SAR spectrum missing imaging method based on factor group sparse regularization of toeplitz structure

Abstract

The application discloses a factor group sparse regularization SAR spectrum missing imaging method based on a Toeplitz structure. Firstly, the echo matrix after interference elimination is transformed to a distance frequency domain and is divided into vectors according to pulses. Then, the pulse vectors are subjected to Toeplitz transformation and are reconstructed into a to-be-optimized matrix. Subsequently, parameters are initialized, and the to-be-optimized matrix is recovered by using a singular value decomposition initialization matrix and a group factor sparse regularization method, and the parameters are iteratively updated. When an iteration termination condition is reached, the obtained estimation matrix is reconstructed into a vector, all the vectors are reshaped into an echo matrix, and finally, a target is imaged by using a backward projection method. The method has the advantages of high imaging quality, wide applicability, fast operation speed and suitability for multiple interference scenarios.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of ultra-wideband radar imaging, and particularly relates to a factor group sparse regularization SAR spectrum missing imaging method based on a Toeplitz structure. BACKGROUND

[0002] Synthetic Aperture Radar (SAR) has the advantages of all-weather, all-day, high resolution, etc., and has been widely used in disaster monitoring, ocean protection, resource exploration, topographic mapping, etc. With the progress of software and hardware technology, the transmission bandwidth of SAR has been significantly enhanced, and the application of ultra-wideband (UWB) signal technology has greatly enhanced the detection capability of SAR.

[0003] UWB signals have been widely used in ultra-high precision imaging, integrated communication and sensing, etc. Radar usually modulates UWB signals in multiple ways to meet different operational requirements. In other words, UWB radar usually needs to transmit a set of discrete sub-bands during use. For example, in a distributed multiple-input multiple-output (MIMO) network, multiple signal sources transmit independent sub-bands, which are then combined to form a UWB signal to enhance resolution. Splitting the UWB signal into multiple sub-bands and transmitting them separately helps to save hardware resources. Sub-band sensing and fusion technology brings new application possibilities to cognitive radar systems. In addition, various interferences in complex spectrum space received by UWB SAR also damage spectrum resources, causing spectrum missing problems.

[0004] However, some current imaging algorithms under spectrum missing have many defects when applied to UWB SAR. For example, the method based on the pole-zero model is difficult to apply under low signal-to-noise ratio conditions. The method based on compressed sensing can solve the problem of low signal-to-noise ratio, but brings an unbearable huge amount of calculation. The statistical method based on the Bayesian model improves the speed of the algorithm, but needs specific prior knowledge of the signal. At the same time, the above methods cannot solve the problem of spectrum recovery under large missing degree. Therefore, it is a key problem to propose a spectrum missing imaging method for UWB SAR under large missing degree. SUMMARY

[0005] To solve the above technical problems, the application provides a factor group sparse regularization SAR spectrum missing imaging method based on a Toeplitz structure.

[0006] The technical scheme adopted by the application is as follows: The technical scheme adopted by the application is as follows:

[0007] S1, transform the echo matrix to be recovered to the frequency domain, and divide it into pulse vectors along the azimuth direction;

[0008] S2, convert the pulse vectors into low-rank matrices using a Toeplitz linear transformation operator;

[0009] S3, restore the low-rank matrices using a factor group sparse regularization method;

[0010] S4, reconstruct the restored estimation matrix into a vector and reshape it into an echo matrix, and transform it to the time domain;

[0011] S5, use a back-projection algorithm to image the target.

[0012] Further, the step S1 is specifically as follows:

[0013] S11, transform the echo matrix to be recovered to the distance frequency domain using distance-direction fast Fourier transform to obtain a spectrum matrix The expression is as follows:

[0014]

[0015] Wherein, FFT ran represents distance-direction fast Fourier transform. represents the echo matrix.

[0016] S12, divide the spectrum matrix into N pulse vectors along the azimuth direction

[0017] The pulse vector The expression is as follows:

[0018]

[0019] Wherein, M represents the length of the pulse sequence.

[0020] Further, the step S2 is specifically as follows:

[0021] Define the Toeplitz transformation operator T: The expression is as follows:

[0022]

[0023] Wherein, C represents the complex field, n1 and n2 represent the number of rows and columns of the matrix after transformation respectively; d represents a hyperparameter, and its size determines the number of rows and columns of the Toeplitz matrix after transformation.

[0024] Then apply the Toeplitz transformation operator T to the N pulse vectors respectively to obtain low-rank matrices The expression is as follows:

[0025]

[0026] Furthermore, step S3 is specifically as follows:

[0027] S31. Indiscriminately classify low-rank matrices. Represented as The best estimate recovered from each low-rank matrix is ​​represented as follows:

[0028] S32. Initialize parameters and use singular value decomposition of the matrix;

[0029] Initialization parameters: regularization parameter α, scaling factor β, penalty factor ρ 0 The dimension parameter p, the number of iterations k=0, and the maximum number of iterations K.

[0030] Initialize the matrix: initialize the matrix Perform singular value decomposition And A 0 =α 1 / 3 US 2 / 3 B 0 =α -1 / 3 S 1 / 3 V H ,

[0031] Where the superscript H denotes the conjugate transpose; matrices A and B denote the pairs of... The decomposition of A, where Y represents the Lagrange multiplier matrix, and A 0 B 0 Y 0 Let A, B, and Y represent the initial iteration values ​​of matrices A, B, and Y, respectively. This represents the best estimated initial value recovered from the low-rank matrix.

[0032] S33, Matrix to be optimized and parameter iterative update;

[0033] (1) Update A k+1 This involves applying a soft threshold to the column vectors of the matrix, as shown in the following expression:

[0034]

[0035] Where, ρ k Let represent the penalty factor for the k-th iteration, and:

[0036]

[0037] in:

[0038]

[0039] where W i represents the i-th column vector in A. The threshold λ is determined by ; τ = p k ‖B k (B k ) H ‖2.

[0040] (2) Update B k+1 , i.e. by taking the partial derivative of the augmented Lagrangian function, the expression is as follows:

[0041]

[0042] where I represents the identity matrix.

[0043] (3) Update , i.e. by taking the partial derivative of the augmented Lagrangian function, the expression is as follows:

[0044]

[0045] (4) Update the Lagrange multiplier Y k+1 , the expression is as follows:

[0046]

[0047] (5) Update the penalty factor p k+1 , the expression is as follows:

[0048] p k+1 = βp k

[0049] S34, determine whether the iteration termination condition is met: when k = K, output Otherwise, return to step S32, let k = k + 1, and continue iteration;

[0050] S35, for each constructed Toeplitz low-rank matrix , perform the optimization recovery of steps S31-S34 to obtain N recovered estimated matrices (low-rank matrices)

[0051] Further, the step S4 is specifically as follows:

[0052] S41, for each , reconstruct it into a pulse vector according to the first row and the first column.

[0053] S42, for each pulse vector obtained in step S41, splice and reshape it into a recovered spectral matrix

[0054] S43, transform the spectrum matrix into an echo matrix by inverse fast Fourier transform along the distance direction

[0055]

[0056] wherein IFFT ran denotes inverse fast Fourier transform along the distance direction.

[0057] Further, the step S5 is specifically as follows:

[0058] S51, scene grid division: divide the imaging area into a grid, and each pixel in the finally formed SAR image represents a grid;

[0059] S52, backward projection: calculate the distance R ij of the radar from each grid point at each azimuth time (the time of transmitting a pulse) ij , and calculate the two-way time delay t ij , then in the data collected at the current azimuth time, find the data with the distance R ij , and the data is the data received by the grid point at this azimuth time;

[0060] S53, coherent superposition: after phase compensation, the data at each azimuth time is superimposed to obtain the imaging result.

[0061] The method of the present application firstly transforms the echo matrix after interference removal to the distance frequency domain, and divides it into vectors according to pulses, then performs Toeplitz transformation on the pulse vectors, reconstructs into a to-be-optimized matrix, initializes parameters, and initializes the matrix by singular value decomposition, recovers the to-be-optimized matrix by using group factor sparse regularization method, and iteratively updates the parameters, after reaching the iteration termination condition, reconstructs the estimated matrix into a vector, and reshapes all the vectors into an echo matrix, finally uses the backward projection method to image the target. The method of the present application solves the limitations of the existing spectrum missing SAR imaging method, has the advantages of high imaging quality, wide applicability, fast operation speed, and is suitable for multiple interference scenarios. BRIEF DESCRIPTION OF DRAWINGS

[0062] Figure 1 It is a flow chart of a factor group sparse regularization SAR spectrum missing imaging method based on Toeplitz structure of the present application.

[0063] Figure 2 It is a comparison chart of singular value distribution of Toeplitz transformation and ordinary reshaped matrix method in the embodiment of the present application.

[0064] Figure 3A recovery effect diagram of a multi-subband spectrum missing in an embodiment of the present application.

[0065] Figure 4 A recovery effect diagram of a large-band spectrum missing in an embodiment of the present application.

[0066] Figure 5 A comparison diagram of imaging effects before and after recovery of a multi-subband spectrum missing in an embodiment of the present application.

[0067] Figure 6 A comparison diagram of imaging effects before and after recovery of a large-band spectrum missing in an embodiment of the present application. DETAILED DESCRIPTION

[0068] The method of the present application is further illustrated below in combination with the accompanying drawings and embodiments.

[0069] As shown in the figure, a flowchart of a factor group sparse regularization SAR spectrum missing imaging method based on Toeplitz structure of the present application, the specific steps are as follows: Figure 1

[0070] S1, transform the echo matrix to be recovered to the frequency domain, and divide it into pulse vectors along the azimuth direction;

[0071] S2, use the Toeplitz linear transformation operator to convert the pulse vectors into low-rank matrices;

[0072] The Toeplitz transformation can reduce the rank of the data matrix and avoid the appearance of all-zero rows and columns, laying a foundation for subsequent signal recovery.

[0073] S3, recover the low-rank matrix using the factor group sparse regularization method;

[0074] The remarkable advantage of the factor group sparse regularization method is that it provides a more stringent approximate rank function, plus strong initialization and convergence effects. This enables non-zero columns to quickly converge and stabilize, producing accurate iterative results with minimal computational overhead.

[0075] S4, reconstruct the recovered estimation matrix into a vector and reshape it into an echo matrix, and transform it to the time domain;

[0076] S5, use the back-projection (BP) algorithm to image the target.

[0077] The basic idea of the BP algorithm is to back-project the radar echo data to each pixel in the imaging area, and then coherently stack the echo at each pixel. The BP algorithm can accurately calculate the distance and phase of the radar and the target, thereby obtaining high-quality imaging results.

[0078] In the present embodiment, the step S1 is specifically as follows: ​

[0079] S11. Use the range-direction Fast Fourier Transform to transform the echo matrix to be recovered to the range frequency domain to obtain the spectrum matrix. The expression is as follows:

[0080]

[0081] Among them, FFT ran This represents the distance to Fast Fourier Transform. This represents the echo matrix.

[0082] S12, The spectrum matrix Divide into N pulse vectors along the azimuth direction

[0083] Pulse Vector The expression is as follows:

[0084]

[0085] Where M represents the pulse sequence length.

[0086] In this embodiment, step S2 is specifically as follows:

[0087] Define the Toeplitz transform operator T: The expression is as follows:

[0088]

[0089] Where C represents the complex field, n1 and n2 represent the number of rows and columns of the transformed matrix, respectively; d represents a hyperparameter whose magnitude determines the number of rows and columns of the transformed Toeplitz matrix.

[0090] Then, the Toeplitz transform operator T is applied to each of the N impulse vectors to obtain the low-rank matrix. The expression is as follows:

[0091]

[0092] like Figure 2 As shown, Figure 2 (a) represents the singular value distribution after the impulse vector is directly reconstructed into a matrix. Figure 2 (b) represents the singular value distribution of the impulse vector after the Toeplitz transformation. Therefore, the Toeplitz transformation improves the low-rank property of the matrix compared to directly reshaping the vector into a matrix.

[0093] In this embodiment, step S3 is specifically as follows:

[0094] S31. Indiscriminately classify low-rank matrices. Represented as Let the best estimate of each low-rank matrix be denoted as

[0095] S32, initialize parameters and decompose the matrix by singular value decomposition;

[0096] Initialize parameters: regularization parameter α, scaling factor β, penalty factor ρ 0 , dimension parameter p, iteration number k = 0 and maximum iteration number K.

[0097] Initialize matrix: decompose the matrix by singular value decomposition and A 0 = α 1 / 3 US 2 / 3 ,B 0 = α -1 / 3 S 1 / 3 V H ,

[0098] wherein the superscript H represents conjugate transpose; the matrices A and B represent decomposition of , Y represents a Lagrange multiplier matrix, A 0 , B 0 , Y 0 respectively represent initial iteration values of the matrices A, B and Y, represent the best estimate of the low-rank matrix recovery initial value.

[0099] S33, update the matrix to be optimized and the parameter iteration;

[0100] (1) update A k+1 , i.e. soft threshold processing is performed on the column vectors of the matrix, and the expression is as follows:

[0101]

[0102] wherein ρ k represents the penalty factor of the kth iteration, and:

[0103]

[0104] wherein:

[0105]

[0106] wherein W i represents the i-th column vector in A. The threshold value λ is represented by ; τ = ρ k ‖B k (B k ) H ‖2.

[0107] (2) Update B k+1 , i.e. by taking the partial derivative of the augmented Lagrangian function, the expression is as follows:

[0108]

[0109] where I represents the unit matrix.

[0110] (3) Update , i.e. by taking the partial derivative of the augmented Lagrangian function, the expression is as follows:

[0111]

[0112] (4) Update the Lagrange multiplier Y k+1 , the expression is as follows:

[0113]

[0114] (5) Update the penalty factor ρ k+1 , the expression is as follows:

[0115] ρ k+1 = βρ k

[0116] S34, determine whether the iteration termination condition is met: when k = K, output Otherwise, return to step S32, let k = k + 1, and continue iteration;

[0117] S35, for each constructed Toeplitz low-rank matrix , perform the optimization recovery of steps S31-S34 to obtain N recovered estimated matrices (low-rank matrices)

[0118] In this embodiment, the step S4 is specifically as follows:

[0119] S41, for each , reconstruct it into a pulse vector according to the first row and the first column.

[0120] S42, for each pulse vector obtained in step S41, splice and reshape it into a recovered spectrum matrix

[0121] S43, transform the spectrum matrix into an echo matrix using the distance direction inverse fast Fourier transform

[0122]

[0123] where IFFT ran represents the distance direction inverse fast Fourier transform.

[0124] In the embodiment, the step S5 is as follows:

[0125] S51, scene grid division: the imaging area is divided into a grid, and each pixel in the finally formed SAR image represents a grid;

[0126] S52, backward projection: the distance R of the radar from each grid point at each azimuth time (the time of transmitting a pulse) is calculated ij , and the two-way time delay t is calculated ij Then, in the data collected at the current azimuth time, the data with the distance R is found according to the two-way time delay t ij , and the data is the data received by the grid point at the azimuth time; ij

[0127] S53, coherent superposition: the data at each azimuth time is phase compensated and then superimposed to obtain the imaging result.

[0128] The simulation results of the embodiment are shown in Figure 3 、 Figure 4 、 Figure 5 、 Figure 6 , Figure 3 (a) a multi-subband missing spectrum diagram, Figure 3 (b) a recovery result diagram under the condition of multi-subband spectrum missing. Figure 4 (a) a large-band missing spectrum diagram, Figure 4 (b) a recovery result diagram under the condition of large-band spectrum missing. Figure 5 (a) a multi-subband spectrum missing direct BP imaging result, Figure 5 (b) a multi-subband spectrum missing recovery BP imaging result. Figure 6 (a) a large-band spectrum missing direct BP imaging result, Figure 6 (b) a large-band spectrum missing recovery BP imaging result. It can be seen from the simulation result diagram that the spectrum can be well recovered under the condition of multi-subband spectrum missing or large-band missing by using the method of the present application, and then BP imaging is performed, the sidelobe is well suppressed, the point target is well focused, and the imaging effect is significantly improved.

[0129] In summary, the method of the present application transforms the echo matrix after interference removal to the range frequency domain, reconstructs each azimuth pulse into a low-rank Toeplitz matrix, iteratively optimizes each low-rank matrix by using the factor group sparse regularization method, reconstructs the plurality of recovered matrices into a vector group, reshapes the vector group into a matrix and transforms the matrix to the time domain, and finally performs backward projection method imaging. The method of the present application solves the limitations of the existing spectrum missing SAR imaging method, has the advantages of high imaging quality, wide applicability, fast operation speed, and is suitable for multiple interference scenarios. ​

[0130] Those skilled in the art will appreciate that the embodiments described herein are presented for purposes of illustration and understanding of the principles of the application and should not be construed as limiting the scope of the application to such specifically outlined embodiments and examples. Various other specific embodiments and examples not described herein will be apparent to those skilled in the art in view of these teachings. The scope of the application should be determined from the claims.

Claims

1. A factor-based sparse regularized SAR spectral missing imaging method based on Toeplitz structure, the specific steps of which are as follows: S1. Transform the echo matrix to be recovered to the frequency domain and divide it into pulse vectors along the azimuth direction; S2. Use the Toeplitz linear transformation operator to convert the impulse vector into a low-rank matrix; S3. Use the factor group sparse regularization method to recover the low-rank matrix; S4. Reconstruct the recovered estimated matrix into a vector, reshape it into an echo matrix, and transform it to the time domain; S5. The target is imaged using a back projection algorithm; The specific steps of S1 are as follows: S11. Use the range-direction Fast Fourier Transform to transform the echo matrix to be recovered to the range frequency domain to obtain the spectrum matrix. The expression is as follows: ； in, This represents the distance to the Fast Fourier Transform; Represents the echo matrix; S12, The spectrum matrix Divided along the azimuth direction into pulse vectors ; Pulse Vector The expression is as follows: ； Where M represents the pulse sequence length; Step S2 is as follows: Define the Toeplitz transform operator The expression is as follows: ； in, Represents the field of complex numbers. and These represent the number of rows and columns of the transformed matrix, respectively. This represents a hyperparameter whose size determines the number of rows and columns of the transformed Toeplitz matrix; Then, the Toeplitz transform operator is applied to each of the N pulse vectors. The low-rank matrix is ​​obtained. The expression is as follows: ； Step S3 is as follows: S31. Indiscriminately classify low-rank matrices. Represented as The best estimate recovered from each low-rank matrix is ​​expressed as: ; S32. Initialize parameters and use singular value decomposition of the matrix; Initialization parameters: regularization parameters scaling factor Punishment factor Dimensional parameters Number of iterations and maximum number of iterations ; Initialize the matrix: initialize the matrix Perform singular value decomposition ,and ; Among them, superscript Represents the conjugate transpose; matrix and Indicates to Decomposition, Represents the Lagrange multiplier matrix. , , Represent matrices respectively , , The initial iteration value, This represents the best estimated initial value recovered from the low-rank matrix; S33, Matrix to be optimized and parameter iterative update; (1) Update This involves applying a soft threshold to the column vectors of the matrix, as shown in the following expression: ； in, Let represent the penalty factor for the k-th iteration, and: ； in: ； in, express The first in Column vector; threshold Depend on express; ; (2) Update That is, by taking the partial derivative of the augmented Lagrange function, the expression is as follows: ； in, Represents the identity matrix; (3) Update That is, by taking the partial derivative of the augmented Lagrange function, the expression is as follows: ； (4) Update the Lagrange multipliers The expression is as follows: ； (5) Update the penalty factor The expression is as follows: ； S34. Determine if the iteration termination condition is met: when When, output Otherwise, return to step S32 and let Continue iterating; S35. For each constructed Toeplitz low-rank matrix All steps S31-S34 are performed for optimization and recovery, resulting in N recovered estimated matrices. .

2. The factor-group sparse regularized SAR spectral missing imaging method based on Toeplitz structure according to claim 1, characterized in that, Step S4 is as follows: S41, for each Reconstruct a pulse vector based on the first row and first column; S42. For each pulse vector obtained in step S41, concatenate and reshape it into the recovered spectrum matrix. ; S43. Transform the spectrum matrix into an echo matrix using the inverse fast Fourier transform in the range direction. ; ； in, This represents the inverse fast Fourier transform of the distance.

3. The factor-group sparse regularized SAR spectral missing imaging method based on Toeplitz structure according to claim 1, characterized in that, Step S5 is as follows: S51, Scene grid subdivision: The imaging area is divided into grids, and each pixel in the final SAR image represents a grid. S52, Back projection: Calculate the distance between the radar and each grid point at each azimuth time. And calculate the two-way delay. Then, based on the data collected at the current location and time, according to the two-way delay... Find the distance also If the data is such that the grid point receives the data at this location at this time; S53. Coherent superposition: The data of each azimuth time are phase compensated and then superimposed to obtain the imaging result.

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