Sparse microwave imaging method and system based on near-end gradient descent method

Abstract

The invention provides a sparse microwave imaging method and system based on a near-end gradient descent method, and relates to the technical field of information processing. The echo signals are converted into a matrix form, and an under-sampling observation signal matrix X, a sampling matrix P and a complete sampling signal matrix S are obtained; the relationship among the matrixes is X = Pcircle S; a rank minimization problem of signal recovery is converted into a nuclear norm minimization problem; constructing a Lagrange function of a nuclear norm minimization problem, and performing iterative solution by applying a near-end gradient descent method to obtain a recovered signal matrix; and substituting the recovered signal matrix into an imaging inversion formula, calculating to obtain a target reflectivity function, and generating an image according to the target reflectivity function. According to the method, the original signal is recovered from the under-sampling observation signal, the reconstruction precision of the imaging matrix is improved, the recovery difficulty of the low-rank matrix is reduced, and the algorithm efficiency is improved at the same time.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of information processing, in particular to a sparse microwave imaging method and system based on proximal gradient descent method. BACKGROUND

[0002] The statements in this section merely provide background technology related to the present disclosure and do not necessarily constitute prior art.

[0003] Microwave imaging technology, especially Synthetic Aperture Radar (SAR) technology, has been widely used in many fields such as security and safety inspection, post-disaster rescue, environmental monitoring and military reconnaissance due to its unique advantages of all-weather and all-day operation, strong penetration and no influence of weather. However, with the increasing requirements of resolution and coverage range in application scenarios, the amount of data that the system needs to collect and process increases dramatically, which brings great pressure and challenges to the data acquisition, transmission and storage system.

[0004] In order to cope with the above challenges, the sparse microwave imaging theory emerged as the times require. This theory is an advanced imaging technology that uses the prior knowledge that the scene targets have sparsity (i.e. only a small number of non-zero values) in a certain domain (such as wave number domain, wavelet domain). The core idea is to collect signals with much less data than required by the Nyquist sampling theorem, and to reconstruct high-quality images using nonlinear optimization algorithms. This greatly relieves the data pressure of the system. Existing sparse imaging algorithms mainly include data recovery algorithms and compressed sensing algorithms, and the present application mainly improves the data recovery algorithms.

[0005] In the data recovery algorithm, the low-rank property of the echo signal matrix is usually used to recover the original signal. Its mathematical model can be summarized as a low-rank matrix recovery problem, i.e. how to recover the complete low-rank matrix from partial observation data. In this optimization problem, the minimization of the rank function is a key NP-hard problem. For this reason, the nuclear norm (sum of singular values of a matrix) is generally used as a convex relaxation of the rank function to solve it.

[0006] However, there is a significant problem in the prior art: although the nuclear norm is convex, it is non-smooth (non-differentiable) itself. This property makes many traditional optimization algorithms that rely on gradient information and are suitable for smooth functions (such as gradient descent method) cannot be directly applied to solve optimization problems containing nuclear norm regularization term. Direct solution often faces problems such as difficulty in convergence, slow convergence speed or poor recovery accuracy. Therefore, how to effectively and efficiently handle the non-smoothness of the nuclear norm has become a key technical problem that needs to be solved in sparse microwave imaging technology. SUMMARY

[0007] To solve the above problems, the application provides a sparse microwave imaging method and system based on a proximal gradient descent method, adopts the proximal gradient descent method, separates the smooth term and the non-smooth term in the function, adopts the singular value thresholding operation on the non-smooth kernel norm part, effectively deals with the optimization problem caused by the kernel norm, and reduces the complexity of the problem.

[0008] The first aspect of the application provides a sparse microwave imaging method based on a proximal gradient descent method, comprising:

[0009] receiving echo signals;

[0010] transforming the echo signals into a matrix form to obtain an undersampling observation signal matrix X, a sampling matrix P and a full sampling signal matrix S; the relationship between the matrices is X=P⊙S;

[0011] transforming the rank minimization problem of signal recovery into a kernel norm minimization problem:

[0012]

[0013] wherein, ||. * represents the kernel norm;

[0014] constructing a Lagrange function of the kernel norm minimization problem, and applying the proximal gradient descent method to iteratively solve, to obtain a recovered signal matrix;

[0015] substituting the obtained recovered signal matrix into an imaging inversion formula to calculate a target reflectivity function, and generating an image according to the target reflectivity function.

[0016] Further, the Lagrange function is:

[0017]

[0018] wherein, rho is a parameter, Y is an auxiliary variable, and ||. * is a kernel norm regularization term, is a non-smooth term, <,> is a Lagrange multiplier term, is a penalty term of constraint relaxation.

[0019] Further, the application of the proximal gradient descent method to iteratively solve comprises:

[0020] iteratively updating variables X and Y, and the update formula is:

[0021]

[0022] wherein, k is the iteration number, eta is the learning rate parameter, and D{} represents the singular value soft thresholding operation.

[0023] Further, the imaging inversion formula is:

[0024]

[0025] wherein FFT2 represents a two-dimensional Fourier transform, IFFT2 represents a two-dimensional inverse Fourier transform, k z is a component of wave number in the z direction.

[0026] Further, the echo signal is a linear frequency modulation signal.

[0027] Further, the expression of the echo signal is:

[0028] s(x ′ ,y ′ ,z ′ =z0) = ∫∫g(x, y, z=0)e -j2kR dxdy;

[0029] wherein g(x0, y0, z=0) is a reflectivity function of the target, R is a distance from the antenna to the target, j is an imaginary unit, k is a wave number, k=2π / λ, and λ is a wavelength of the echo signal.

[0030] Further, the fully sampled signal matrix The sampling matrix P∈{0,1} m×n , wherein m and n are respectively the number of sampling position points of the acquisition system along the horizontal and vertical directions.

[0031] The second aspect of the present application provides a sparse microwave imaging system based on a proximal gradient descent method, comprising:

[0032] A signal receiving module is configured to receive an echo signal.

[0033] A matrix conversion module is configured to convert the echo signal into a matrix form to obtain an undersampled observation signal matrix X, a sampling matrix P, and a fully sampled signal matrix S; the relationship between the matrices is X=P⊙S.

[0034] An optimization problem construction module is configured to convert a rank minimization problem of signal recovery into a kernel norm minimization problem:

[0035]

[0036] wherein ||·||0 represents a kernel norm. *

[0037] A proximal gradient solving module is configured to construct a Lagrange function of the kernel norm minimization problem and apply a proximal gradient descent method to iteratively solve, so as to obtain a recovered signal matrix.

[0038] ​An imaging module is configured to substitute the restored signal matrix into an imaging inversion formula to calculate a target reflectivity function, and generate an image based on the target reflectivity function.

[0039] A third aspect of the present application provides a sparse microwave imaging device based on proximal gradient descent method, the device comprising a memory and a processor; the memory is used to store a computer program; the processor is used to realize the above-mentioned sparse microwave imaging method based on proximal gradient descent method when the computer program is executed.

[0040] A fourth aspect of the present application provides a computer readable storage medium, the storage medium stores a computer program, when the computer program is executed by a processor, the above-mentioned sparse microwave imaging method based on proximal gradient descent method is realized.

[0041] Compared with the prior art, the sparse microwave imaging method and system based on proximal gradient descent method provided by the present application have the following beneficial effects:

[0042] (1) For the technical problem that the nuclear norm is non-smooth and non-differentiable, which leads to the fact that the traditional optimization algorithm cannot be directly applied, the proximal gradient descent method is used for iterative solution in the present application, the complex optimization problem is decomposed into a smooth item (multiplier item and penalty item) and a non-smooth item (nuclear norm item) by constructing a Lagrange function, and they are processed respectively. For the non-smooth nuclear norm item, the proximal operator is directly used to solve its proximal mapping, thereby effectively avoiding the difficulty of direct derivation. This method realizes the stable convergence of the optimization process, and significantly improves the practicability and reliability of the algorithm in the low-rank matrix recovery problem.

[0043] (2) The present application can more accurately restore the original signal matrix from the undersampled observation data which is much lower than the Nyquist rate. Through accurate nuclear norm minimization, the distortion and detail loss phenomenon in the signal recovery process is effectively suppressed, and finally a higher quality image reconstruction is realized, which improves the clarity and accuracy of the imaging.

[0044] (3) The proximal gradient descent method used in the present application has a clear structure, and the iterative formula is simple. The complex matrix recovery problem is converted into a series of simple gradient updates and singular value decomposition threshold operations. This reduces the dependence on special optimization solvers, facilitates implementation on general computing devices, and improves the execution efficiency of the algorithm. BRIEF DESCRIPTION OF DRAWINGS

[0045] The accompanying drawings, which form a part of this disclosure, are used to provide a further understanding of the present disclosure, and the illustrative embodiments of the present disclosure and their descriptions are used to explain the present disclosure, and do not constitute improper limitations on the present disclosure.

[0046] Figure 1 is a step flow chart of the sparse microwave imaging method based on the proximal gradient descent method provided by the embodiment one of the present application.

[0047] Figure 2 is a whole process schematic diagram from signal acquisition to completion of imaging provided by the embodiment one of the present application. It includes pre-processing of echo signals, analyzing the relationship between under-sampling signals and full-sampling signals, reconstructing the signal S by solving the kernel norm minimization problem, applying the proximal gradient descent method to solve the optimization problem, and finally bringing the obtained signal matrix back to the inversion formula to obtain the target reflectivity function and complete imaging.

[0048] Figure 3 is a specific implementation schematic diagram of the proximal gradient descent method provided by the embodiment one of the present application. First, the parameters are initialized, and a new function is constructed. The partial derivative of X is solved by the new function, the singular value soft thresholding operation is applied, and the update formula of X is obtained. The partial derivative of the Lagrange function is solved with respect to Y, and the gradient descent method is applied to obtain the update formula of Y. Through continuous updating, convergence judgment is carried out, and finally the target recovery matrix is obtained.

[0049] Figure 4 is a schematic diagram of the sparse microwave imaging system based on the proximal gradient descent method provided by the embodiment two of the present application. DETAILED DESCRIPTION

[0050] It should be noted that the following detailed description is exemplary and is intended to provide further explanation of the present application. Unless otherwise specified, all technical and scientific terms used in the present application have the same meaning as generally understood by those skilled in the art to which the present application belongs.

[0051] It should be noted that the terms used herein are only for the purpose of describing specific embodiments and are not intended to limit the exemplary embodiments according to the present application. As used herein, the singular form is intended to include the plural form unless the context clearly indicates otherwise, and furthermore, it should be understood that the terms "comprise" and "have" and any variations thereof are intended to cover non-exclusive inclusion, for example, a process, method, system, product or device comprising a series of steps or units does not have to be limited to only those steps or units clearly listed, but can include other steps or units not clearly listed or inherent to these processes, methods, products or devices.

[0052] The embodiments in the present application and the features in the embodiments can be combined with each other without conflict.

[0053] All data acquisition of the present embodiment is based on compliance with laws and regulations and user consent, and legal application of data.

[0054] Term explanation:

[0055] proximal: in iterative update, force the new solution to keep "proximity" with the previous step solution, while minimizing the influence of non-smooth terms.

[0056] gradient: gradient is a vector, which represents the directional derivative of a function at a point, along which the function changes fastest.

[0057] sparse: refers to the sparsity of the signal, that is, in a certain representation space, only a few components are non-zero, while most components are zero or close to zero.

[0058] nuclear norm and low rank: nuclear norm refers to the sum of singular values of a matrix, which mainly serves to constrain low rank. There is a certain correlation between the rows of the matrix, which is generally low rank. Low rank matrix has a lot of redundant information, which can be used to recover missing data or extract features. The value of the matrix is non-convex, and its convex approximation needs to be found to approximate. The convex approximation of the rank of the matrix is the nuclear norm.

[0059] smooth function: refers to a function that is continuous and differentiable in the domain.

[0060] Example 1

[0061] As shown in Figures 1 to 3 , the present application provides a sparse microwave imaging method based on proximal gradient descent method, comprising:

[0062] receiving echo signals;

[0063] convert the echo signals into matrix form to obtain an undersampled observation signal matrix X, a sampling matrix P and a fully sampled signal matrix S; the relationship between the matrices is X=P⊙S;

[0064] convert the rank minimization problem of signal recovery into a nuclear norm minimization problem:

[0065]

[0066] where, || * represents the nuclear norm;

[0067] construct the Lagrangian function of the nuclear norm minimization problem, and apply proximal gradient descent method for iterative solution to obtain the recovered signal matrix;

[0068] substitute the obtained recovered signal matrix into the imaging inversion formula to calculate the target reflectivity function, and generate an image according to the target reflectivity function.

[0069] The method provided by the application solves the problem that the traditional optimization algorithm cannot handle the non-smoothness of the nuclear norm as a whole, realizes stable recovery of the under-sampled signal by separating the smooth term and the non-smooth term and combining the idea of the proximal gradient method, converts the rank minimization problem of signal recovery into the nuclear norm minimization problem of convex optimization, reduces the difficulty of solving, and provides an operable mathematical basis for subsequent iterative solving, and completely covers the whole process from signal receiving to image generation, ensuring the practicability and landability of the method.

[0070] Specifically, the Lagrange function is:

[0071]

[0072] wherein p is a parameter, Y is an auxiliary variable, ‖‖ * is a non-smooth term, and <,> is a Lagrange multiplier term. is a penalty term of constraint relaxation.

[0073] By constructing the Lagrange function, the complex constrained optimization problem is converted into an unconstrained optimization problem, providing a mathematical carrier for the application of the proximal gradient descent method. The non-smooth term and the smooth term are clearly separated, making it possible to use the singular value soft thresholding for the non-smooth term and the gradient descent for the smooth term, and ensuring the stability of the optimization process.

[0074] Specifically, the application of the proximal gradient descent method for iterative solving includes:

[0075] The iteration update variables X and Y, and the update formula is:

[0076]

[0077] wherein k is the iteration number, η is the learning rate parameter, and D{} represents the singular value soft thresholding operation.

[0078] The non-smooth term is directly processed by the singular value soft thresholding operation (D) in view of the non-smoothness of the nuclear norm, avoiding the non-convergence problem caused by the inability to derive in the traditional algorithm; X and Y are updated by iteration to gradually approach the optimal solution, ensuring the accuracy of signal recovery; the iteration formula is simple and intuitive, reducing the difficulty of algorithm implementation and improving the calculation efficiency.

[0079] Specifically, the imaging inversion formula is:

[0080]

[0081] wherein FFT2 represents a two-dimensional Fourier transform, IFFT2 represents a two-dimensional inverse Fourier transform, k z is the component of the wave number in the z direction.

[0082] The abstract signal matrix is converted into a target reflectivity function with clear physical meaning, which is directly related to the generation of the final image, ensuring that the signal recovery result can be landed as a visual image; through the application of Fourier transform pair, accurate mapping from echo signal to target scattering characteristics is realized, reducing distortion in the signal conversion process and improving imaging clarity.

[0083] Specifically, the echo signal is a linear frequency modulation signal. The linear frequency modulation signal has the characteristics of high range resolution and strong anti-interference. The imaging process optimized for this type of signal can further improve the accuracy of signal recovery.

[0084] Specifically, the expression of the echo signal is:

[0085] s(x ′ ,y ′ ,z ′ =z0)=∫∫g(x,y,z=0)e -j2kR dxdy;

[0086] Wherein, g(x0,y0,z=0) is the reflectivity function of the target, R is the distance from the antenna to the target, j is the imaginary unit, k is the wave number, k=2π / λ, λ is the wavelength of the echo signal. The mapping relationship between the echo signal and the target reflectivity function is clear, the signal modeling error is reduced, and the accuracy of subsequent matrix recovery and imaging is improved.

[0087] Specifically, the fully sampled signal matrix Sampling matrix Wherein, m and n are respectively the number of sampling position points of the acquisition system along the horizontal and vertical directions. The mathematical form of the matrix is standardized, the relationship of X=P⊙S (⊙ is Hadamard product) is clearer (the position of "1" in P corresponds to the sampled data, and "0" corresponds to the missing data of undersampling), the matrix dimension (m×n) is clear, which provides specific basis for engineering implementation of the algorithm (such as hardware storage, calculation resource allocation), and improves the operability of the method.

[0088] In one specific embodiment, the sparse microwave imaging algorithm based on the proximal gradient descent method provided by the application is realized as follows:

[0089] First, it is assumed that the used signal is a linear frequency modulation signal. The received echo signal can be represented as:

[0090] s′(x ′ ,y ′ ,z ′ =z0)=g(x0,y0,z=0)e -jkR (1)

[0091] Where g(x0, y0, z=0) is the reflectivity function of the object, and R is the distance from the antenna to the target. Since the echo signal can be regarded as the superposition of multiple scattering points in the scene, the echo signal can be expressed as:

[0092] s(x ′ ,y ′ ,z ′ =z0)=∫∫g(x,y,z=0)e -j2kR dxdy(2)

[0093] A spherical wave can be represented as a superposition of a series of plane waves:

[0094]

[0095] Finally, the reflectivity function of the target can be expressed as:

[0096]

[0097] We express the above formula in matrix form:

[0098]

[0099] The goal of the above equation is to solve for the imaging matrix G. Since the target occupies only a small portion of the imaging region, we can treat the echo signal as low-rank, and signal reconstruction can be performed using low-rank constraints. We assume the fully sampled signal matrix is... The sampling matrix is ​​P∈{0,1} m×n Given an undersampled observation signal X, we can obtain:

[0100] X=P⊙S(6)

[0101] Since the signal satisfies the low-rank constraint, the reconstruction of signal S can be equivalent to:

[0102]

[0103] The goal is to find a matrix S that is consistent with X at known observation locations and has the smallest possible rank. This problem is NP-hard, and the above equation can be simplified to:

[0104]

[0105] Among them || * Let represent the nuclear norm. The above equation transforms the rank minimization problem into a nuclear norm minimization problem, which is a convex optimization problem. Solving this equation requires constructing a Lagrangian function, but differentiating the nuclear norm is very difficult. Therefore, this patent proposes using the proximal gradient descent method to solve the equation, constructing the following Lagrangian function:

[0106]

[0107] where, ρ is a parameter, Y is an auxiliary variable. The first term is a nuclear norm regularization term, the second term is a non-smooth term; the second term and the third term are regarded as smooth terms, and a new function is constructed, that is,

[0108]

[0109] According to the proximal gradient descent method, the partial derivative of g(X) with respect to X is obtained, and the gradient is:

[0110]

[0111] The first term in the above formula comes from the derivative of the inner product <Y, P⊙(S-X)>, and the second term comes from the chain rule of the square term. By solving the proximal gradient update, we can obtain

[0112]

[0113] where η is the learning rate parameter, and D{} represents the singular value soft thresholding operation. Let L ρ The partial derivative of (X, Y) with respect to Y is obtained as follows:

[0114]

[0115] The gradient descent of Y can be solved as follows:

[0116] Y k+1 = Y k + ρ[P⊙(S-X)] (14)

[0117] The update formula of X and Y can be obtained as follows:

[0118]

[0119] Finally, the optimized observation matrix X is obtained, which is brought back to equation (6) to solve the recovered signal, and then imaging is performed.

[0120] Embodiment Two

[0121] As shown in Figure 4 The embodiment provides a sparse microwave imaging system based on a proximal gradient descent method, which comprises:

[0122] A signal receiving module is configured to receive a return signal.

[0123] A matrix conversion module is configured to convert the return signal into a matrix form to obtain an undersampled observation signal matrix X, a sampling matrix P and a fully sampled signal matrix S. The relationship among the matrices is X=P⊙S.

[0124] An optimization problem construction module is configured to convert the rank minimization problem of signal recovery into a core norm minimization problem.

[0125]

[0126] wherein, || * represents a core norm;

[0127] A proximal gradient solving module is configured to construct a Lagrange function of the core norm minimization problem and apply a proximal gradient descent method to iteratively solve the problem to obtain a recovered signal matrix.

[0128] An imaging module is configured to substitute the recovered signal matrix into an imaging inversion formula to calculate a target reflectivity function and generate an image based on the target reflectivity function.

[0129] Embodiment three

[0130] The embodiment provides a sparse microwave imaging device based on a proximal gradient descent method, which comprises a memory and a processor; the memory is configured to store a computer program; and the processor is configured to implement the sparse microwave imaging method based on the proximal gradient descent method when the computer program is executed.

[0131] The processor is connected with the memory, and one or more computer programs are stored in the memory; when the electronic device is running, the processor executes the one or more computer programs stored in the memory, so that the electronic device executes the method in the embodiment one.

[0132] It should be understood that, in the embodiment, the processor can be a central processing unit CPU, and the processor can also be other general-purpose processors, digital signal processors DSPs, application-specific integrated circuits ASICs, ready-to-program gate arrays FPGA or other programmable logic devices, discrete gates or transistor logic devices, discrete hardware components, etc. The general-purpose processor can be a microprocessor or the processor can also be any conventional processor.

[0133] The memory can include read-only memory and random access memory, and provide instructions and data for the processor, and a part of the memory can also include non-volatile random access memory. For example, the memory can also store device type information.

[0134] In the implementation process, each step of the above method can be completed by integrated logic circuits of hardware in the processor or instructions in the form of software.

[0135] The method in the embodiment one can be directly embodied as being completed by a hardware processor, or being completed by a combination of hardware and software modules in the processor. The software modules can be located in a storage medium in the art, such as a random access memory, a flash memory, a read-only memory, a programmable read-only memory, an electrically erasable programmable memory, a register, or the like. The storage medium is located in a memory, and a processor reads information in the memory and combines hardware to complete the steps of the above method. To avoid repetition, no further detailed description is given here.

[0136] Those skilled in the art can understand that the units and algorithm steps of the examples described in combination with the embodiments can be realized in electronic hardware or a combination of computer software and electronic hardware. Whether the functions are realized in hardware or software depends on the specific application and design constraints of the technical solution. Those skilled in the art can use different methods to implement the described functions for each specific application, but such implementation should not be considered beyond the scope of the present application.

[0137] Embodiment four

[0138] Another embodiment of the present application provides a computer readable storage medium storing a computer program, when the computer program is executed by a processor, realizing the sparse microwave imaging method based on the proximal gradient descent method as described above.

[0139] Those skilled in the art can understand that all or part of the processes in the above-mentioned embodiment methods can be completed by a computer program instructing related hardware. The program can be stored in a computer readable storage medium, and when the program is executed, the processes of the above-mentioned embodiment methods can be included. The storage medium can be a magnetic disk, an optical disk, a read-only memory (ROM) or a random access memory (RAM), etc. In the present application, the units described as separate components can or can not be physically separated, and the components displayed as units can or can not be physical units, i.e. they can be located in one place or distributed on multiple network units. Some or all of the units can be selected to achieve the purpose of the embodiment of the present application according to actual needs. In addition, the functional units in each embodiment of the present application can be integrated in one processing unit, or each unit can exist physically, or two or more units can be integrated in one unit. The integrated unit can be realized in the form of hardware or in the form of a software functional unit.

[0140] Although the present application has been disclosed with reference to the above embodiments, the scope of the present application is not limited to the above. Various changes and modifications can be made to the present application without departing from the spirit and scope of the present application, and these changes and modifications should fall within the scope of the present application.

Claims

1. A sparse microwave imaging method based on proximal gradient descent, characterized in that, include: Receive echo signal; The echo signal is converted into matrix form to obtain the undersampled observation signal matrix X, the sampling matrix P, and the fully sampled signal matrix S; the relationship between the matrices is X = P ⊙ S; The problem of minimizing the rank of signal recovery is transformed into the problem of minimizing the nuclear norm: Among them, || * Represents the nuclear norm; The Lagrangian function of the nuclear norm minimization problem is constructed, and the proximal gradient descent method is applied iteratively to obtain the recovered signal matrix; The recovered signal matrix is ​​substituted into the imaging inversion formula to calculate the target reflectivity function, and an image is generated based on the target reflectivity function.

2. The method as described in claim 1, characterized in that, The Lagrange function is: Where ρ is a parameter, Y is an auxiliary variable, and ||| * Here are the nuclear norm regularization terms, non-smooth terms, and <,. are Lagrange multipliers. This is a penalty for lax constraints.

3. The method as described in claim 1, characterized in that, The iterative solution using the proximal gradient descent method includes: The iterative update formula for variables X and Y is as follows: Where k is the number of iterations, η is the learning rate parameter, and D{} represents the singular value soft thresholding operation.

4. The method as described in claim 1, characterized in that, The imaging inversion formula is as follows: Where FFT2 represents the two-dimensional Fourier transform, IFFT2 represents the two-dimensional inverse Fourier transform, and k z Let be the wavenumber component in the z-direction.

5. The method as described in claim 1, characterized in that, The echo signal is a linear frequency modulated signal.

6. The method as described in claim 5, characterized in that, The expression for the echo signal is: s(x ′ ,y ′ ,z ′ =z0)=∫∫g(x,y,z=0)e -j2kR dxdy; Where g(x,y,z=0) is the reflectivity function of the target, R is the distance from the antenna to the target, j is the imaginary unit, k is the wave number, k=2π / λ, and λ is the wavelength of the echo signal.

7. The method as described in claim 1, characterized in that, The fully sampled signal matrix Sampling matrix P∈{0,1} m×n , where m and n are the number of sampling points along the horizontal and vertical directions of the acquisition system, respectively.

8. A sparse microwave imaging system based on proximal gradient descent, characterized in that, include: The signal receiving module is used to receive echo signals; The matrix transformation module is used to convert the echo signal into matrix form to obtain the undersampled observation signal matrix X, the sampling matrix P, and the fully sampled signal matrix S; the relationship between the matrices is X = P ⊙ S; The optimization problem building module is used to transform the rank minimization problem of signal recovery into a nuclear norm minimization problem: Among them, || * Represents the nuclear norm; The proximal gradient solution module is used to construct the Lagrangian function of the nuclear norm minimization problem and apply the proximal gradient descent method to iteratively solve the problem to obtain the recovered signal matrix. The imaging module is used to substitute the recovered signal matrix into the imaging inversion formula to calculate the target reflectivity function, and generate an image based on the target reflectivity function.

9. A sparse microwave imaging device based on proximal gradient descent, characterized in that, The device includes a memory and a processor; the memory is used to store a computer program; the processor is used to implement the sparse microwave imaging method based on proximal gradient descent as described in any one of claims 1 to 7 when the computer program is executed.

10. A computer-readable storage medium, characterized in that, The storage medium stores a computer program, which, when executed by a processor, implements the sparse microwave imaging method based on proximal gradient descent as described in any one of claims 1 to 7.

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