Three-dimensional electromagnetic inverse scattering imaging inversion method, device and system based on prime dual gradient method and medium

Abstract

The invention provides a three-dimensional electromagnetic inverse scattering imaging inversion method, device and system based on a primal dual gradient method and a medium, and relates to the technical field of image processing, and the method comprises the steps: determining an inversion imaging model based on a contraction integral model of Fourier basis expansion regularization; constructing an optimization model for inversion imaging based on the inversion imaging model and L2 / 3 regularization; obtaining electromagnetic field data to be inverted, inputting the electromagnetic field data to be inverted into the optimization model, and solving the optimization model by using a primal dual gradient method in an alternate iteration manner to obtain an inverted image; according to the invention, the definition of inversion imaging can be improved.

Description

Technical Field

[0001] The present application relates to, but is not limited to, the field of image processing technology, and in particular to a three-dimensional electromagnetic inverse scattering imaging inversion method, device, system, and medium based on a primal-dual gradient method. Background Art

[0002] Electromagnetic inverse scattering imaging is a non-invasive imaging method that uses known incident and scattered field information to infer the target object's shape, position, and constitutive properties (such as dielectric constant and conductivity). This technique, a key branch of electromagnetics, is widely used in medicine, civil engineering, industry, and the military.

[0003] For the inverse scattering problem, the solution must exist and be unique. Since most inverse scattering data is obtained through instrument measurement, there will be noise pollution, measurement instrument errors, measurement inaccuracies and other factors in the measurement process, which will lead to unstable and discontinuous measurement data. Therefore, the inverse scattering problem is ill-posed. In addition, the inverse scattering problem is also nonlinear. This is precisely because the inverse scattering problem involves the relationship between the scattered field and the scatterer parameters. Obviously, this relationship is not linear. The main reason is that there is a multiple scattering effect. There are many scattering particles in the scatterer. When the dielectric constant of the scatterer doubles, the scattered field does not simply double. Secondly, expensive phase measurement instruments lead to the lack of phase information in the measurement data of the inverse scattering problem. The lack of phase information further deepens the nonlinearity of the problem.

[0004] The nonlinearity and ill-posedness of the inverse scattering problem are two of the greatest challenges. The goal is to invert high-quality images from noisy scattered and incident field data. Addressing these nonlinearities and ill-posedness to improve the clarity of inversion imaging is a long-standing challenge in the industry. Summary of the Invention

[0005] The following is a summary of the subject matter described in detail herein. This summary is not intended to limit the scope of the claims.

[0006] The embodiments of the present application provide a three-dimensional electromagnetic inverse scattering imaging inversion method, device, system and medium based on the primal-dual gradient method to improve the clarity of the inversion imaging.

[0007] In a first aspect, an embodiment of the present application provides a three-dimensional electromagnetic inverse scattering imaging inversion method based on a primal-dual gradient method, comprising the following steps:

[0008] S100, determining the inversion imaging model based on the shrinkage integral model with Fourier basis expansion regularization;

[0009] In S100, the inversion imaging model is determined based on the shrinkage integral model of Fourier basis expansion regularization, including: the determined shrinkage integral model, whose data equation and state equation are

[0010]

[0011] in, is the scattered field, is the incident field, I l is a source of comparison, is the integral operator with the dyadic Green's function G as the integral kernel, R(r) is the modified contrast function, β(r) is the auxiliary parameter, χ(r) is the contrast function, D is the region of interest, S is the receiver area, l Representative l Second incidence.

[0012] The full contrast source is expanded using the discrete Fourier basis using the Fourier basis regularization. The specific expression is:

[0013]

[0014] in, is a source of comparison, is a vectorized discrete Fourier basis, IDFT represents the inverse discrete Fourier transform performed by the 3D fast Fourier transform algorithm, is the 3D Fourier coefficient tensor.

[0015] The inversion imaging model is determined based on the shrinkage integral model of Fourier basis expansion regularization. The inversion imaging model is:

[0016]

[0017] in, and The distribution represents the data equation residuals and the state equation residuals, and τ>0 is the weight factor.

[0018] S200, based on the inversion imaging model and L 2 / 3 Regularization is used to construct an optimization model for inversion imaging;

[0019] In S200, the inversion imaging model and L 2 / 3 Regularization builds an optimization model for inversion imaging, including:

[0020] Based on the inversion imaging model and the L 2 / 3 Regularized construction of a primal-dual gradient method shrinkage integral model, and using the non-convex regularized shrinkage integral model as an optimization model for inversion imaging;

[0021] The non-convex regularized shrinkage integral model is

[0022]

[0023] in, Indicates L 2 / 3 norm.

[0024] S300 , obtaining electromagnetic field data to be inverted, inputting the electromagnetic field data to be inverted into the optimization model, and solving the optimization model by alternating iteration using the primal-dual gradient method to obtain an inverted image.

[0025] In S300, the electromagnetic field data to be inverted is input into the optimization model, and the optimization model is solved by alternating iteration using the primal-dual gradient method to obtain an inverted image, including:

[0026] S310, initializing the number of iterations, stopping criteria, regularization parameter μ, auxiliary parameter β and weight factor τ;

[0027] S320: Input the data to be inverted into the optimization model, and convert the optimization model into the following dual model:

[0028]

[0029] Among them, P and Q are dual variables.

[0030] S330, sequentially solving the solution that maximizes P, the solution that maximizes Q, and the solution that minimizes The solution and minimization The solution;

[0031] S340, accumulate the number of iterations and the stopping criteria of the current generation, and determine whether the preset maximum number of iterations and the stopping criteria are reached. If so, execute S350; if not, minimize The solution is used as the data to be inverted, and S320 is executed;

[0032] S350, will minimize The solution is output as the inverted image.

[0033] In S330, the solution to maximize P, the solution to maximize Q, and the solution to minimize P in the dual model are obtained in sequence. The solution and minimization The solution includes:

[0034] The terms containing the dual parameters P and Q in the dual model are extracted according to the following formula to obtain:

[0035]

[0036] in, is the dual update variable, and γ1 is the dual auxiliary parameter.

[0037] The dual model contains the original parameters according to the following formula Extract the items and get:

[0038]

[0039] Among them, γ2 is the original auxiliary parameter.

[0040] In order to obtain better initial values for the above problem, solve P, Q, Optimize the subproblem and get P, Q, Closed-form solution to the optimization subproblem:

[0041]

[0042] in, is the relative Fourier coefficient exist The derivative at .

[0043] Use the solutions of the above three optimization subproblems to update the modified contrast function The sub-problem is

[0044]

[0045] In order to solve the above sub-problems, we introduce L 2 / 3 The proximity operator of the function, and then we get Optimize the solution of the subproblem:

[0046]

[0047] in,

[0048] In a second aspect, an embodiment of the present application further provides a three-dimensional electromagnetic inverse scattering imaging inversion device based on the primal dual gradient method, wherein the three-dimensional electromagnetic inverse scattering imaging inversion device based on the primal dual gradient method comprises:

[0049] The first module determines the inversion imaging model based on the shrinkage integral model regularized by Fourier basis expansion;

[0050] The second module is based on the inversion imaging model and L 2 / 3 Regularization is used to construct an optimization model for inversion imaging;

[0051] The third module obtains the electromagnetic field data to be inverted, inputs the electromagnetic field data to be inverted into the optimization model, and solves the optimization model by alternating iteration using the primal dual gradient method to obtain the inverted imaging.

[0052] In a third aspect, an embodiment of the present application also provides a three-dimensional electromagnetic inverse scattering imaging inversion system based on the primal dual gradient method, comprising: a collector, a memory, a processor, a display, and a computer program stored on the memory and runnable on the processor. When the processor executes the computer program, it implements a three-dimensional electromagnetic inverse scattering imaging inversion method based on the primal dual gradient method as described in the first aspect.

[0053] In a fourth aspect, an embodiment of the present application also provides a three-dimensional electromagnetic inverse scattering imaging inversion medium based on the primal dual gradient method, which is a computer-readable storage medium storing computer-executable instructions. The computer-executable instructions are used to execute a three-dimensional electromagnetic inverse scattering imaging inversion method based on the primal dual gradient method as described in the first aspect.

[0054] The embodiments of the present application include the following beneficial effects: In the embodiments provided in the present application, the inversion imaging model and L 2 / 3 Combined with regularization, the constructed optimization model parameters are more flexible and more suitable for electromagnetic inverse scattering reconstruction and inversion imaging clarity. The optimization model can be solved by alternating iterations using the primal-dual gradient method to obtain the inverted image. This application can improve the clarity of the inversion imaging.

[0055] Other features and advantages of the present application will be described in the following description, and in part will become apparent from the description, or will be understood by practicing the present application. The purposes and other advantages of the present application can be achieved and obtained through the structures particularly pointed out in the description, claims and drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] The accompanying drawings are used to provide a further understanding of the technical solution of the present application and constitute a part of the specification. Together with the embodiments of the present application, they are used to explain the technical solution of the present application and do not constitute a limitation on the technical solution of the present application.

[0057] Figure 1 This is a flowchart of a three-dimensional electromagnetic inverse scattering imaging inversion method based on the primal-dual gradient method provided by one embodiment of the present application;

[0058] Figure 2 This is an inversion comparison diagram of two dielectric cube structures provided by an embodiment of the present application;

[0059] Figure 3 This is an inversion comparison diagram of two dielectric sphere targets provided by an embodiment of the present application;

[0060] Figure 4This is a structural diagram of a three-dimensional electromagnetic inverse scattering imaging inversion device based on the primal-dual gradient method provided by one embodiment of the present application;

[0061] Figure 5 This is a structural diagram of a three-dimensional electromagnetic inverse scattering imaging inversion system based on the primal-dual gradient method provided in one embodiment of the present application. DETAILED DESCRIPTION

[0062] In order to make the purpose, technical solutions and advantages of this application more clear, the following further describes this application in detail with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain this application and are not intended to limit this application.

[0063] It should be noted that although the device schematics illustrate functional module divisions and the flowcharts illustrate logical sequences, in certain circumstances, the steps shown or described may be performed in a sequence that differs from the module divisions in the device or the sequence in the flowcharts. The terms "first," "second," and the like in the specification, claims, or accompanying drawings are used to distinguish similar items and are not necessarily used to describe a specific sequence or precedence.

[0064] refer to Figure 1 The present application provides a three-dimensional electromagnetic inverse scattering imaging inversion method based on the primal-dual gradient method, which includes the following steps:

[0065] S100, determining the inversion imaging model based on the shrinkage integral model with Fourier basis expansion regularization;

[0066] In S100, the inversion imaging model is determined based on the shrinkage integral model of Fourier basis expansion regularization, including: the determined shrinkage integral model, whose data equation and state equation are

[0067]

[0068] in, is the scattered field, is the incident field, I l is a source of comparison, is the integral operator with the dyadic Green's function G as the integral kernel, R(r) is the modified contrast function, β(r) is the auxiliary parameter, χ(r) is the contrast function, D is the region of interest, S is the receiver area, l Representative l Second incidence.

[0069] The full contrast source is expanded using the discrete Fourier basis using the Fourier basis regularization. The specific expression is:

[0070]

[0071] in, is a source of comparison, is a vectorized discrete Fourier basis, IDFT represents the inverse discrete Fourier transform performed by the 3D fast Fourier transform algorithm, is the 3D Fourier coefficient tensor.

[0072] The inversion imaging model is determined based on the shrinkage integral model of Fourier basis expansion regularization. The inversion imaging model is:

[0073]

[0074] in, and The distribution represents the data equation residuals and the state equation residuals, and τ>0 is the weight factor.

[0075] S200, based on the inversion imaging model and L 2 / 3 Regularization is used to construct an optimization model for inversion imaging;

[0076] In S200, the inversion imaging model and L 2 / 3 Regularization builds an optimization model for inversion imaging, including:

[0077] Based on the inversion imaging model and the L 2 / 3 Regularization constructs a non-convex regularized shrinkage integral model, and uses the non-convex regularized shrinkage integral model as an optimization model for inversion imaging;

[0078] The non-convex regularized shrinkage integral model is

[0079]

[0080] in, Indicates L 2 / 3 norm.

[0081] S300 , obtaining electromagnetic field data to be inverted, inputting the electromagnetic field data to be inverted into the optimization model, and solving the optimization model by alternating iteration using the primal-dual gradient method to obtain an inverted image.

[0082] In S300, the electromagnetic field data to be inverted is input into the optimization model, and the optimization model is solved by alternating iteration using the primal-dual gradient method to obtain an inverted image, including:

[0083] S310, initializing the number of iterations, stopping criteria, regularization parameter μ, auxiliary parameter β and weight factor τ;

[0084] S320: Input the data to be inverted into the optimization model, and convert the optimization model into the following dual model:

[0085]

[0086] Among them, P and Q are dual variables.

[0087] S330, sequentially solving the solution that maximizes P, the solution that maximizes Q, and the solution that minimizes The solution and minimization The solution;

[0088] S340, accumulate the number of iterations and the stopping criteria of the current generation, and determine whether the preset maximum number of iterations and the stopping criteria are reached. If so, execute S350; if not, minimize The solution is used as the data to be inverted, and S320 is executed;

[0089] S350, will minimize The solution is output as the inverted image.

[0090] In S330, the solution to maximize P, the solution to maximize Q, and the solution to minimize P in the dual model are obtained in sequence. The solution and minimization The solution includes:

[0091] The terms containing the dual parameters P and Q in the dual model are extracted according to the following formula to obtain:

[0092]

[0093] in, is the dual update variable, and γ1 is the dual auxiliary parameter.

[0094] The dual model contains the original parameters according to the following formula Extract the items and get:

[0095]

[0096] Among them, γ2 is the original auxiliary parameter.

[0097] In order to obtain better initial values for the above problem, solve P, Q, Optimize the subproblem and get P, Q, Closed-form solution to the optimization subproblem:

[0098]

[0099] in, is the relative Fourier coefficient exist The derivative at .

[0100] Use the solutions of the above three optimization subproblems to update the modified contrast function Its sub-problem is

[0101]

[0102] In order to solve the above sub-problems, we introduce L 2 / 3 The proximity operator of the function, and then we get Optimize the solution of the subproblem:

[0103]

[0104] in,

[0105] In the embodiment provided in the present application, the contraction integral equation (CIE) regularized by Fourier bases-expansion (FBE) can effectively preserve the edge information of the image; the inversion imaging model and L 2 / 3 Regularization is combined to construct an optimization model. The primal-dual hybrid gradient method (PDHG) is used to alternately iteratively solve the optimization model, transforming the unconstrained non-convex problem into an equality-constrained problem, which is equivalent to multi-block iterative optimization, thereby solving the inverted imaging. The optimization model constructed in this application and the corresponding solution method can improve the clarity of the inverted imaging.

[0106] In order to invert the three-dimensional electromagnetic inverse scattering imaging, the number of iterations, the stopping criterion, the regularization parameter μ, the auxiliary parameter β and the weight factor τ are first initialized; then, the electromagnetic field data to be inverted is input into the optimization model, and the optimization model is solved: each time the solution is completed, the number of iterations of the current generation and the stopping criterion are accumulated once, and the steps of solving the minimization model are continued to be iteratively executed until the preset maximum number of iterations and the stopping criterion are reached, and the optimization model is solved. The solution to the optimization subproblem is output as an inverted image.

[0107] In some embodiments, the number of iterations is preset to k=2, and the stopping criterion is the real part structural similarity SSIM-Re tolerance value of 10 -5 .

[0108] As an optional embodiment, in S300, the method further includes: outputting evaluation index parameters for evaluating the inverted image, wherein the evaluation index parameters include the scattered field error Err sct , structural similarity SSIM;

[0109]

[0110] in, represents the true relative permittivity distribution, Represents the reconstruction result. θ>0、 and ξ>0 control the relative importance of the three terms, are the brightness, contrast, and structure components, and the SSIM values of the real and imaginary parts are denoted as SSIM-Re and SSIM-Im, respectively.

[0111] In this embodiment, the scattered field error Err stc , structural similarity SSIM to evaluate the inverted image quality.

[0112] The following is a classic electromagnetic inverse scattering imaging experiment conducted based on the model proposed in the examples of this application, and verifies the efficiency and feasibility of the non-convex regularized shrinkage integral model (NRCIE). First, the electromagnetic field data to be inverted is obtained, and the electromagnetic field data to be inverted is input into the optimization model to obtain the inverted image.

[0113] In some embodiments, the inversion object is a structure of two dielectric cubes. In some embodiments, two dielectric sphere targets are used as target objects, and the original data is recorded as After determining the original data, load the original data to be inverted.

[0114] It should be noted that all target objects are experimental data from the Fresnel Institute, which are used as inversion targets for the experiment. The purpose of this embodiment is to image the electromagnetic field data after inversion.

[0115] like Figure 2 As shown, Figure 2 Inversion images of two dielectric cube structures using different algorithms, each set of images corresponds to a different algorithm. (a) Real and imaginary parts of three cross-sections of the real structure (z = 14.4 mm, y = 5.6 mm, x = 5.6 mm); (b) Inversion image using the MR-CSI method; (c) Inversion image using the MR-TSOM-NIE method; and (d) Inversion image using the PD-FBE-WCIE method. It can be seen that the images inverted by the proposed method are superior in quality to those inverted by other methods, with greater detail. The images inverted by the MR-CSI method have noticeable excessive shadows, while the images inverted by the MR-TSOM-NIE method are relatively blurry, demonstrating the importance of method selection in the experiment.

[0116] like Figure 3As shown, Figure 3 Figure 1 shows the inversion images of two dielectric sphere targets using different algorithms. Figure (a) shows the real and imaginary parts of three cross-sections of the actual structure (z = 14.4 mm, y = 5.6 mm, and x = 5.6 mm). Figure (b) shows the inversion image using the MR-CSI method, Figure (c) shows the inversion image using the MR-TSOM-NIE method, and Figure (d) shows the inversion image using the FBE-WCIE-PD method. Image analysis shows that the inversion method proposed in this paper is more effective than the MR-CSI and MR-TSOM-NIE methods, and the inversion results are clearer.

[0117] Referring to Table 1, by comparing the experimental results of all the methods in Table 1, it can be found that the various indicators of the FBE-WCIE-PD method proposed in this application have been significantly improved, and the comprehensive experimental results are better.

[0118] Table 1: Experimental results of each method;

[0119]

[0120] The above two experiments verify that the NRCIE model provided in this application can be significantly improved. Compared with the existing MR-CSI method and MR-TSOM-NIE method, the NRCIE model has obvious numerical advantages.

[0121] In addition, reference Figure 4 In some embodiments, a three-dimensional electromagnetic inverse scattering imaging inversion device based on the primal dual gradient method is further provided. The three-dimensional electromagnetic inverse scattering imaging inversion device based on the primal dual gradient method includes:

[0122] The first module determines the inversion imaging model based on the shrinkage integral model regularized by Fourier basis expansion;

[0123] The second module is based on the inversion imaging model and L 2 / 3 Regularization is used to construct an optimization model for inversion imaging;

[0124] The third module obtains the electromagnetic field data to be inverted, inputs the electromagnetic field data to be inverted into the optimization model, and solves the optimization model by alternating iteration using the primal dual gradient method to obtain the inverted imaging.

[0125] The device embodiments described above are merely illustrative. The units described as separate components may or may not be physically separate, i.e., they may be located in one place or distributed across multiple network units. Some or all of the modules may be selected based on actual needs to achieve the objectives of this embodiment.

[0126] In addition, refer to Figure 5 An embodiment of the present application also provides a three-dimensional electromagnetic inverse scattering imaging inversion system based on the primal-dual gradient method, which includes: a collector 11, a memory 12, a processor 13, a display 14, and a computer program stored in the memory 12 and executable on the processor 13.

[0127] The processor 13 and the memory 12 may be connected via a bus or other means.

[0128] The inversion data required to implement a three-dimensional electromagnetic inverse scattering imaging inversion method based on the original dual gradient method in the above embodiment is extracted by the collector 11, and the non-transient software program and instructions are stored in the memory 12. When executed by the processor 13, a three-dimensional electromagnetic inverse scattering imaging inversion method based on the original dual gradient method in the above embodiment is executed, and finally imaging visualization is realized through the display 14.

[0129] In addition, an embodiment of the present application also provides a three-dimensional electromagnetic inverse scattering imaging inversion medium based on the primal dual gradient method, and the computer-readable storage medium stores computer-executable instructions, which are executed by a processor or controller, for example, by a processor in the above-mentioned electronic device embodiment, so that the above-mentioned processor can execute a three-dimensional electromagnetic inverse scattering imaging inversion method based on the primal dual gradient method in the above-mentioned embodiment.

[0130] It will be understood by those skilled in the art that all or some of the steps and systems in the methods disclosed above can be implemented as software, firmware, hardware, and appropriate combinations thereof. Certain physical components can be implemented in software form by a processor (such as a central processing unit, a digital signal processor, or a microprocessor), or as hardware or an application-specific integrated circuit. Such software can be distributed on computer-readable media, including computer storage media (non-transitory) and communication media (transitory). According to knowledge in the art, computer storage media refers to all media used to store volatile and non-volatile information, such as RAM, ROM, flash memory, CD-ROM, DVD, magnetic tape, etc., and includes removable and non-removable forms. In addition, communication media generally contain computer-readable instructions and other information, which are transmitted via a carrier or transmission mechanism. The above is a specific description of the preferred implementation of the present application, but the present application is not limited to the above-mentioned embodiments. Those skilled in the art can also make various equivalent modifications or substitutions without violating the spirit of the present application, and these equivalent modifications or substitutions are all included in the scope defined by the claims of the present application.

Claims

1. A three-dimensional electromagnetic inverse scattering imaging inversion method based on the primal dual gradient method, characterized in that: The following steps are involved: S100, determining the inversion imaging model based on the shrinkage integral model with Fourier basis expansion regularization; S200, based on the inversion imaging model and L 2 / 3 Regularization is used to construct an optimization model for inversion imaging; S300 , obtaining electromagnetic field data to be inverted, inputting the electromagnetic field data to be inverted into the optimization model, and solving the optimization model by alternating iteration using the primal-dual gradient method to obtain an inverted image.

2. The three-dimensional electromagnetic inverse scattering imaging inversion method based on the primal-dual gradient method according to claim 1, characterized in that: In S100, the inversion imaging model is determined based on the shrinkage integral model with Fourier basis expansion regularization, including: The data equation and state equation of the determined shrinkage integral model are: in, is the scattered field, is the incident field, I l is a source of comparison, is the integral operator with the dyadic Green's function G as the integral kernel, R(r) is the modified contrast function, β(r) is the auxiliary parameter, χ(r) is the contrast function, D is the region of interest, S is the receiver area, l Representative l Second incidence. The full contrast source is expanded using the discrete Fourier basis using the Fourier basis regularization. The specific expression is: in, is a source of comparison, is the vectorized discrete Fourier basis, are Fourier coefficients, IDFT represents the inverse discrete Fourier transform performed by the 3D fast Fourier transform algorithm, is the 3D Fourier coefficient tensor. The inversion imaging model is determined based on the shrinkage integral model of Fourier basis expansion regularization. The inversion imaging model is: in, and The distribution represents the data equation residuals and the state equation residuals, and τ>0 is the weight factor.

3. The three-dimensional electromagnetic inverse scattering imaging inversion method based on the primal-dual gradient method according to claim 2, characterized in that: In S200, based on the inversion imaging model and L 2 / 3 Regularization builds an optimization model for inversion imaging, including: Based on the inversion imaging model and the L 2 / 3 Regularization constructs a non-convex regularized shrinkage integral model, and uses the non-convex regularized shrinkage integral model as an optimization model for inversion imaging; The non-convex regularized shrinkage integral model is in, Indicates L 2 / 3 norm.

4. The three-dimensional electromagnetic inverse scattering imaging inversion method based on the primal-dual gradient method according to claim 3, characterized in that: In S300, the data to be inverted is input into the optimization model, and the optimization model is solved by alternating iteration using the primal-dual gradient method to obtain an inverted image, including: S310, initializing the number of iterations, stopping criteria, regularization parameter μ, auxiliary parameter β and weight factor τ; S320: Input the data to be inverted into the optimization model, and convert the optimization model into the following dual model: Among them, P and Q are dual variables. S330, sequentially solving the solution that maximizes P, the solution that maximizes Q, and the solution that minimizes The solution and minimization The solution; S340, accumulate the number of iterations and the stopping criteria of the current generation, and determine whether the preset maximum number of iterations and the stopping criteria are reached. If so, execute S350; if not, minimize The solution is used as the data to be inverted, and S320 is executed; S350, will minimize The solution is output as the inverted image. In S330, the solution to maximize P, the solution to maximize Q, and the solution to minimize P in the dual model are obtained in sequence. The solution and minimization The solution includes: The terms containing the dual parameters P and Q in the dual model are extracted according to the following formula to obtain: in, is the dual update variable, and γ1 is the dual auxiliary parameter. The dual model contains the original parameters according to the following formula Extract the items and get: Among them, γ2 is the original auxiliary parameter. In order to obtain better initial values for the above problem, solve P, Q, Optimize the subproblem and get P, Q, Closed-form solution to the optimization subproblem: in, is the relative Fourier coefficient exist The derivative at . Use the solutions of the above three optimization subproblems to update the modified contrast function Its sub-problem is In order to solve the above sub-problems, we introduce L 2 / 3 The proximity operator of the function, and then we get Optimize the solution of the subproblem: in,

Images