Electrical capacitance tomography reconstruction method integrating measurement principles, data and domain knowledge
By combining the measurement principles of fusion capacitive tomography, data-driven priors and sparse priors, combined with deep learning and alternating direction multiplier method, the problem of low image quality in the prior art is solved, and image reconstruction with higher accuracy and robustness is achieved.
Patent Information
- Application Number
- CN202410042458.7
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2024-01-11
- Publication Date
- 2025-03-14
- Estimated Expiration
- 2044-01-11
AI Technical Summary
The existing capacitive tomography technology has low image quality in complex multiphase flow environments, making it difficult to achieve high-precision reconstruction, especially in details reconstruction and artifact reduction.
A capacitive tomography reconstruction method that combines the principle of measurement, data-driven priors and sparse priors is designed. The data-driven priors are predicted by deep learning models, and the reconstruction model with weighted L1-2 norms is solved by combining the alternating direction multiplier method and the forward-backward splitting algorithm.
The expression ability and noise and artifact processing effects of the imaging model are improved, and the detail accuracy and robustness of image reconstruction are significantly improved.
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Figure CN118196158B_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to nondestructive testing technology, and involves fields such as capacitance tomography technology, image processing, deep learning and multiphase flow parameter measurement, and specifically to a capacitance tomography reconstruction method that integrates measurement principles, data and domain knowledge. Background Art
[0002] Electrical Capacitance Tomography (ECT) is considered to be a measurement technology that is expected to solve the challenges of measuring multiphase flow process parameters in the fields of energy storage, hydrogen production, carbon neutrality, chemical engineering, carbon capture and storage. With the continuous development of theoretical research and application, ECT technology has made continuous progress in software and hardware facilities, imaging algorithms and application fields. However, in practical applications, especially in complex multiphase flow environments, ECT technology still faces the problem of low image quality. Therefore, designing efficient reconstruction algorithms to improve imaging quality has become a widely-watched research hotspot and key challenge in this field.
[0003] At present, a variety of algorithms are used to solve inverse problems, including Landweber algorithm, nonlinear imaging algorithm, SplitBregman algorithm, conjugate gradient method, iterative method, non-iterative algorithm, etc. Although these algorithms have their own advantages, they generally face the limitation of being unable to achieve high-precision reconstruction. Especially in terms of detail reconstruction and artifact reduction, the existing methods still have obvious shortcomings. In this context, fusing diverse and complementary prior information becomes an effective way to improve the quality of reconstruction. The key challenge is that once these prior information are determined, how to effectively fuse them to reduce reconstruction errors and achieve higher imaging accuracy and reliability. This is not only the core issue in the development of electrical capacitance tomography technology, but also the key to promoting its application in the field of multiphase flow parameter measurement. Summary of the invention
[0004] To solve the above technical problems, the present invention proposes a capacitance tomography reconstruction method that integrates measurement principles, data, and domain knowledge by designing a more effective imaging model and introducing data-driven image priors, thereby improving reconstruction quality and robustness. The invention has made useful explorations in the integration of measurement mechanisms and deep learning, and provides new ideas for releasing the potential of capacitance tomography technology in multiphase flow parameter measurement and improving imaging quality.
[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows:
[0006] The capacitance tomography reconstruction method integrating measurement principle, data and domain knowledge comprises the following steps:
[0007] Step 1: Collect training samples and train deep learning models;
[0008] Step 2: Reconstruct the initial image based on the capacitance vector;
[0009] Step 3: Use the trained deep learning model to infer the data-driven prior DDP;
[0010] Step 4: Design a reconstruction model that integrates measurement principle, data-driven prior DDP and sparse prior;
[0011] Step 5: Solve the reconstruction model and complete image reconstruction.
[0012] Furthermore, the step 1 comprises:
[0013] The deep convolutional neural network DCNN includes an output layer, an input layer, and a hidden layer; the hidden layer consists of a convolutional layer, a pooling layer, an activation function, and a fully connected layer; before using the deep convolutional neural network DCNN, training samples are collected to train the model parameters in a supervised manner to solve the following optimization problem:
[0014]
[0015] In the formula, min{·} represents the minimum value of the function; It means the function sums from i=1 to i=l; l represents the total number of samples; is a pair of training samples, c i and θ i are the i-th input sample and output sample respectively; ψ(·) represents the forward propagation operator, whose form is determined by the deep convolutional neural network DCNN structure; Θ represents the model parameters; Represents the square of the L2 norm.
[0016] Furthermore, the step 2 comprises:
[0017] The image reconstruction model is approximately as follows:
[0018] Af=z+r (2)
[0019] Where A, f, z and r are the dimensionless normalized sensitivity matrix, dielectric constant, capacitance vector and noise vector, respectively; their dimensions are m×n, n×1, m×1 and m×1, respectively; m and n both represent the dimensions of the matrix.
[0020] Furthermore, the step 3 comprises:
[0021] Based on the collected training data, a deep convolutional neural network (DCNN) is used to predict the data-driven prior DDP, which is defined as:
[0022] f DDP =ψ(f0 |Θ) (3)
[0023] In the formula, f DDP represents the data-driven prior DDP; ψ(f 0 |Θ) represents the forward propagation operator in the deep learning model; f 0 is the initial image of the input; Θ represents the model parameters; | represents given or conditional on;
[0024] Furthermore, the step 4 comprises:
[0025] The regularization method is used to integrate the measurement mechanism and image priors. The regularization method is expressed as the following mathematical model:
[0026]
[0027] In the formula, g j (f), R(f), α j >0 and ρ are the regularization term, data fidelity term, regularization parameter, and the number of regularization terms, respectively;
[0028] Use L1 norm as data fidelity term:
[0029] R(f,y)=||Af-z||1 (5)
[0030] Where ||·||1 is the L1 norm;
[0031] Using sparse prior to improve reconstruction quality, L 1-2 The norm is the sparsity regularization term, defined as:
[0032] L 1-2 =||f||1-||f||2 (6)
[0033] Where, L 1-2 Indicates L 1-2 norm;
[0034] Weighted L 1-2 The norm increases flexibility and is defined as:
[0035] g(f)=||Df||1-||Df||2 (7)
[0036] In the formula, g represents L 1-2 norm; D is a non-negative diagonal matrix that will change with iterations; the initial weight matrix is represented by the data-driven prior DDP;
[0037] According to equations (4), (5) and (7), the new imaging model is expressed as:
[0038]
[0039] In the formula, α is the regularization parameter.
[0040] Furthermore, the step 5 comprises:
[0041] The imaging model equation (8) is solved by using the fused alternating direction multiplier method ADMM;
[0042] According to the optimization principle of differential convex function, f k It represents the image of the kth step of iteration. When Df k When is not a zero vector, equation (8) is relaxed to:
[0043] min{||Af-z||1+α||Df||1- <f,αD1f k / ||Df k ||2>} (9)
[0044] In the formula, diag means diagonal matrix; D nn represents an element with n rows and n columns; <·> represents the inner product; the superscript k defines the number of iterations;
[0045] Equation (9) can be rewritten as:
[0046] min{||Af-z||1+α||Df||1+<f,v k+1 >} (10)
[0047] In the formula, v represents the gradient variable; v k+1 Defined as:
[0048] v k+1 =-αD1f k / ||Df k ||2 (11)
[0049] Introducing a new auxiliary variable d, equation (10) can be rewritten as:
[0050]
[0051] In the formula, st represents the constraint condition;
[0052] According to the principle of fusion alternating direction multiplier method ADMM, equation (12) is rewritten as:
[0053]
[0054] In the formula, is a function of d and f, defined as:
[0055]
[0056] In the formula, the superscript T is defined as the transposition operator; u is a matrix, and its update format is:
[0057] u k+1 =u k +ρ(Af k+1 -zd k+1 ) (15)
[0058] Separate equation (13) into two more easily solvable subproblems:
[0059]
[0060] According to equations (13) and (16), the subproblem about d is expressed as:
[0061]
[0062] The subproblem about f is expressed as:
[0063]
[0064] Rewriting equation (18) into a compact form:
[0065]
[0066] The solution of equation (17) is:
[0067] d k+1 =shrink(Af k -z+u k / ρ,1 / ρ) (20)
[0068] Where shrink(·,·) represents the soft threshold algorithm;
[0069] Solving equation (19) using the forward-backward splitting algorithm FBSA leads to the following subproblems:
[0070]
[0071] Among them, δ is a positive real number; s is a variable; s k+1 Defined as:
[0072] s k+1 =f k -δ(ρA T (Af k -zd k+1 +u k / ρ)+v k+1 ) (twenty two)
[0073] The following acceleration formats are introduced:
[0074] fk+1 =f k+1 / 2 +(o k -1)(f k+1 / 2 -f k ) / o k+1 (twenty three)
[0075] In the formula, o represents an auxiliary variable, and its update format is defined as:
[0076]
[0077] In the formula, o 1 =1, indicating the acceleration step length of the first step.
[0078] Compared with the prior art, the present invention has the following beneficial effects:
[0079] The present invention discloses a capacitance tomography reconstruction method, which integrates the measurement principle of capacitance tomography, data-driven prior DDP and sparse prior. This fusion not only enhances the expression ability of the imaging model, but also improves the processing effect of noise and artifacts. In particular, the adaptive weighted L 1-2 The introduction of the norm increases the sparsity of the model, which helps to reconstruct image details more accurately. In addition, in order to effectively implement this imaging model, the present invention has developed a new algorithm. The algorithm combines ADMM (alternating direction multiplier method) and FBSA (forward-backward splitting algorithm), which effectively improves the solution efficiency. The evaluation results show that compared with the existing reconstruction algorithms, this new algorithm shows significant advantages in detail reconstruction, artifact removal and robustness.
[0080] In summary, the capacitance tomography reconstruction method disclosed in the present invention, which integrates measurement principles, data, and domain knowledge, has opened up a new path for the development of capacitance tomography technology by combining measurement mechanism with deep learning technology. This not only provides a more accurate and robust imaging method for multiphase flow parameter measurement, but also provides new ideas for improving the overall performance and application scope of capacitance tomography technology. BRIEF DESCRIPTION OF THE DRAWINGS
[0081] Figure 1 1 is a diagram of the actual dielectric constant distribution in an embodiment of the present invention; wherein (a)-(e) are five dielectric constant distributions; and the gradient color bar represents the dielectric constant value.
[0082] Figure 2(a)-Figure 2(o)It is an image reconstruction result diagram in an embodiment of the present invention; wherein, Figure 2(a) is Kaczmarz, Figure 2(b) is L1R, Figure 2(c) is Landweber, Figure 2(d) is SK, Figure 2(e) is TVR, Figure 2(f) is PRPCG, Figure 2(g) is LRR, Figure 2(h) is L1 / 2R, Figure 2(i) is TV-L1-L2, Figure 2(j) is ENR, Figure 2(k) is DLAR, Figure 2(l) is L2R, Figure 2(m) is OIOR, Figure 2(n) is LBP, and Figure 2(o) is TSVD; the gradient color bar represents the dielectric constant value. DETAILED DESCRIPTION
[0083] The present invention is further described below in conjunction with the accompanying drawings and specific embodiments.
[0084] The capacitance tomography reconstruction method of the present invention integrating measurement principles, data, and domain knowledge comprises the following steps:
[0085] Step 1: Collect training samples and train deep learning models;
[0086] Step 2: Reconstruct the initial image based on the capacitance vector;
[0087] Step 3: Use the trained deep learning model to infer DDP (Data Driven Prior);
[0088] Step 4: Design a reconstruction model that integrates measurement principle, DDP and sparse prior;
[0089] Step 5: Solve the reconstruction model and complete image reconstruction.
[0090] Specifically, in step 1, training the deep learning model includes:
[0091] DCNN (Deep Convolutional Neural Network) includes output layer, input layer and hidden layer. The hidden layer usually consists of convolution layer, pooling layer, activation function and fully connected layer. Before using DCNN, it is necessary to collect training samples to train model parameters in a supervised manner, and solve the following optimization problem:
[0092]
[0093] In the formula, min{·} represents the minimum value of the function; It means the function sums from i=1 to i=l; l represents the total number of samples; is a pair of training samples, c i and θ iare the i-th input sample and output sample respectively; ψ(·) represents the forward propagation operator, whose form is determined by the deep convolutional neural network DCNN structure; Θ represents the model parameters; Represents the square of the L2 norm.
[0094] Algorithm I in Table 1 summarizes the training steps of DCNN. DCNN can be trained offline, and after training, DDP can be quickly inferred.
[0095] Table 1
[0096]
[0097] Specifically, the step 2 includes:
[0098] In electrical capacitance tomography, image reconstruction is a key step to estimate the dielectric constant distribution from the measured capacitance data. The image reconstruction model can be approximated as follows:
[0099] Af=z+r (2)
[0100] A, f, z and r are dimensionless normalized sensitivity matrix, dielectric constant, capacitance vector and noise vector respectively; their dimensions are m×n, n×1, m×1 and m×1 respectively; m and n both represent the dimensions of the matrix. Equation (2) is an ill-posed inverse problem and is more difficult to solve.
[0101] Specifically, the step 3 includes:
[0102] Based on the collected training data, DDP is predicted using DCNN and can be defined as:
[0103] f DDP =ψ(f 0 |Θ) (3)
[0104] In the formula, f DDP represents the data-driven prior DDP; ψ(f 0 |Θ) represents the forward propagation operator in the deep learning model; f 0 is the initial image of the input; Θ represents the model parameters; | represents given or conditional on.
[0105] Specifically, step 4 includes:
[0106] The new reconstruction model needs to integrate the measurement mechanism and image priors. The regularization method has unique advantages in dealing with this task, which can be expressed as the following mathematical model:
[0107]
[0108] In the formula, g j (f), R(f), α j>0 and ρ are the regularization term, data fidelity term, regularization parameter, and the number of regularization terms, respectively;
[0109] Use L1 norm as data fidelity term:
[0110] R(f,y)=||Af-z||1 (5)
[0111] Where ||·||1 is the L1 norm;
[0112] Using sparse prior to improve reconstruction quality, L 1-2 The norm is the sparsity regularization term, defined as:
[0113] L 1-2 =||f||1-||f||2 (6)
[0114] Where, L 1-2 Indicates L 1-2 norm;
[0115] Weighted L 1-2 The norm increases flexibility and is defined as:
[0116] g(f)=||Df||1-||Df||2 (7)
[0117] In the formula, g represents L 1-2 norm; D is a non-negative diagonal matrix that will change with iterations; the initial weight matrix is represented by the data-driven prior DDP;
[0118] According to equations (4), (5) and (7), the new imaging model is expressed as:
[0119]
[0120] In the formula, α is the regularization parameter.
[0121] Equation (8) is a non-smooth Difference-of-Convex Functions (DC) optimization problem, which is more challenging to solve than the conventional reconstruction model.
[0122] Specifically, the step 5 includes:
[0123] ADMM (Alternating Direction Method of Multipliers) is very advantageous in solving non-smooth optimization problems. The present invention uses this method to solve the imaging model equation (8).
[0124] According to the optimization principle of differential convex function, f k It represents the image of the kth step of iteration. When Dfk When is not a zero vector, equation (8) is relaxed to:
[0125] min{||Af-z||1+α||Df||1- <f,αD1f k / ||Df k ||2>} (9)
[0126] In the formula, diag means diagonal matrix; D nn represents an element with n rows and n columns; <·> represents the inner product; the superscript k defines the number of iterations.
[0127] Equation (9) can be rewritten as:
[0128] min{||Af-z||1+α||Df||1+<f,v k+1 >} (10)
[0129] In the formula, v represents the gradient variable; v k+1 Defined as:
[0130] v k+1 =-αD1f k / ||Df k ||2 (11)
[0131] By introducing a new auxiliary variable d, equation (10) can be rewritten as:
[0132]
[0133] In the formula, st represents the constraint condition.
[0134] According to the ADMM principle, equation (12) can be rewritten as:
[0135]
[0136] In the formula, is a function of d and f, defined as:
[0137]
[0138] In the formula, the superscript T is defined as the transposition operator; u is a matrix, and its update format is:
[0139] u k+1 =u k +ρ(Af k+1 -zd k+1 ) (15)
[0140] Equation (13) can be separated into two sub-problems that are easier to solve:
[0141]
[0142] According to equations (13) and (16), the subproblem about d can be expressed as:
[0143]
[0144] Similarly, the subproblem about f can be expressed as:
[0145]
[0146] Rewriting equation (18) into a compact form:
[0147]
[0148] The solution of equation (17) is:
[0149] d k+1 =shrink(Af k -z+u k / ρ,1 / ρ) (20)
[0150] Where shrink(·,·) represents the soft threshold algorithm.
[0151] Equation (19) is an unconstrained non-smooth optimization problem. The present invention uses the forward-backward splitting algorithm FBSA to solve the equation, resulting in the following sub-problems:
[0152]
[0153] Among them, δ is a positive real number; s is a variable; s k+1 Defined as:
[0154] s k+1 =f k -δ(ρA T (Af k -zd k+1 +u k / ρ)+v k+1 ) (twenty two)
[0155] Capacitance tomography measurement technology is often used for fast-changing objects, such as multiphase flow, combustion and other processes, which require fast image reconstruction. In order to accelerate the convergence of the algorithm, the following acceleration format is introduced:
[0156] f k+1 =f k+1 / 2 +(o k -1)(f k+1 / 2 -f k ) / o k+1 (twenty three)
[0157] In the formula, o represents an auxiliary variable, and its update format is defined as:
[0158]
[0159] In the formula, o 1 =1, indicating the acceleration step length of the first step.
[0160] According to the above derivation process, the new algorithm is summarized in Algorithm II of Table 2. By executing Algorithm II, image reconstruction can be achieved.
[0161] Table 2
[0162]
[0163]
[0164] The reconstruction algorithm proposed in the present invention is summarized in Algorithm III of Table 3, referred to as Deep Learning Assisted Reconstruction (DLAR), which mainly includes the steps of DCNN training, DDP prediction and image reconstruction.
[0165] Table 3
[0166]
[0167] The feasibility and effectiveness of the DLAR algorithm are quantitatively evaluated using numerical simulation methods, and its performance is analyzed. Figure 1 The complex dielectric constant distribution in the image is reconstructed. Fourteen popular imaging algorithms are compared. Table 4 shows these algorithms, including iterative methods (Landweber, Kaczmarz, PRPCG, and TVR), non-iterative methods (TSVD, OIOR, LBP, and L2R), sparsity algorithms (SK, L1R, and L 1 / 2 R), low-rank regularization (LRR), and combined regularization (ENR and TV-L1-L2). The step size of Kaczmarz, PRPCG, Landweber, and SK algorithms is 1. Figure 2(a)-Figure 2(e) The iteration numbers of the reconstruction targets in are 269, 240, 1933, 322, and 400, respectively. In the Landweber method, the iteration numbers are 324, 328, 2877, 504, and 432, respectively. 1 / 2In the R algorithm, the regularization parameters are 0.008, 0.01, 0.003, 0.01, and 0.02, respectively. In the ENR method, the regularization parameters based on the L1 and L2 norms are 0.01 and 0.0001, respectively. In the TV-L1-L2 algorithm, the regularization parameters based on the L1 and TV norms are 0.01 and 0.0003, respectively. The singular value cutoff threshold of the TSVD method is 0.003, and the number of offline iterations of the OIOR algorithm is 500. In the DLAR algorithm, α = 0.001, and when the residual reaches ||f k+1 -f k || / ||f k ||≤10 -4 When , the iteration terminates. Figure 2(a)-Figure 2(o) and Table 5 are the reconstructed images and image errors (%), respectively. Among them, Figure 2(a) is Kaczmarz, Figure 2(b) is L1R, Figure 2(c) is Landweber, Figure 2(d) is SK, Figure 2(e) is TVR, Figure 2(f) is PRPCG, Figure 2(g) is LRR, Figure 2(h) is L1 / 2R, Figure 2(i) is TV-L1-L2, Figure 2(j) is ENR, Figure 2(k) is DLAR, Figure 2(l) is L2R, Figure 2(m) is OIOR, Figure 2(n) is LBP, and Figure 2(o) is TSVD; the gradient color bar indicates the dielectric constant value.
[0168] Table 4
[0169]
[0170] Table 5
[0171]
[0172] It can be observed from Figure 2 and Table 5 that for Figure 1 The DLAR algorithm achieves high-quality reconstruction and accurately reconstructs the spatial distribution of the medium. The image quality is much better than that of the compared algorithms. This result once again confirms the effectiveness of the DLAR algorithm.
[0173] From the data in Table 5, it can be observed that Figure 1 The image errors of the DLAR algorithm are 0.29% and 1.62×10 -2 %, 0.13%, 1.17×10 -2 % and 0.13%, which are smaller than the image errors of the other 14 algorithms. The LBP algorithm produces the largest image error. Figure 1 The image errors of the reconstruction targets in the image are 25.86%, 15.63%, 16.73%, 27.33% and 21.92% respectively. The evaluation results show that the new algorithm can improve the reconstruction quality and robustness compared with the popular reconstruction algorithms.
[0174] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.
Claims
1. A capacitance tomography reconstruction method integrating measurement principles, data, and domain knowledge, characterized in that: The following steps are involved: Step 1: Collect training samples and train the deep learning model, including: The deep convolutional neural network DCNN includes an output layer, an input layer, and a hidden layer; the hidden layer consists of a convolutional layer, a pooling layer, an activation function, and a fully connected layer; before using the deep convolutional neural network DCNN, training samples are collected to train the model parameters in a supervised manner to solve the following optimization problem: (1) In the formula, Indicates the minimum value of the function; Indicates that the function arrive Sum; l represents the total number of samples; is a pair of training samples, and They are Input and output samples; Represents the forward propagation operator, whose form is determined by the deep convolutional neural network DCNN structure; represents the model parameters; represents the square of L2 norm; Step 2: Reconstruct the initial image based on the capacitance vector, including: The image reconstruction model is approximately as follows: (2) In the formula, , , and are the dimensionless normalized sensitivity matrix, dielectric constant, capacitance vector, and noise vector; their dimensions are , , and ; m and n both represent the dimensions of the matrix; Step 3: Use the trained deep learning model to infer the data-driven prior DDP, including: Based on the collected training data, a deep convolutional neural network (DCNN) is used to predict the data-driven prior DDP, which is defined as: (3) In the formula, represents data-driven prior DDP; Represents the forward propagation operator in the deep learning model; is the initial input image; represents the model parameters; Indicates that the given conditions are; Step 4: Design a reconstruction model that integrates the measurement principle, data-driven prior DDP, and sparse prior, including: The regularization method is used to integrate the measurement mechanism and image priors. The regularization method is expressed as the following mathematical model: (4) In the formula, , , and are the regularization term, data fidelity term, regularization parameter, and the number of regularization terms; Use L1 norm as data fidelity term: (5) In the formula, is the L1 norm; Using sparse prior to improve reconstruction quality, L 1-2 The norm is the sparsity regularization term, defined as: (6) Where, L 1-2 Indicates L 1-2 norm; Weighted L 1-2 The norm increases flexibility and is defined as: (7) In the formula, Indicates L 1-2 norm; is a non-negative diagonal matrix that will change with iterations; the initial weight matrix is represented by the data-driven prior DDP; According to equations (4), (5) and (7), the new imaging model is expressed as: (8) In the formula, is the regularization parameter; Step 5: Solve the reconstruction model and complete image reconstruction, including: The alternating direction multiplier method ADMM is used to solve the imaging model equation (8); According to the optimization principle of differential convex function, Represents the image of the k-th step of iteration. When is not a zero vector, equation (8) is relaxed to: (9) In the formula, ;diag represents a diagonal matrix; Represents elements of n rows and n columns; denotes inner product; superscript The number of iterations is defined; Equation (9) can be rewritten as: (10) In the formula, represents the gradient variable; Defined as: (11) Introducing new auxiliary variables , rewrite equation (10) as: (12) In the formula, Indicates constraints; According to the principle of fusion alternating direction multiplier method ADMM, equation (12) can be rewritten as: (13) In the formula, is a function of d and f, defined as: (14) In the formula, the superscript Defined as the transpose operator; is a matrix, and its update format is: (15) Separate equation (13) into two more easily solvable subproblems: (16) According to equations (13) and (16), The sub-problem is expressed as: (17) about The sub-problem is expressed as: (18) Rewriting equation (18) into a compact form: (19) The solution of equation (17) is: (20) In the formula, represents the soft threshold algorithm; Solving equation (19) using the forward-backward splitting algorithm FBSA leads to the following subproblems: (21) in, is a positive real number; is a variable; Defined as: (22) The following acceleration formats are introduced: (23) In the formula, Represents auxiliary variables, whose update format is defined as: (24) In the formula, , represents the acceleration step length of the first step.