CN110275166B - ADMM-based rapid sparse aperture ISAR self-focusing and imaging method - Google Patents

ADMM-based rapid sparse aperture ISAR self-focusing and imaging method Download PDF

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CN110275166B
CN110275166B CN201910629019.5A CN201910629019A CN110275166B CN 110275166 B CN110275166 B CN 110275166B CN 201910629019 A CN201910629019 A CN 201910629019A CN 110275166 B CN110275166 B CN 110275166B
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张双辉
黎湘
刘永祥
霍凯
姜卫东
田彪
张新禹
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National University of Defense Technology
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Abstract

The invention belongs to the field of radar signal processing, and particularly relates to an ADMM-based fast sparse aperture ISAR self-focusing and imaging method. The method comprises the following steps: s1 modeling the sparse aperture radar echo; s2 reconstructing a target ISAR image X through ADMM; s3 estimates the initial phase error phi by the minimum entropy criterion. The invention has the following beneficial effects: by the method, the initial phase error can be effectively estimated from the sparse aperture radar echo, ISAR self-focusing is realized, side lobe and grating lobe influence caused by undersampling can be effectively eliminated, and a high-resolution ISAR image is obtained; the robustness is strong, and ISAR self-focusing and imaging can still be realized under the condition of low signal-to-noise ratio; the method has high operation efficiency, can be applied to a real-time ISAR imaging system, and has important engineering application value.

Description

ADMM-based rapid sparse aperture ISAR self-focusing and imaging method
Technical Field
The invention belongs to the field of radar signal processing, and particularly relates to a rapid sparse aperture ISAR (inertial navigation methods of Multipliers, ADMM) self-focusing and imaging Method based on an Alternating Direction multiplier.
Background
The Inverse Synthetic Aperture Radar (ISAR) can acquire a high-resolution two-dimensional image of a moving target, has the advantages of all-time, all-weather and strong penetration compared with optical equipment, becomes important detection equipment for space target identification, and is widely applied to the fields of space target monitoring, missile defense, radar astronomy and the like. At present, for a complete radar echo signal (or called a full aperture signal), a traditional ISAR imaging method can obtain a high-resolution image of a moving target, but for a sparse aperture signal, the traditional method basically fails, and an effective imaging means is still lacked.
In an ISAR system, a plurality of factors can cause the generation of sparse aperture signals, such as environmental noise and receiver noise, which cause the low signal-to-noise ratio of partial radar echoes and can not meet imaging conditions; under the strong countermeasure condition, partial echo fails due to active and passive interference and the like, and the pulse is in a non-uniform sampling form to form a sparse aperture. In addition, the working mode of the widely applied multifunctional radar also generates sparse aperture signals, generally speaking, in order to meet the functional requirements of multi-target detection, tracking, imaging and identification, the multifunctional radar has to switch the limited radar transmitting energy back and forth among different targets, so that the signals irradiating the same target are sparse and non-uniform, and thus the sparse aperture signals are formed. The performance of the traditional ISAR imaging method is seriously influenced by the sparse aperture, on one hand, the non-uniform sampling causes that the traditional range-Doppler (RD) ISAR imaging method based on Fast Fourier Transform (FFT) is influenced by stronger sidelobe and grating lobe, and the image resolution is reduced; on the other hand, under the condition of sparse aperture, the coherence between radar echo pulses is reduced, so that the traditional ISAR self-focusing method is invalid, and the initial phase error caused by target translation cannot be effectively compensated, so that the ISAR image defocuses. With the development of a compressed sensing technology, the existing method introduces the compressed sensing technology into ISAR imaging, makes full use of the sparsity of ISAR images, and reconstructs well-focused ISAR images from undersampled radar echoes through a sparse recovery algorithm. However, the existing sparse aperture ISAR imaging methods all face the problem of low operation efficiency, cannot meet the requirements of real-time ISAR imaging systems, and the operation efficiency becomes a bottleneck restricting the engineering application of the sparse aperture ISAR imaging technology, so that a solution is urgently needed.
Disclosure of Invention
The invention aims to solve the technical problems that under the condition of sparse aperture, the performance of the traditional RD ISAR imaging method is reduced, the ISAR image quality is reduced, and although the ISAR imaging method based on the compressive sensing technology can obtain a high-quality ISAR image, the operation efficiency is low, and the actual engineering requirements are difficult to meet.
The idea of the invention is to aim at sparse aperture ISARThe problem of low operation efficiency of an image method is that an ADMM-based rapid sparse aperture ISAR self-focusing and imaging method is provided. The method carries out generalized modeling on the sparse aperture radar echo through l1Modeling the sparse characteristics of the ISAR image by norm regularization constraint, and solving the model based on l by using an ADMM method on the basis1And optimizing the norm regularization constraint. In order to improve the operational efficiency of the method, the characteristics of a Fourier matrix, a phase error matrix and a down-sampling matrix are fully utilized, matrix inversion with low operational efficiency in the iterative process is converted into matrix element division, and the iterative solution along the direction of a distance unit is avoided by adopting a two-dimensional batch processing mode, so that the operational efficiency is further improved. And finally, estimating initial phase errors in radar echoes based on a minimum entropy criterion in an iteration process, and jointly realizing sparse aperture ISAR self-focusing.
The technical scheme adopted by the invention for solving the technical problems is as follows: an ADMM-based fast sparse aperture ISAR self-focusing and imaging method comprises the following steps (for convenience and conciseness, unified stipulation: a matrix or vector is represented by bold letters, and an arbitrary vector a, a is represented by bold letters)iThe ith element representing a, for an arbitrary matrix A, Ai,jElement (i, j) of A):
s1 models sparse aperture radar returns:
under the irradiation of high-frequency radar signals, a metal target can be generally equivalent to the sum of a plurality of discrete scattering points, and the target is assumed to contain P scattering points, wherein the coordinate of the P scattering point relative to the rotation center of the target is (x)p,yp) Then, the one-dimensional range image sequence of the object can be represented as a superposition of the one-dimensional range images of the P scattering points:
Figure BDA0002127499350000021
wherein
Figure BDA0002127499350000022
A one-dimensional range image sequence of the target is represented,
Figure BDA0002127499350000023
tmrespectively representing fast time (i.e. intra-pulse time) and slow time (i.e. inter-pulse time), σpBackscattering coefficients, j, f, representing the p-th scattering point of the objectcC is an imaginary number unit, a radar transmission signal carrier frequency and an electromagnetic wave propagation speed respectively, omega represents the equivalent rotating speed of the target after translational compensation, phi (t)m) Indicating an initial phase error that includes both the phase error introduced by the target translation and the ambient noise phase error. Equation (1) can be further discretized as:
Figure BDA0002127499350000024
wherein h (m, n) represents a target discrete one-dimensional range profile sequence, and n and m are respectively a fast time sequence number and a slow time sequence number: n is 1,2, …, N, M is 1,2, …, M, N, M are the total number of fast and slow times, respectively. PrRepresenting the radar transmit signal pulse repetition frequency.
Under the condition of sparse aperture, radar echo signals are in an undersampling form along a slow time dimension, and if the radar echo comprises L pulses (L is less than M) and a vector formed by combining sequence numbers of the pulses is V, the radar echo has
Figure BDA0002127499350000025
On this basis, the target discrete one-dimensional range profile sequence under the sparse aperture condition can be represented as:
h＝ESFx+n (3)
wherein
Figure BDA0002127499350000026
Matrix for representing discrete one-dimensional range profile of target
Figure BDA0002127499350000027
Stacking the formed vectors along the columns, i.e., h ═ Vec (h), where Vec (·) denotes vectorization of the matrix stack along the columns;
Figure BDA0002127499350000028
is a block initial phase error matrix:
Figure BDA0002127499350000029
wherein INRepresenting an N x N identity matrix,
Figure BDA00021274993500000210
the kronecker product of the matrix is represented,
Figure BDA00021274993500000211
represent the initial phase error matrix: e ═ diag [ exp (j φ)]Wherein diag (·) denotes a diagonal matrix composed of vectors, whose diagonal elements are composed of vector elements in parentheses, and phi denotes an initial phase error vector;
Figure BDA00021274993500000212
a block downsampling matrix:
Figure BDA00021274993500000213
wherein
Figure BDA00021274993500000214
For the down-sampling matrix:
Figure BDA00021274993500000215
is a block fourier matrix:
Figure BDA00021274993500000216
wherein
Figure BDA00021274993500000217
Is an M-order Fourier matrix;
Figure BDA00021274993500000218
representing target ISAR images
Figure BDA00021274993500000219
The vectors formed by stacking along the columns, i.e., x ═ vec (x);
Figure BDA00021274993500000220
is the noise vector after column stacking.
For convenience of representation, the expression (3) is a vector expression of a sparse aperture target discrete one-dimensional range profile sequence, but in the actual algorithm implementation process, the processing is still performed in a matrix form. Subsequent steps will give a computational expression in the form of a matrix.
S2 reconstructs the target ISAR image X by ADMM:
the ISAR image only contains a small number of target scattering points, has a simple background and presents stronger sparse characteristics, and therefore the ISAR image reconstruction under the sparse aperture condition can be realized by using a sparse recovery method. The invention adopts1The norm regularization method restrains the sparse characteristics of the ISAR image, and under the condition, the sparse aperture ISAR image reconstruction is equivalent to solving the following optimization problem:
Figure BDA0002127499350000031
wherein | · | purple1、||·||2Are each l1、l2A norm; λ is a regularization parameter, which determines the sparsity of the obtained ISAR image. The method comprises the following specific steps:
s2.1, solving the optimization problem shown in the formula (4) by an ADMM method:
the ADMM method equates the optimization problem shown in formula (4) to the following optimization problem by introducing an auxiliary variable z:
Figure BDA0002127499350000032
the extended Lagrange expression is as follows:
Figure BDA0002127499350000033
where α is the lagrange multiplier and ρ is a penalty factor. The ADMM method solves the following three sub-problems to achieve the solution of equation (4):
Figure BDA0002127499350000034
where k represents the kth iteration. Respectively obtain Lρ(x, z, α) the first and second equations in equation (7) can be solved by taking the first derivative of x and z to zero, as shown in the following equation:
Figure BDA0002127499350000035
wherein S (-) represents a soft threshold function, and for any variable y and parameter a, the following are:
Figure BDA0002127499350000036
s2.2, converting the updated expression shown in the expression (8) into a matrix form:
in the equation (8), the first equation includes inversion of a matrix having a size of MN × MN, and the operation efficiency is low. To simplify the operation, the consideration matrix E, F, S has the following properties:
EHE＝EEH＝ILN (9)
FHF＝FFH＝IMN (10)
Figure BDA0002127499350000041
wherein ILN、IMNRespectively, unit matrixes with the size of LNXLN and MN,
Figure BDA0002127499350000042
is a diagonal matrix whose diagonal elements are
Figure BDA0002127499350000043
Substituting equations (9) - (11) for the first equation in equation (8) can obtain:
Figure BDA0002127499350000044
at this point, the matrix to be inverted has been converted into a diagonal matrix, and the inversion thereof can be achieved by matrix element division. Thus, the vector form of equation (12) can be re-expressed as a matrix form, as shown in the following equation:
Figure BDA0002127499350000045
wherein Z, A are in the form of matrices of the auxiliary variable z and lagrange multiplier alpha respectively,
Figure BDA0002127499350000046
the elements representing the two matrices are divided separately. Mask is a sampling matrix, and the sampling matrix is a sampling matrix,
Figure BDA0002127499350000047
the value of the element is 1 or 0, which respectively represents the extraction or the abandonment of the element at the corresponding position, and the sampling matrix is multiplied with the complete target one-dimensional range profile sequence according to each element, so that the zero-filling sparse aperture one-dimensional range profile sequence can be obtained. 1M×NA full 1 matrix of size M × N is shown. Since the equation (13) includes only the matrix multiplication with the maximum size of M × N, the operation efficiency is significantly improved compared to the first equation of the equation (8). In addition, f and f in formula (13)HThe method can be realized by FFT and fast inverse Fourier transform (IFFT) respectively to further improve the operation efficiency.
Likewise, the second and third equations in equation (8) can be further expressed in matrix form:
Figure BDA0002127499350000048
Figure BDA0002127499350000049
s3 estimates the initial phase error phi by the minimum entropy criterion:
estimating an initial phase error phi through a minimum entropy criterion in an iteration process to realize ISAR self-focusing under a sparse aperture condition, wherein the following formula is shown as follows:
Figure BDA00021274993500000410
wherein the content of the first and second substances,
Figure BDA00021274993500000411
the entropy of the ISAR image obtained from the (k + 1) th iteration is represented and defined as follows:
Figure BDA00021274993500000412
wherein sum (-) indicates the summation of each element of the matrix in the brackets, and is the Hadamard product of the matrix, i.e. each element of the matrix is multiplied separately; c is total energy of the ISAR image: c ═ sum (| X)(k+1)|2) Which is independent of the initial phase error phi; const denotes a constant independent of the initial phase error phi. The method comprises the following specific steps:
s3.1 calculation
Figure BDA0002127499350000051
Initial phase error phi of the l-th pulselDerivative of (a):
the optimization problem shown in the formula (16) is solved, and firstly, the calculation is carried out
Figure BDA0002127499350000052
Initial phase error phi of the l-th pulselThe derivative of (a) is represented by the following formula:
Figure BDA0002127499350000053
where Re (. cndot.) denotes the real part of the complex number in parentheses, X(k+1)*Representing ISAR image X obtained from the k +1 th iteration(k+1)Conjugation of (1).
S3.2 calculating ISAR image X obtained by k +1 iteration(k+1)Initial phase error phi of the l-th pulselDerivative of (a):
the ISAR image X obtained in the k +1 th iteration is included in the formula (18)(k+1)Initial phase error phi of the l-th pulselThe derivative of (c) is calculated as follows:
Figure BDA0002127499350000054
wherein 0(l-1)×N、0(L-l)×NRespectively, all-zero matrices of sizes (L-1) xN and (L-L) xN.
S3.3 calculating the initial phase error phi obtained by the k +1 iterationl (k+1)：
Substituting formula (19) for formula (18) and making the derivative
Figure BDA0002127499350000055
The initial phase error phi of the first pulse can be obtainedlIteratively updating the expression:
Figure BDA0002127499350000056
where angle (·) denotes the phase taking the imaginary number in parentheses.
Therefore, the fast sparse aperture ISAR self-focusing and imaging process based on the alternative direction multiplier is as follows: and (5) iterating the equations (13) - (15) and the equation (20) in a loop until convergence, wherein X obtained by the equation (13) is the reconstructed ISAR image.
Before iterative solution, firstly, parameters need to be initialized, wherein an auxiliary variable Z, a Lagrange multiplier A and an initial phase error matrix e can be initialized to be an all-zero matrix, and a regularization parameter lambda is initialized to be mu var (H), wherein var (·) is a square error operator, mu is more than 0.005 and less than 0.01, and a penalty factor rho is 1.
The invention has the following beneficial effects: by the method, the initial phase error can be effectively estimated from the sparse aperture radar echo, ISAR self-focusing is realized, side lobe and grating lobe influence caused by undersampling can be effectively eliminated, and a high-resolution ISAR image is obtained; the robustness is strong, and ISAR self-focusing and imaging can still be realized under the condition of low signal-to-noise ratio; the method has high operation efficiency, can be applied to a real-time ISAR imaging system, and has important engineering application value.
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FIG. 1 is a flow chart of an embodiment of the present invention;
FIG. 2(a) is a plan view of the aircraft; (b) a model of the scattering points of the aircraft;
FIG. 3 full pore size conditions: (a) a one-dimensional range profile sequence; (b) an ISAR image;
FIG. 4 ISAR images obtained by different algorithms under different sparse aperture conditions;
FIG. 5 compares the performance of the algorithm at different sampling rates: (a) a correlation coefficient; (b) the entropy of the image; (c) calculating time;
FIG. 6 is an ISAR image obtained by different algorithms under different signal-to-noise ratio (SNR) conditions;
FIG. 7 compares the performance of the algorithm under different SNR conditions: (a) a correlation coefficient; (b) the entropy of the image; (c) calculating time;
FIG. 8(a) an aircraft optical image; (b) a vehicle-mounted radar;
FIG. 9 full aperture conditions: (a) a one-dimensional range profile sequence; (b) an ISAR image;
FIG. 10 shows ISAR images obtained by different algorithms under different sparse aperture conditions.
Detailed Description
The invention is further illustrated with reference to the accompanying drawings:
FIG. 1 is a general process flow of the present invention.
The invention discloses an ADMM-based rapid sparse aperture ISAR imaging method, which comprises the following steps:
s1 modeling the sparse aperture radar echo;
s2 reconstructing a target ISAR image X through ADMM;
s3 estimates the initial phase error phi by the minimum entropy criterion.
Firstly, simulation data is adopted to carry out experiments, and in the experimental process, a scattering point model shown in fig. 2 (b) is constructed according to a plane diagram of an airplane shown in fig. 2 (a). Assuming that a target rotates at a constant speed of 0.06rad/s along a rotation center after the radar echo is subjected to translation compensation; the radar parameters are set as follows: let the center frequency be 9GHz, the bandwidth be 800MHz, the pulse be 100 mus, and the pulse repetition frequency be 150 Hz. The number of echo pulses of the full-aperture radar is set to be 256, and each pulse comprises 256 sampling points.
Firstly, ISAR imaging processing is carried out on full-aperture radar echo data, and the ISAR imaging processing is used as a reference of an imaging result under a sparse aperture condition. Under the full aperture condition, the target one-dimensional range image sequence and the ISAR image are respectively shown in fig. 3(a) and fig. 3(b), wherein, the ISAR imaging adopts the traditional RD imaging method (s.zhang, y.liu, and x.li, "Fast entry minimized based automatic imaging technique for ISAR imaging, IEEE trans.signal process, vol.63, No.13, pp. 34253434, jul.2015). As can be seen, under the full aperture condition, the conventional RD method can obtain an ISAR image with good focusing effect.
Further, partial data are extracted from the full-aperture one-dimensional range profile sequence in three ways, namely random, segmentation and mixing, respectively, so as to simulate a sparse aperture one-dimensional range profile sequence, as shown in the first column of fig. 4. In the experimental process, the echo sampling rate is set to be 0.25, the initial phase error follows Gaussian distribution with the mean value being zero and the variance being pi, and the SNR of the radar echo is set to be 15 dB. An RD method, An AFSBL method (L.ZHao, L.Wang, G.Bi, and L.Yang, "An auto focus technique for high resolution analysis adaptive aperture image, IEEE trans. Geosci.Remote Sens., vol.52, No.10, pp. 63926403, Oct.2014.) and the ADMM method provided by the invention are respectively adopted to reconstruct a target ISAR image from three kinds of sparse aperture data, and imaging results are respectively shown in the second, third and fourth columns of FIG. 4. It can be known from the figure that, in the three methods, the ISAR image obtained by the RD method is basically defocused, while the ADMM method provided by the invention obtains the ISAR image with the best focusing effect, and particularly under the mixed sparse condition, the ISAR image obtained by the ADMM method is obviously clearer than that obtained by the AFSBL method.
Under the condition of different sampling rates, 100 monte carlo experiments are respectively carried out, and the correlation coefficient, the image entropy and the change curve of the calculation time with respect to the sampling rate of the ISAR image obtained by the three algorithms and the reference ISAR image shown in fig. 3(b) are recorded, and are respectively shown in fig. 5(a), fig. 5(b) and fig. 5 (c). As can be seen from the figure, the ADMM method provided by the invention obtains the highest correlation coefficient and the lowest image entropy under all the conditions of the given sampling rate, and the calculation time is kept within 10s and is improved by 10-20 times compared with the AFSBL method.
Fig. 6 shows ISAR imaging results of the three methods under the condition that the radar echo SNRs are 10dB, 5dB, and 0dB, respectively, where the sparse aperture one-dimensional range profile sequence is obtained by random sampling, and the sampling rate is 0.25. It can be known from the figure that when the SNR is lower than 5dB, the RD and AFSBL methods are basically invalid, and the ISAR image focusing cannot be realized, but the ADMM method provided by the invention obtains an ISAR image with a good focusing effect under three SNR conditions, which indicates that the noise robustness is strong.
The average correlation coefficient, the image entropy and the calculation time obtained by the three methods in 100 Monte Carlo experiments under different SNR conditions are respectively shown in FIG. 7(a), FIG. 7(b) and FIG. 7 (c). As can be seen from the figure, the method obtains the highest correlation coefficient and the lowest image entropy again, improves the operation efficiency by nearly 10 times compared with the AFSBL method, and further verifies the stronger robustness and higher operation efficiency.
And further performing algorithm performance verification by adopting measured data of a certain airplane. The optical image of the airplane is shown in fig. 8(a), and is measured by using a certain onboard X-band radar system, as shown in fig. 8 (b). The transmitted radar signal parameters are as follows: the center frequency was 9GHz, the bandwidth was 1GHz, the pulse width was 100 mus, and the pulse repetition frequency was 100 Hz. The full aperture radar echo contains 256 pulses, each pulse containing 512 sampling points. Fig. 9(a) and 9(b) show a target one-dimensional range image sequence and an ISAR image under the full aperture condition, respectively, as a reference of an ISAR imaging result under the sparse aperture condition.
Data are extracted from the full-aperture one-dimensional range profile shown in fig. 8(a) by three ways of randomness, segmentation and mixing to simulate a sparse aperture one-dimensional range profile sequence, wherein the sampling rate is 0.25. Fig. 10 shows a target one-dimensional image sequence under three sparse aperture conditions and an ISAR image obtained by the three methods, and it can be seen from the figure that the focusing effect of the ISAR image obtained by the ADMM method provided by the present invention is better than that of the image obtained by the RD and AFSBL methods, and is closer to the ISAR image shown in fig. 9(a), and further verifies the better algorithm performance.
The experimental results show that the ISAR imaging under the sparse aperture condition can be realized, the focusing effect of the obtained image is better than that of the traditional RD method and the representative sparse aperture ISAR imaging method, the robustness to noise is strong, the method is still applicable under the condition of low signal-to-noise ratio, the operation efficiency is improved by 10-20 times compared with that of the representative method, and the method has higher engineering application value.

Claims (3)

1. An ADMM-based fast sparse aperture ISAR self-focusing and imaging method is characterized by comprising the following steps:
s1 models sparse aperture radar returns:
under the irradiation of high-frequency radar signals, a metal target can be generally equivalent to the sum of a plurality of discrete scattering points, and the target is assumed to contain P scattering points, wherein the coordinate of the P-th scattering point relative to the rotation center of the target is (x)p,yp) Then, the one-dimensional range image sequence of the object can be represented as a superposition of the one-dimensional range images of the P scattering points:
Figure FDA0002720903460000011
wherein
Figure FDA0002720903460000012
A one-dimensional range image sequence of the target is represented,
Figure FDA0002720903460000013
tmrespectively representing fast and slow times, σpBackscattering coefficients, j, f, representing the p-th scattering point of the objectcC is an imaginary number unit, a radar transmission signal carrier frequency and an electromagnetic wave propagation speed respectively, omega represents the equivalent rotating speed of the target after translational compensation, phi (t)m) Representing an initial phase error which comprises a phase error introduced by target translation and an environmental noise phase error; formula (1) canFurther discretization is as follows:
Figure FDA0002720903460000014
wherein h (m, n) represents a target discrete one-dimensional range profile sequence, and n and m are respectively a fast time sequence number and a slow time sequence number: n 1,2, …, N, M1, 2, …, M, N, M are the total number of fast and slow times, PrRepresenting a radar transmit signal pulse repetition frequency;
under the condition of sparse aperture, radar echo signals are in an undersampling form along a slow time dimension, and if the radar echo comprises L pulses at the moment, L is less than M, and a vector formed by combining sequence numbers of the pulses is V, then the radar echo has
Figure FDA0002720903460000015
On this basis, the target discrete one-dimensional range profile sequence under the sparse aperture condition can be represented as:
h＝ESFx+n (3)
wherein
Figure FDA0002720903460000016
Matrix for representing discrete one-dimensional range profile of target
Figure FDA0002720903460000017
Stacking the formed vectors along the columns, i.e., h ═ Vec (h), where Vec (·) denotes vectorization of the matrix stack along the columns;
Figure FDA0002720903460000018
is a block initial phase error matrix:
Figure FDA0002720903460000019
wherein INRepresenting an N x N identity matrix,
Figure FDA00027209034600000110
the kronecker product of the matrix is represented,
Figure FDA00027209034600000111
represent the initial phase error matrix: e ═ diag [ exp (j φ)]Wherein diag (·) denotes a diagonal matrix composed of vectors, whose diagonal elements are composed of vector elements in parentheses, and phi denotes an initial phase error vector;
Figure FDA00027209034600000112
a block downsampling matrix:
Figure FDA00027209034600000113
wherein
Figure FDA00027209034600000114
For the down-sampling matrix:
Figure FDA00027209034600000115
Figure FDA00027209034600000116
is a block fourier matrix:
Figure FDA00027209034600000117
wherein
Figure FDA00027209034600000118
Is an M-order Fourier matrix;
Figure FDA00027209034600000119
representing target ISAR images
Figure FDA00027209034600000120
The vectors formed by stacking along the columns, i.e., x ═ vec (x);
Figure FDA00027209034600000121
is the noise vector after the column stacking;
s2 reconstructs the target ISAR image X by ADMM:
by means of1The norm regularization method restrains the sparse characteristics of the ISAR image, and under the condition, the sparse aperture ISAR image reconstruction is equivalent to solving the following optimization problem:
Figure FDA00027209034600000122
wherein | · | purple1、||·||2Are each l1、l2A norm; lambda is a regularization parameter, and the sparsity degree of the obtained ISAR image is determined; the method comprises the following specific steps:
s2.1, solving the optimization problem shown in the formula (4) by an ADMM method:
the ADMM method equates the optimization problem shown in formula (4) to the following optimization problem by introducing an auxiliary variable z:
Figure FDA0002720903460000021
the extended Lagrange expression is as follows:
Figure FDA0002720903460000022
wherein alpha is a Lagrange multiplier and rho is a penalty factor; the ADMM method solves the following three sub-problems to achieve the solution of equation (4):
Figure FDA0002720903460000023
wherein k represents the kth iteration; respectively obtain Lρ(x, z, α) the first and second equations in equation (7) can be solved by taking the first derivative of x and z to zero, as shown in the following equation:
Figure FDA0002720903460000024
wherein S (-) represents a soft threshold function, and for any variable y and parameter a, the following are:
Figure FDA0002720903460000025
s2.2, converting the updated expression shown in the expression (8) into a matrix form:
in the formula (8), the first equation includes inversion of a matrix with a size of MN × MN, and the operation efficiency is low; to simplify the operation, consider that the matrix E, F, S has the following properties:
EHE＝EEH＝ILN (9)
FHF＝FFH＝IMN (10)
Figure FDA0002720903460000026
wherein ILN、IMNRespectively, unit matrixes with the size of LNXLN and MN,
Figure FDA0002720903460000027
is a diagonal matrix whose diagonal elements are
Figure FDA0002720903460000028
Substituting equations (9) - (11) for the first equation in equation (8) can obtain:
Figure FDA0002720903460000029
at the moment, the matrix to be inverted is converted into a diagonal matrix, and the inversion of the diagonal matrix can be realized by matrix element division; thus, the vector form of equation (12) can be re-expressed as a matrix form, as shown in the following equation:
Figure FDA0002720903460000031
wherein Z, A are in the form of matrices of the auxiliary variable z and lagrange multiplier alpha respectively,
Figure FDA0002720903460000032
the elements of the two matrixes are divided respectively, Mask is a sampling matrix,
Figure FDA0002720903460000033
the value of the element is 1 or 0, which respectively represents the extraction or the abandonment of the element at the corresponding position, and the sampling matrix is multiplied with the complete target one-dimensional range profile sequence according to each element, so as to obtain the zero-filling sparse aperture one-dimensional range profile sequence, 1M×NRepresenting a full 1 matrix of size mxn;
likewise, the second and third equations in equation (8) can be further expressed in matrix form:
Figure FDA0002720903460000034
A(k+1)＝A(k)+ρ(X(k+1)-Z(k+1))； (15)
s3 estimates the initial phase error phi by the minimum entropy criterion:
estimating an initial phase error phi through a minimum entropy criterion in an iteration process to realize ISAR self-focusing under a sparse aperture condition, wherein the following formula is shown as follows:
Figure FDA0002720903460000035
wherein the content of the first and second substances,
Figure FDA0002720903460000036
the entropy of the ISAR image obtained from the (k + 1) th iteration is represented and defined as follows:
Figure FDA0002720903460000037
wherein sum (-) indicates the summation of each element of the matrix in the brackets, and is the Hadamard product of the matrix, i.e. each element of the matrix is multiplied separately; c is total energy of the ISAR image: c ═ sum (| X)(k+1)|2) Which is independent of the initial phase error phi; const represents a constant independent of the initial phase error φ; the method comprises the following specific steps:
s3.1 calculation
Figure FDA0002720903460000038
Initial phase error phi of the l-th pulselDerivative of (a):
the optimization problem shown in the formula (16) is solved, and firstly, the calculation is carried out
Figure FDA0002720903460000039
Initial phase error phi of the l-th pulselThe derivative of (c) is shown as follows:
Figure FDA00027209034600000310
where Re (. cndot.) denotes the real part of the complex number in parentheses, X(k+1)*Representing ISAR image X obtained from the k +1 th iteration(k+1)Conjugation of (1);
s3.2 calculating ISAR image X obtained by k +1 iteration(k+1)Initial phase error phi of the l-th pulselDerivative of (a):
the ISAR image X obtained in the k +1 th iteration is included in the formula (18)(k+1)Initial phase error phi of the l-th pulselThe derivative of (c) is calculated as follows:
Figure FDA0002720903460000041
wherein 0(l-1)×N、0(L-l)×NAll-zero matrices of sizes (L-1) xN and (L-L) xN are respectively represented;
s3.3 calculating the initial phase error phi obtained by the k +1 iterationl (k+1)：
Substituting formula (19) for formula (18) and making the derivative
Figure FDA0002720903460000042
The initial phase error phi of the first pulse can be obtainedlIteratively updating the expression:
Figure FDA0002720903460000043
where angle (·) denotes the phase taking the imaginary number in parentheses;
therefore, the fast sparse aperture ISAR self-focusing and imaging process based on the alternative direction multiplier is as follows: and (5) iterating the equations (13) - (15) and the equation (20) in a loop until convergence, wherein X obtained by the equation (13) is the reconstructed ISAR image.
2. An ADMM-based fast sparse aperture ISAR auto-focusing and imaging method according to claim 1, wherein: before iterative solution, parameters are initialized, wherein an auxiliary variable Z, a Lagrange multiplier A and an initial phase error matrix e can be initialized to be an all-zero matrix, and a regularization parameter lambda is initialized to be mu var (H), wherein var (·) is a variance operator, mu is more than 0.005 and less than 0.01, and a penalty factor rho is 1.
3. An ADMM-based fast sparse aperture ISAR auto-focusing and imaging method according to claim 1 or 2, wherein: in S2.2, f and f in formula (13)HThe method can be realized by FFT and IFFT respectively to further improve the operation efficiency.
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