A tomographic SAR three-dimensional imaging method based on atomic norm minimization

Abstract

The application discloses a kind of based on atom norm minimization tomographic SAR three-dimensional imaging method, comprising: constructing the atom set and atom norm of complete tomographic SAR observation signal;Construct the atom set and atom norm of incomplete tomographic SAR observation signal;According to the atom set and atom norm of complete tomographic SAR observation signal and incomplete tomographic SAR observation signal constructed, construct the tomographic SAR imaging model based on atom norm minimization;According to tomographic SAR imaging model and semi-positive definite programming theory, construct the semi-positive definite programming solution of atom norm optimization problem, realizes SAR three-dimensional imaging using semi-positive definite programming solution.The application uses atom norm minimization method to carry out tomographic SAR inversion, compared with the optimization method based on L1 norm has stronger sparsity, super-resolution ability and more accurate estimation accuracy.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of three-dimensional imaging of radar, and particularly relates to a tomographic SAR three-dimensional imaging method based on atomic norm minimization. BACKGROUND

[0002] Synthetic aperture radar (SAR) is an active microwave remote sensing system with side-looking imaging geometry, which can map the backscattering properties of a three-dimensional scene onto a two-dimensional azimuth-range plane. Tomographic synthetic aperture radar (TomoSAR) is an extension of the synthetic aperture principle to the elevation direction, which uses a stack of SAR acquisitions to separate the contributions of the scatterers in each resolution cell and reconstruct the reflectivity function along the elevation direction for three-dimensional imaging.

[0003] Compressive sensing (CS) can break through the Nyquist sampling theorem to achieve accurate recovery of signals. The CS theory points out that for sparse signals, low-dimensional observations can be obtained at a rate much lower than the Nyquist sampling rate, and the original signal can be accurately recovered through a sparse signal reconstruction algorithm. The tomographic SAR imaging method based on the compressive sensing framework can reduce the number of data acquisitions required. However, due to the limited equidistance property and non-coherence, CS cannot guarantee sufficient sparse reconstruction in the presence of noise, so the correct number of scatterers cannot be obtained. SUMMARY

[0004] The application aims to provide a tomographic SAR three-dimensional imaging method based on atomic norm minimization, which eliminates the dense sampling grid effect based on the atomic norm method, directly recovers the sparse signal in the continuous frequency space, optimizes and enhances the sparsity, has strong robustness to noise, and has superior super-resolution capability.

[0005] The application provides a tomographic SAR three-dimensional imaging method based on atomic norm minimization, comprising the following steps:

[0006] S1: constructing an atomic set and an atomic norm of a complete tomographic SAR observation signal;

[0007] S2: constructing an atomic set and an atomic norm of an incomplete tomographic SAR observation signal;

[0008] S3: constructing a tomographic SAR imaging model based on atomic norm minimization according to the atomic set and the atomic norm of the complete tomographic SAR observation signal and the incomplete tomographic SAR observation signal;

[0009] S4: Constructing the semidefinite programming solution of the atomic norm optimization problem according to the tomographic SAR imaging model and the semidefinite programming theory, and realizing the three-dimensional SAR imaging by using the semidefinite programming solution.

[0010] Further, the complete tomographic SAR observation signal, the atomic set and the atomic norm in the step S1 are defined by the following formula:

[0011]

[0012]

[0013]

[0014] Wherein, f l represents the normalized frequency, β(f l ) represents the discrete reflectivity signal, z m represents the tomographic observation signal at the aperture m, z is the complete observation vector, [M] represents the complete index set {0,…,M-1}, a(f l ) represents the observation vector, represents the atomic set, represents the atomic norm on the atomic set .

[0015] Further, the atomic set and the atomic norm of the incomplete tomographic SAR observation signal in the step S2 are defined by the following formula:

[0016]

[0017]

[0018] Wherein, Ω represents the subset , a Ω (f) is the sub-vector of a(f) on the subset Ω, z Ω represents the incomplete observation vector, represents the atomic set .

[0019] Further, the tomographic SAR imaging model based on the atomic norm minimization in the step S3 is realized by the following formula:

[0020]

[0021] Wherein, μ is a specified tuning parameter, y Ω is the noisy incomplete observation data, z represents the complete noise-free observation data, z Ω represents the observation data on the subset Ω, The added noise reduction term is denoted as the atomic norm. The formula is also called the atomic norm denoising model.

[0022] Further, according to the semi-definite programming theory, the atomic norm minimization problem in the step S4 can be realized by the following semi-definite optimization method:

[0023]

[0024]

[0025] Wherein, tr(·) represents the trace of a matrix, and T(u) represents a Toeplitz matrix in the following form:

[0026]

[0027] Wherein, u i represents the i-th element of the vector u.

[0028] The present application is to overcome the problem that the grid intensive subdivision in the L1 norm minimization method cannot guarantee the sparsity of the solution, and proposes an atomic norm sparse optimization method for continuous frequency estimation driven by the atomic norm concept of continuous signal. Based on the atomic norm minimization method, the sparse inversion of the continuous signal can be realized, the abnormal values caused by the grid subdivision are eliminated, and the sparsity is enhanced. The method based on the atomic norm minimization is introduced into the field of sparse three-dimensional imaging of radar, and through data simulation and real SAR data, the sparsity, super-resolution ability and estimation accuracy of the atomic norm minimization method applied to TomoSAR inversion are demonstrated. The present application uses atomic norm minimization for tomographic SAR inversion, and compared with the optimization method based on L1 norm, has stronger sparsity, super-resolution ability and more accurate estimation accuracy.

[0029] Advantages: Compared with the existing three-dimensional imaging method, the present application has the following advantages:

[0030] 1. The atomic norm minimization method can eliminate the grid effect in the compressive sensing method, directly recover the sparse signal in the continuous frequency space, and optimize and enhance the sparsity, has strong robustness to noise, and can effectively eliminate abnormal values.

[0031] 2. The atomic norm minimization method provided by the present application has superior super-resolution ability and provides more accurate scatterer position information in the super-resolution area. BRIEF DESCRIPTION OF DRAWINGS

[0032] Figure 1 The flowchart of the method of the present application is shown in the figure;

[0033] Figure 2Distribution maps of 13 non-uniform spatial baselines and 45 uniform spatial baselines;

[0034] Figure 3 To reconstruct the elevation signal map of a single scatterer by minimizing the L1 norm and atomic norm;

[0035] Figure 4 A comparison of the probability of detecting K sparse scatterers in single-scatterer simulation data by minimizing the L1 norm and atomic norm;

[0036] Figure 5 To reconstruct the elevation signal map of the two scatterers by minimizing the L1 norm and atomic norm;

[0037] Figure 6 A comparison of the probability of detecting K sparse scatterers in two-scatterer simulation data by minimizing the L1 norm and atomic norm;

[0038] Figure 7 A graph showing the relationship between the probability of detecting a two-scatterer with atomic norm minimization and the normalized distance α.

[0039] Figure 8 The relationship between the root mean square error of the scatterer and the normalized distance α is shown in the figure; where (a) is the root mean square error of the scatterer elevation; and (b) is the root mean square error of the scatterer reflectivity.

[0040] Figure 9 Images of the study area in Pudong New Area, Shanghai; where (a) is an optical image; and (b) is an average SAR intensity image.

[0041] Figure 10 This is the average SAR intensity image of Zhengda Wudaokou Square;

[0042] Figure 11 For L1 norm minimization and atomic norm minimization Figure 10 Comparison of tomographic signals at scattering point P1; where (a) represents L1 norm minimization; (b) represents atomic norm minimization;

[0043] Figure 12 For L1 norm minimization and atomic norm minimization Figure 10 Comparison of tomographic signals at scattering points P2, P3, and P4; where (a), (c), and (e) are the results of minimizing the L1 norm for P2, P3, and P4, respectively; and (b), (d), and (f) are the results of minimizing the atomic norm for P2, P3, and P4, respectively.

[0044] Figure 13 , Figure 14 , Figure 15 and Figure 16 Analysis and comparison of inversion results for minimizing L1 norm and atomic norm.Figures 13-16 respectively are the height maps of detecting single scatterer and double scatterer binding, detecting single scatterer, detecting strong scatterer in double scatterer and detecting weak scatterer in double scatterer; wherein (a) is L1 norm minimization; (b) is atomic norm minimization;

[0045] Figure 17 are the height maps of detecting single scatterer and double scatterer binding, detecting single scatterer, detecting strong scatterer in double scatterer and detecting weak scatterer in double scatterer; wherein (a) is L1 norm minimization; (b) is atomic norm minimization; DETAILED DESCRIPTION

[0046] The present application is further illustrated by the following drawings and specific examples, which are intended to be illustrative only and not limiting of the scope of the application. Various modifications of the application in addition to those described herein will become apparent to those skilled in the art from the foregoing description. Such modifications are intended to fall within the scope of the appended claims.

[0047] The present application provides a tomographic SAR three-dimensional imaging method based on atomic norm minimization, as shown in the following formula: Figure 1 The method comprises the following steps:

[0048] Step 1: Construct the atomic set and atomic norm of the complete tomographic SAR observation signal.

[0049] The specific formula of the complete tomographic SAR observation signal, the atomic set and the atomic norm is as follows:

[0050]

[0051]

[0052]

[0053] Wherein, f l represents the normalized frequency, β(f l ) represents the discrete reflectivity signal, z m represents the tomographic observation signal at aperture m, z is the complete observation vector, [M] represents the complete index set {0,…,M-1}, a(f l represents the observation vector, represents the atomic set, represents the atomic norm on the atomic set .

[0054] Step 2: Construct the atomic set and atomic norm of the incomplete tomographic SAR observation signal.

[0055] The atomic set of the incomplete tomographic SAR observation signal and the specific formula of the atomic norm are as follows:

[0056]

[0057]

[0058] Wherein, Ω represents a subset , a Ω (f) is a sub-vector of a(f) on the subset Ω, z Ω represents an incomplete observation vector, represents an atomic set on which the atomic norm is.

[0059] Step 3: Construct a tomographic SAR imaging model based on atomic norm minimization.

[0060] The tomographic SAR imaging model based on atomic norm minimization is specifically:

[0061]

[0062] Wherein, μ is a specified tuning parameter, y Ω is noise-incomplete observation data, z represents complete noise-free observation data, z Ω represents observation data on the subset Ω, is an added noise reduction term. The formula is also called an atomic norm noise reduction model.

[0063] Step 4: Construct a semi-definite programming solution of the atomic norm optimization problem, and realize SAR three-dimensional imaging by using the semi-definite programming solution.

[0064] According to the semi-definite programming theory, the specific implementation of the semi-definite programming solution of the atomic norm minimization problem is:

[0065]

[0066]

[0067] Wherein, tr(·) represents the trace of a matrix, and T(u) represents a Toeplitz matrix in the following form:

[0068]

[0069] Wherein, u i represents the i-th element of the vector u.

[0070] Based on the above, in order to verify the actual effect of the method of the present application, the above scheme is applied in an example in this embodiment, and the specific implementation is as follows:

[0071] This example verifies the 3D tomographic SAR imaging algorithm based on atomic norm minimization by using a data set of 13 TerraSAR-X stripmap images of Shanghai region from August 2009 to February 2010. The parameters of the data set are shown in Table 1.

[0072] Table 1 Parameters of the data set

[0073] Imaging mode Strip Polarization VV Imaging area Shanghai Observation number 13 Collection time 2009.08～2010.02 Spatial baseline [m] -172～212 Center frequency f c [GHz]]]> 9.65 incident angle θ0 [deg] 26.4 ​ 559.32 Distance resolution p r [m]]]> 1.2 Azimuth resolution p a [m]]]> 3.3

[0074] Figure 2 13 actual non-uniform baseline profiles located between -172m and 212m and 45 virtual uniform baseline profiles located between -220m and 220m.

[0075] Figure 3 The reconstruction results of a single scatterer signal with height position of 50m, amplitude of 1.0 and signal-to-noise ratio of 10dB by using L1 norm and atomic norm minimization. It can be seen that the atomic norm minimization can accurately detect a scatterer, while the L1 norm minimization detects an extra outlier besides the main peak of the reconstructed signal.

[0076] Figure 4 The probability comparison chart of detecting K sparse scatterers by L1 norm minimization and atomic norm minimization for single scatterer simulation data. It can be seen that the probability of detecting a single scatterer by atomic norm minimization is 100%, while the probability of detecting a single scatterer by L1 norm minimization is only 60%.

[0077] Figure 5 The height signal reconstruction of two scatterers by L1 norm minimization and atomic norm minimization. Figure 6 The probability comparison of detecting K sparse scatterers by L1 norm minimization and atomic norm minimization for double scatterer simulation data. Figure 7 The relationship between the probability of detecting double scatterers by atomic norm minimization and the normalized distance a. Figure 8 The relationship between the root mean square error of the reconstructed scatterer and the normalized distance a, wherein (a) is the root mean square error of the height of the scatterer; (b) is the root mean square error of the reflectivity of the scatterer. Figure 9 The images of the research region of Pudong New Area in Shanghai, wherein (a) is an optical image; (b) is an average SAR intensity image. Figure 10 The average SAR intensity image of the Zhongda Wadaokou Square. Figure 11 The comparison of the tomographic signals of the scattering point P1, wherein (a) is L1 norm minimization; (b) is atomic norm minimization. Figure 10 The comparison of the tomographic signals of the scattering point P1, wherein (a) is L1 norm minimization; (b) is atomic norm minimization. Figure 12 The comparison of the tomographic signals of the scattering point P1, wherein (a) is L1 norm minimization; (b) is atomic norm minimization. Figure 10Comparison of tomographic signals of scattering points P1, P3 and P4, wherein (a), (c) and (e) are L1 norm minimization of P2, P3 and P4 respectively; (b), (d) and (f) are atomic norm minimization of P2, P3 and P4 respectively.

[0078] Figure 13 For detecting the height map of single scatterer and strong scatterer in double scatterer, wherein (a) is L1 norm minimization; (b) is atomic norm minimization, it can be seen that atomic norm minimization obtains more accurate estimation. Figure 14 For detecting the height map of single scatterer, wherein (a) is L1 norm minimization; (b) is atomic norm minimization, it can be seen that L1 norm minimization detects less single scatterer than atomic norm minimization, and forms an incomplete point cloud of single scatterer. Figure 15 、 Figure 16 For detecting the height map of strong scatterer and weak scatterer in double scatterer, wherein (a) is L1 norm minimization; (b) is atomic norm minimization, it can be seen that L1 norm minimization detects the height of weak scatterer in building wall higher than that of strong scatterer, which is unreasonable, while atomic norm minimization detects the height of weak scatterer in roof higher than that of strong scatterer in wall. Figure 17 For detecting the height map of scatterer by atomic norm minimization, wherein (a) is the height map of single scatterer and strong scatterer in double scatterer; (b) is the height map of single scatterer; (c) is the height map of strong scatterer in double scatterer; (d) is the height map of weak scatterer in double scatterer.

[0079] The embodiment also provides a tomographic SAR three-dimensional imaging system based on atomic norm minimization, which comprises a network interface, a memory and a processor; wherein the network interface is used for receiving and sending signals in the process of transceiving information with other external network elements; the memory is used for storing computer program instructions capable of running on the processor; and the processor is used for executing the steps of the consensus method when running the computer program instructions.

[0080] The embodiments also provide a computer storage medium storing a computer program, which, when executed by a processor, can implement the above-described method. The computer readable medium can be considered tangible and non-transitory. Non-limiting examples of non-transitory tangible computer-readable media include nonvolatile memory circuits (e.g., flash memory devices, erasable programmable read only memory (EPROM), or electrically erasable programmable read only memory (EEPROM)), volatile memory circuits (e.g., static random access memory (SRAM), dynamic random access memory (DRAM), or variants thereof), magnetic storage media (e.g., magnetic tape, magnetic hard drive), optical storage media (e.g., optical disk, optical tape, etc.), and the like. The computer program includes processor-executable instructions stored on at least one non-transitory computer-readable medium. The computer program can also include or rely on stored data. The computer program can include the Basic Input-Output system (BIOS) that interacts with the hardware of the special purpose computer, device drivers that interact with particular devices of the special purpose computer, one or more operating systems, user applications, background services, background applications, etc.

[0081] Those skilled in the art will appreciate that embodiments of the present application can be devised for a method, a system, or a computer program product. Accordingly, the present application can take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects. Furthermore, the present application can take the form of a computer program product on one or more computer readable storage media (including, but not limited to, disk storage, CD-ROMs, optical storage devices, etc.) embodying computer readable program code.

[0082] The present application is described in reference to the flowchart illustrations and / or block diagrams of methods, apparatus (systems) and computer program products according to embodiments of the application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general purpose computer, special purpose computer, embedded processor or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart illustrations and / or block diagrams block or blocks. Figure 1 The flowchart illustrations and / or block diagrams in the figures illustrate the architecture, functionality, and operation of possible implementations of apparatuses, methods and computer program products according to various embodiments of the present application. In this regard, each block in the flowchart illustrations and / or block diagrams can represent a module, segment, or portion of instructions, which comprises one or more executable procedures or functions. In some embodiments, the flowchart illustrations and / or block diagrams can include Figure 1 The flowchart illustrations and / or block diagrams in the figures illustrate the architecture, functionality, and operation of possible implementations of apparatuses, methods and computer program products according to various embodiments of the present application. In this regard, each block in the flowchart illustrations and / or block diagrams can represent a module, segment, or portion of instructions, which comprises one or more executable procedures or functions. In some embodiments, the flowchart illustrations and / or block diagrams can include

[0083] The computer program instructions can also be loaded into a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart illustrations and / or block diagrams. Figure 1 The flowchart illustrations and / or block diagrams in the figures illustrate the architecture, functionality, and operation of possible implementations of apparatuses, methods and computer program products according to various embodiments of the present application. In this regard, each block in the flowchart illustrations and / or block diagrams can represent a module, segment, or portion of instructions, which comprises one or more executable procedures or functions. In some embodiments, the flowchart illustrations and / or block diagrams can include Figure 1the function(s) specified in the block or blocks.

[0084] These computer program instructions can also be loaded into computer or other programmable data processing devices, so that a series of operation steps are performed on the computer or other programmable data processing devices to generate computer-implemented processes, so that the instructions executed on the computer or other programmable devices provide a process for implementing the flowchart Figure 1 the flowchart or flowcharts and / or a block Figure 1 the function(s) specified in the block or blocks.

Claims

1. A tomographic SAR three-dimensional imaging method based on atomic norm minimization, characterized in that, The method comprises the following steps: S1: constructing an atomic set and an atomic norm of a complete tomographic SAR observation signal; S2: constructing an atomic set and an atomic norm of an incomplete tomographic SAR observation signal; S3: constructing a tomographic SAR imaging model based on atomic norm minimization according to the atomic set and the atomic norm of the complete tomographic SAR observation signal and the incomplete tomographic SAR observation signal; S4: constructing a semi-definite programming solution of an atomic norm optimization problem according to the tomographic SAR imaging model, and realizing SAR three-dimensional imaging by using the semi-definite programming solution.

2. The tomographic SAR 3-D imaging method based on atomic norm minimization according to claim 1, characterized in that, The complete tomographic SAR observation signal, the atomic set and the atomic norm in the step S1 are defined by the following formula: where f l denotes the normalized frequency, β(f l ) denotes the discrete reflectivity signal, z m denotes the tomographic observation signal at aperture m, z is the complete observation vector, [M] denotes the complete index set {0,..., M-1}, a(f l ) denotes the observation vector, denotes the atom set, denotes the atom set over which the atom norm is taken.

3. The tomographic SAR 3-D imaging method based on atomic norm minimization according to claim 1, characterized in that, The atomic set and the atomic norm of the incomplete tomographic SAR observation signal in the step S2 are defined by the following formula: where Ω denotes a subset a Ω (f) is a subvector of a(f) over the subset Ω, z Ω denotes an incomplete observation vector, denotes an atom set over the subset Ω.

4. The tomographic SAR 3-D imaging method based on atomic norm minimization according to claim 1, characterized in that, The tomographic SAR imaging model based on atomic norm minimization in the step S3 is realized by the following formula: where μ is a specified tuning parameter, y Ω is the noisy incomplete observation data, z Ω denotes the observation data on the subset Ω, is the added denoising term.

5. The tomographic SAR 3-D imaging method based on atomic norm minimization according to claim 1, characterized in that, According to the semi-definite programming theory, the atomic norm minimization problem in the step S4 can be realized by the following semi-definite optimization method: Wherein, tr(·) represents the trace of a matrix, and T(u) is a Toeplitz matrix.

6. The tomographic SAR 3-D imaging method based on atomic norm minimization according to claim 5, characterized in that, The T(u) is a Toeplitz matrix in the following form: where u i denotes the i-th element of the vector u.

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