to the k-th iteration of the subproblem w. Let w be the solution obtained in the k-th iteration; Step 4): Update s; To solve the subproblem about s, we transform it into solving the following optimization problem: ； Due to the problem of quadratic s-subs It is differentiable, and the solution in closed form is... Solve the problem; ； in, express The transpose of the matrix, Represents the identity matrix. express The transpose of the matrix, The equation represents the closed-form solution of s obtained in the k-th iteration; The solution is obtained using Fast Fourier Transform; due to the nonnegativity constraint... , Equation Make an estimate; ； in, Let represent the solution s obtained in the k-th iteration; Step 5): Update the Lagrange multipliers Based on ADMM, the Lagrange multipliers are updated using the following formula: As shown; ;in, This represents the value of p in the input of the k-th iteration. This represents the value of p obtained in the k-th iteration.

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