CN112684445A - MIMO-ISAR three-dimensional imaging method based on MD-ADMM - Google Patents

MIMO-ISAR three-dimensional imaging method based on MD-ADMM Download PDF

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CN112684445A
CN112684445A CN202011402759.4A CN202011402759A CN112684445A CN 112684445 A CN112684445 A CN 112684445A CN 202011402759 A CN202011402759 A CN 202011402759A CN 112684445 A CN112684445 A CN 112684445A
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CN112684445B (en
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张双辉
邓理康
黎湘
刘永祥
张新禹
霍凯
姜卫东
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National University of Defense Technology
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Abstract

The invention belongs to the field of radar imaging, and particularly relates to an MD-ADMM-based MIMO-ISAR three-dimensional imaging method, which comprises the following steps: s1 modeling the MIMO-ISAR moving target echo; s2 modeling the MIMO-ISAR three-dimensional sparse imaging problem; s3 MIMO-ISAR three-dimensional image sparse reconstruction based on MD-ADMM. The invention has the following beneficial effects: the method can realize the three-dimensional sparse imaging of the MIMO-ISAR, can effectively reduce the storage and calculation burden and improve the three-dimensional imaging calculation efficiency and robustness under the condition of three-dimensional sparse data, further obtains a three-dimensional image with good focus, and has important engineering application value for target radar imaging, feature extraction and target identification under the condition of multi-dimensional data limitation.

Description

MIMO-ISAR three-dimensional imaging method based on MD-ADMM
Technical Field
The invention belongs to the field of Radar imaging, and particularly relates to a multi-Input multi-Output Inverse Synthetic Aperture Radar (MIMO-ISAR) three-dimensional imaging Method based on a multi-dimensional alternating direction Method (MD-ADMM).
Background
Inverse Synthetic Aperture Radar (ISAR) may generate a two-dimensional image of a target, typically for target recognition and classification. Compared with a two-dimensional image, the three-dimensional image can obtain more target information, and is more beneficial to target identification.
The MIMO-ISAR imaging technology is used for processing multi-path echo data in a limited imaging time and replacing spatial sampling with time sampling so as to improve the image resolution. However, the imaging requires a large amount of data processing, requires a large number of array elements, and has high hardware cost. When the number of array elements is reduced or complete echoes are lost due to noise or hardware, if the Fourier transform method is continuously adopted for three-dimensional imaging, the image quality is seriously reduced.
The compressive sensing theory is based on the sparsity of images, and high-resolution images can be recovered by using a small amount of data sampling, so that the compressive sensing theory is widely applied to the field of image processing. But the traditional compressed sensing method needs to convert the multidimensional signal into a one-dimensional vector. However, this entails the problem of an excessively large dimension of the measurement matrix and the signal, which leads to a drastic increase in the memory and computational burden. The currently known multi-dimensional signal-based sparse recovery method has relatively low computational complexity and memory consumption, but has more parameters to be adjusted, and has a general effect of recovering an image under a low signal-to-noise ratio condition. Therefore, the method has important engineering application value for improving the image recovery effect under the multidimensional sparse condition.
Disclosure of Invention
The invention aims to solve the technical problems that under the condition of MIMO-ISAR multi-dimensional sparse data, the traditional multi-dimensional sparse recovery method has large storage and calculation burden and poor robustness and is difficult to meet the engineering application requirements.
The invention provides an MD-ADMM-based MIMO-ISAR three-dimensional imaging method aiming at the problems that a traditional compressed sensing method is large in storage and calculation consumption and a known multi-dimensional sparse recovery method is poor in robustness under the condition of MIMO-ISAR multi-dimensional sparse data. The method firstly establishes an MIMO-ISAR model into a sparse recovery problem of three-dimensional data. And an MD-ADMM sparse recovery method is further adopted for three-dimensional imaging, so that the storage burden is reduced, and the robustness and the calculation efficiency are improved. The method can finally obtain the three-dimensional ISAR image of the target through loop iteration.
The technical scheme adopted by the invention for solving the technical problems is as follows: an MD-ADMM-based MIMO-ISAR three-dimensional imaging method comprises the following steps:
s1 models the MIMO-ISAR echo:
assuming that the transmitting array transmits a stepped frequency signal, the transmitting frequency fv＝fc+ (v-1) Δ f, wherein fcFor the center frequency of the transmitted signal, Δ f is the frequency step, v denotes the number of the stepped frequency signal: v ═ 1,2,. V, where V is the total number of transmitted stepped frequency signals; then, when the u-th equivalent transceiving array element takes the w-th snapshot, the echo passing through the q-th scattering point on the target and sorted by the signal can be represented as:
Figure BDA0002812986600000021
u represents the equivalent transceiving array element number: u1, 2.. U, where U denotes the total number of equivalent transmit/receive array elements, q denotes the scattering point number: q1, 2.. Q, where Q is the total number of scattering points, w is the snapshot number: w1, 2,. W, W representing the total number of snapshots; c is the speed of light, σqIs the scattering intensity of the qth scattering point,
Figure BDA0002812986600000022
representing the distance from the u-th equivalent array element to the q-th scattering point at the w-th snapshot time; through translation compensation (Sharp, Cheng Meng, Wangtong radar imaging technology [ M)]Beijing electronics industry Press, 2005) the following formula (1) can be expressed as:
Figure BDA0002812986600000023
wherein
Figure BDA0002812986600000024
Representing the distance from the u-th equivalent array element to the w-th snapshot time target rotation center; in formula (2)
Figure BDA0002812986600000025
According to the literature (Wang Y, Li X.3-D Imaging base on Combination of the ISAR Technique and a MIMO Radar System [ J]IEEE Transactions on GeoTance and Remote Sensing,2018,56(10): 6033-:
Figure BDA0002812986600000026
wherein
Figure BDA0002812986600000027
And thetawRespectively are included angles (shown in figure 2) between a connecting line of the u-th equivalent array element and the w-th snapshot time target rotation center and the Z axis and the Y axis,
Figure BDA0002812986600000028
is the coordinate of the q scattering point in the initial reference coordinate system; when the observation time is short,
Figure BDA0002812986600000029
θ w0, substituting the formula (3) into the formula (2) to obtain:
Figure BDA00028129866000000210
wherein d is the equivalent transmit-receive array element interval, R0Is the distance from the target to the array element, omega is the target equivalent rotation speed, TpIs the pulse width;
s2, modeling the MIMO-ISAR three-dimensional sparse imaging problem:
in the formula (4), a three-dimensional image can be obtained by performing Fourier transform along u, v and w respectively; the fourier transform-based signal tensor model can be expressed as
Figure BDA00028129866000000211
Wherein the ingredientl(1, 2,3), representing the n-modal product of the tensor and matrix,
Figure BDA00028129866000000212
in the complex field
Figure BDA00028129866000000213
A three-dimensional signal with a medium dimension of UxV xW;
Figure BDA00028129866000000214
representing a complex field
Figure BDA00028129866000000215
The middle dimension is a U multiplied by V multiplied by W three-dimensional image,
Figure BDA00028129866000000216
Figure BDA00028129866000000217
representing a full Fourier transform matrix, wherein
Figure BDA00028129866000000218
Respectively expressed in a plurality of fields
Figure BDA00028129866000000219
A matrix with a median dimension of UxU, VxV, WxW;
when the number of array elements is reduced or a complete echo is missing due to noise or hardware (equivalent to sparsely sampling three dimensions of the echo), the image quality is seriously degraded if the fourier transform is continuously adopted, and therefore the formula (5) can be expressed as follows:
Figure BDA00028129866000000220
wherein
Figure BDA00028129866000000221
Presentation pair
Figure BDA00028129866000000222
And after sparse sampling, the signals M, N and K respectively represent the sampling number of three dimensions of the signals.
Figure BDA0002812986600000031
Respectively, the dimensions of the partial Fourier transform matrix are M × U, N × U and K × W. Let F(1)＝T1F1，F(2)＝T2F2，F(3)＝T3F3Wherein
Figure BDA0002812986600000032
A sampling matrix is represented. Let G, H, J denote the pair tensors respectively
Figure BDA0002812986600000033
A three-dimensional sampling sequence, where G ∈ [1]T,H∈[1,...,V]T,J∈[1,...,W]TThe sequence lengths are respectively M, N and K; then T1,T2,T3Can be respectively expressed as:
Figure BDA0002812986600000034
where M1, 2,. M, N1, 2,. N, K1, 2,. K represent the sampling numbers in three dimensions, respectively.
S3 MIMO-ISAR three-dimensional image sparse reconstruction based on MD-ADMM:
quantizing the MIMO-ISAR tensor model vector in S2 into a one-dimensional form, and establishing the model based on l1If the norm minimization model is directly solved by a compressed sensing method, the measurement matrix and the signal dimension of the norm minimization model are too large, so that great calculation and storage burden is caused. So the section adopts the calculation based on MD-ADMMThe sparse imaging method is characterized in that a perception matrix is decomposed into tensor modal products, tensor element division is used for replacing matrix inversion, calculation and storage burden is reduced remarkably, and the method comprises the following specific steps:
s3.1 vectorizing and establishing the tensor model in S2 based on l1Norm minimization model:
unfolding equation (5) into a one-dimensional form as follows:
Figure BDA0002812986600000035
wherein
Figure BDA0002812986600000036
Where vec (·) denotes vectorizing the tensor. Hypothetical image
Figure BDA0002812986600000037
Is sparse, is based on l according to the compressed sensing principle1The norm minimization optimization problem can be expressed as
Figure BDA0002812986600000038
Wherein
Figure BDA0002812986600000039
Representing an estimate of the vector x, λ is a regular coefficient.
S3.2 the model in S3.1 is optimized by the ADMM method:
according to the ADMM algorithm principle, an auxiliary variable z and a primary variable l are introduced1The norm minimization problem can be equivalent to the following equality constrained optimization problem:
Figure BDA00028129866000000310
further solving the constraint optimization problem shown in the formula (10) by an augmented Lagrange method, as shown in the following formula:
Figure BDA00028129866000000311
wherein α is a dual variable, ρ is a penalty coefficient, and the problem can be decomposed into the following three subproblems in the iterative process:
Figure BDA0002812986600000041
wherein (·)(k)Representing the updated variable values for the kth iteration, the first two equations of equation (12) can be solved by making Lρ(x, z, α) is obtained with a first order partial derivative of x and z equal to zero as shown in the following equation:
Figure BDA0002812986600000042
where ST (·) is a soft threshold function, which is expressed as ST (x, a) ═ x/| x |) max (| x | -a, 0). F is to be(1)＝T1F1,F(2)＝T2F2,F(3)＝T3F3Substitution (13) can give:
Figure BDA0002812986600000043
wherein B is1＝T1 HT1,B2＝T2 HT2,B3＝T3 HT3By simplification, the following can be obtained:
Figure BDA0002812986600000044
writing equation (15) in tensor form:
Figure BDA0002812986600000045
wherein 1 isU×V×WA three-dimensional tensor representing elements all having 1 dimension U x V x W,
Figure BDA0002812986600000046
the division of the elements representing the tensor is,
Figure BDA0002812986600000047
the value of the sample indicating the three-dimensional direction of the docking echo is set to 0 or 1, which indicates whether the sample is taken or not.
S3.3 reconstruction of three-dimensional images of objects by iterative looping
Figure BDA0002812986600000048
Equation (13) can also be written in the form of a tensor as follows:
Figure BDA0002812986600000049
Figure BDA0002812986600000051
the value of the penalty coefficient rho is set to be 1, and the value range of the regular coefficient lambda is [2,6]]In the invention, the value of lambda is 5, and the combined iteration formulas (16), (17) and (18) are combined until the relative error of the ISAR images in two adjacent iterations
Figure BDA0002812986600000052
Less than a set threshold (e.g., 10)-4) Then, the three-dimensional ISAR image of the target can be obtained
Figure BDA0002812986600000053
The initial parameter settings are as follows:
Figure BDA0002812986600000054
and
Figure BDA0002812986600000055
is set to a three-dimensional tensor in which all elements are 0.
The invention has the following beneficial effects: the method can realize the three-dimensional sparse imaging of the MIMO-ISAR, can effectively reduce the storage and calculation burden and improve the three-dimensional imaging calculation efficiency and robustness under the condition of three-dimensional sparse data, further obtains a three-dimensional image with good focus, and has important engineering application value for target radar imaging, feature extraction and target identification under the condition of multi-dimensional data limitation.
Drawings
FIG. 1 is a flow chart of an embodiment of the present invention;
FIG. 2
Figure BDA0002812986600000056
And thetawA corner schematic diagram;
FIG. 3(a) a three-dimensional scatter plot of the target; (b) a top view imaging the target at full aperture; (c) a front view imaging the target at full aperture; (d) a view map of the target under full aperture conditions;
FIG. 4 is a comparison of images obtained by different algorithms under different sparsity conditions: (a) images obtained by different algorithms with the sparsity of 50 percent; (b) images obtained by different algorithms with the sparsity of 33.3 percent; (c) sparsity is 25% of images obtained by different algorithms;
FIG. 5 is a comparison of images obtained by different algorithms under the same sparsity and different signal-to-noise ratios: (a) images obtained by different algorithms with sparsity of 25% and signal-to-noise ratio of-5 dB; (b) images obtained by different algorithms with sparsity of 25% and signal-to-noise ratio of 0 dB; (c) images obtained by different algorithms with sparsity of 25% and signal-to-noise ratio of 10 dB;
FIG. 6 shows ISAR images obtained by different algorithms under two-dimensional random sampling conditions for Yak-42 aircraft measured data: (a) imaging results after two-dimensional FFT under the full aperture condition; (b) an RD algorithm imaging result; (c) the imaging result of the MD-SL0 algorithm; (d) imaging results of the invention;
FIG. 7 is an ISAR image obtained by different algorithms under the condition of adding Gaussian white noise two-dimensional random sampling to Yak-42 aircraft measured data: (a) imaging results after two-dimensional FFT under the full aperture condition; (b) an RD algorithm imaging result; (c) the imaging result of the MD-SL0 algorithm; (d) the invention provides an imaging result.
Detailed Description
The invention is further illustrated with reference to the accompanying drawings:
FIG. 1 is a process flow of the present invention. The invention discloses a multi-dimensional ADMM-based MIMO-ISAR three-dimensional imaging method, which comprises the following steps:
s1 modeling the MIMO-ISAR moving target echo;
s2 modeling the MIMO-ISAR three-dimensional sparse imaging problem;
s3 MIMO-ISAR three-dimensional image sparse reconstruction based on MD-ADMM.
In FIG. 2, in S1
Figure BDA0002812986600000057
And thetawAnd (5) a corner schematic diagram.
Fig. 3(a) is a three-dimensional scatter diagram of a simulated airplane target, and fig. 3(b) (c) (d) is a three-dimensional view of the simulated airplane target under a full aperture condition. The aircraft flies perpendicular to the 10-transmitter 6-receiver MIMO linear array at a speed v of 200 m/s. The radar emission signal parameters are as follows: the center frequency is 10GHz, the bandwidth is 150MHz, the slow time sampling frequency is 80Hz, and the number of the step frequency signals is 60. The 10-transmitter 6-receiver array can be equivalent to 60 transceiving shared array elements, the full aperture data of each equivalent transceiving shared array element comprises 60 pulses, and each pulse comprises 60 sampling points.
And respectively and randomly extracting 30, 20 and 15 pulses in three dimensions of the full-aperture data to simulate sparse echo data with sparsity of 50%, 33.3% and 25% in the three dimensions. And respectively smoothing l by using a conventional range-Doppler (RD) method and a multi-dimensional SL0 method0Norm (MD-SL0) method (Hu X, Tong N, Wang H, et al, multiple-input-multiple-output radial super resolution on multiple smooth 0[ J-SL 0 ]]Journal of Applied Remote Sensing,2016,10(3):035017.) and the present invention performed ISAR imaging on this sparse data, the resulting ISAR images are shown in FIG. 4(a) (b) (c), respectively. As can be seen from FIG. 4, the side lobes and grating lobes are generated under the sparse echo conditionUnder the influence, ISAR images obtained by the RD method are seriously defocused. The MD-SL0 method and the image obtained by the method have good effect, which shows that the method effectively inhibits the sidelobe and grating lobe interference introduced by sparse echo. However, the MD-SL0 algorithm has smaller image entropy and shorter calculation time, and shows better imaging performance.
TABLE 1
Figure BDA0002812986600000061
Table 1 shows the entropy and computation time of the ISAR images obtained by the two methods under random sparse sampling conditions to further compare the performances of the two methods. As can be seen from the table, the image entropy obtained by the method is lower, the calculation time is shorter, and the ISAR image obtained by the method is better in focusing effect and higher in calculation efficiency.
TABLE 2
Figure BDA0002812986600000062
Randomly extracting 20 pulses from three dimensions in the original data to simulate sparse echo data with sparsity of 25%, and adding white Gaussian noise with signal-to-noise ratio of-5 dB, 0dB and 10dB mean value of zero. And ISAR imaging is carried out on the sparse data by respectively adopting a traditional RD method, an MD-SL0 method and the method of the invention, and the obtained ISAR three-dimensional images are respectively shown as (a) (b) (c) of FIG. 5. As can be seen from fig. 5(a), (b), and (c), the ISAR image obtained by the RD method is severely defocused under the influence of side lobes, grating lobes, and noise generated by the sparse echo. The degree of focus of the image obtained by the MD-SL0 method is also reduced to a certain degree due to the influence of noise. The method is minimally affected by noise and has the highest imaging quality, and therefore the robustness is better compared with the MD-SL0 method.
Table 2 shows the entropy and the calculation time of the ISAR images obtained by the two methods under the condition of sparsity of 25% and different signal-to-noise ratios, so as to further compare the performances of the two methods. As can be seen from the table, the image entropy obtained by the method is lower, the calculation time is shorter, and the robustness to noise is better.
And further partially verifying the performance of the algorithm by using the Yak-42 airplane measured data. The radar signal parameters are as follows: the center frequency is 5.52GHz, the bandwidth is 400MHz, and the pulse width is 25.6 mus. The full aperture radar echo contains 256 pulses, each pulse containing 256 sampling points. And (3) extracting 96 pulses and 128 fast time signals by adopting a random sampling mode to simulate two-dimensional echo data with the sparsity of 37.5% and 50% respectively. Fig. 6(a) is a reference of an ISAR image of an object under the complete data condition as an ISAR imaging result under the random sampling condition. Fig. 6(b), fig. 6(c) and fig. 6(d) respectively show that the RD algorithm, the MD-SL0 method and the method of the present invention obtain ISAR images. From fig. 5, it can be known that the image obtained by the RD algorithm is greatly defocused, and compared with the MD-SL0 method, the method of the present invention has fewer error points and sharper image.
FIG. 7 shows the result of adding Gaussian white noise with a signal-to-noise ratio of 0dB to the two-dimensional echo. Fig. 7(a) is a reference of an ISAR image of an object under the complete data condition as an ISAR imaging result under the random sampling condition. Fig. 7(b), fig. 7(c) and fig. 7(d) respectively show that the RD algorithm, the MD-SL0 method and the method of the present invention obtain ISAR images. As can be seen from FIG. 7, compared with the MD-SL0 method, the method has the advantages of less image error points, better image focusing effect and better noise robustness.
In conclusion, the invention can effectively realize imaging under the condition of MIMO-ISAR multi-dimensional data sparse sampling, and compared with the existing MD-SL0 algorithm, the algorithm has higher calculation efficiency, better robustness and stronger engineering practicability.

Claims (7)

1. An MD-ADMM-based MIMO-ISAR three-dimensional imaging method is characterized by comprising the following steps:
s1 models the MIMO-ISAR echo:
assuming that the transmitting array transmits a stepped frequency signal, the transmitting frequency fv＝fc+ (v-1) Δ f, wherein fcFor the center frequency of the transmitted signal, Δ f is the frequency step, v denotes the number of the stepped frequency signal: v ═ 1,2,. V, where V is the total number of transmitted stepped frequency signals; the u-th equivalent transceiving array elementAt the w-th snapshot, the echo after sorting the q-th scattering point on the target by the signal can be represented as:
Figure FDA0002812986590000011
u represents the equivalent transceiving array element number: u is 1,2 … U, where U represents the total number of equivalent transceiver array elements, q represents the scattering point number: q is 1,2 … Q, where Q is the total number of scattering points, w is the snapshot number: w is 1,2, … W, W represents total number of snapshots; c is the speed of light, σqIs the scattering intensity of the qth scattering point,
Figure FDA0002812986590000012
representing the distance from the u-th equivalent array element to the q-th scattering point at the w-th snapshot time; the equation (1) after translation compensation can be expressed as:
Figure FDA0002812986590000013
wherein
Figure FDA0002812986590000014
Representing the distance from the u-th equivalent array element to the w-th snapshot time target rotation center; in the formula (2)
Figure FDA0002812986590000015
Can be expressed as:
Figure FDA0002812986590000016
wherein
Figure FDA0002812986590000017
And thetawRespectively are included angles between a connecting line of the u-th equivalent array element and the w-th target rotation center at the moment of snapshot and the Z axis and the Y axis,
Figure FDA0002812986590000018
is the coordinate of the q scattering point in the initial reference coordinate system; when the observation time is short,
Figure FDA0002812986590000019
θw0, substituting the formula (3) into the formula (2) to obtain:
Figure FDA00028129865900000110
wherein d is the equivalent transmit-receive array element interval, R0Is the distance from the target to the array element, omega is the target equivalent rotation speed, TpIs the pulse width;
s2, modeling the MIMO-ISAR three-dimensional sparse imaging problem:
in the formula (4), a three-dimensional image can be obtained by performing Fourier transform along u, v and w respectively; the fourier transform-based signal tensor model can be expressed as
Figure FDA00028129865900000111
Wherein the ingredientl(1, 2,3), representing the n-modal product of the tensor and matrix,
Figure FDA00028129865900000112
in the complex field
Figure FDA00028129865900000113
A three-dimensional signal with a medium dimension of UxV xW;
Figure FDA00028129865900000114
representing a complex field
Figure FDA00028129865900000119
The middle dimension is a U multiplied by V multiplied by W three-dimensional image,
Figure FDA00028129865900000115
Figure FDA00028129865900000116
representing a full Fourier transform matrix, wherein
Figure FDA00028129865900000117
Respectively expressed in a plurality of fields
Figure FDA00028129865900000118
A matrix with a median dimension of UxU, VxV, WxW;
equation (5) can be expressed as:
Figure FDA0002812986590000021
wherein
Figure FDA0002812986590000022
Presentation pair
Figure FDA0002812986590000023
After sparse sampling, M, N and K respectively represent the sampling number of three dimensions of the signal;
Figure FDA0002812986590000024
respectively representing partial Fourier transform matrixes with dimensions of M multiplied by U, N multiplied by U and K multiplied by W; let F(1)＝T1F1，F(2)＝T2F2，F(3)＝T3F3Wherein
Figure FDA0002812986590000025
Representing a sampling matrix; let G, H, J denote the pair tensors respectively
Figure FDA0002812986590000026
Three dimensional sampling sequence, where G ∈ [1, …, U]T,H∈[1,…,V]T,J∈[1,…,W]TThe sequence lengths are respectively M, N and K; then T1,T2,T3Can be respectively expressed as:
Figure FDA0002812986590000027
wherein M is 1,2, … M, N is 1,2, … N, K is 1,2, … K respectively represent sampling numbers of three dimensions;
s3 MIMO-ISAR three-dimensional image sparse reconstruction based on MD-ADMM:
the method comprises the following steps of adopting a sparse imaging method based on MD-ADMM to perceive matrix decomposition into tensor modal products, and substituting tensor element division for matrix inversion, wherein the method specifically comprises the following steps:
s3.1 vectorizing and establishing the tensor model in S2 based on l1Norm minimization model:
unfolding equation (5) into a one-dimensional form as follows:
Figure FDA0002812986590000028
wherein
Figure FDA0002812986590000029
Where vec (·) denotes vectorizing the tensor; hypothetical image
Figure FDA00028129865900000210
Is sparse, is based on l according to the compressed sensing principle1The norm minimization optimization problem can be expressed as
Figure FDA00028129865900000211
Wherein
Figure FDA00028129865900000212
Represents an estimate of the vector x, λ being a regular coefficient;
s3.2 the model in S3.1 is optimized by the ADMM method:
according to the ADMM algorithm principle, an auxiliary variable z and a primary variable l are introduced1The norm minimization problem can be equivalent to the following equality constrained optimization problem:
Figure FDA00028129865900000213
further solving the constraint optimization problem shown in the formula (10) by an augmented Lagrange method, as shown in the following formula:
Figure FDA00028129865900000214
wherein α is a dual variable, ρ is a penalty coefficient, and the problem can be decomposed into the following three subproblems in the iterative process:
Figure FDA0002812986590000031
wherein (·)(k)Representing the updated variable values for the kth iteration, the first two equations of equation (12) can be solved by making Lρ(x, z, α) is obtained with a first order partial derivative of x and z equal to zero as shown in the following equation:
Figure FDA0002812986590000032
where ST (-) is a soft threshold function; f is to be(1)＝T1F1,F(2)＝T2F2,F(3)＝T3F3Substitution (13) can give:
Figure FDA0002812986590000033
wherein B is1＝T1 HT1,
Figure FDA0002812986590000034
The method can be simplified to obtain:
Figure FDA0002812986590000035
writing equation (15) in tensor form:
Figure FDA0002812986590000036
wherein 1 isU×V×WA three-dimensional tensor representing elements all having 1 dimension U x V x W,
Figure FDA0002812986590000037
the division of the elements representing the tensor is,
Figure FDA0002812986590000038
the sampling represents the three-dimensional direction of the docking echo, and the value of the sampling is set to 0 or 1, which respectively represents whether the sampling is achieved or not;
s3.3 reconstruction of three-dimensional images of objects by iterative looping
Figure FDA0002812986590000039
Equation (13) can also be written in the form of a tensor as follows:
Figure FDA00028129865900000310
Figure FDA00028129865900000311
the combined iteration of the equations (16) (17) (18) is carried out until the relative error of the ISAR images in two adjacent iterations
Figure FDA0002812986590000041
When the image is smaller than the set threshold, the three-dimensional ISAR image of the target can be obtained
Figure FDA0002812986590000042
2. The MD-ADMM-based MIMO-ISAR three-dimensional imaging method according to claim 1, wherein: in S3.2, the expression of the soft threshold function ST (·) is ST (x, a) ═ x/| max (| x | -a, 0).
3. The MD-ADMM-based MIMO-ISAR three-dimensional imaging method according to claim 1, wherein: initial parameters
Figure FDA0002812986590000043
And
Figure FDA0002812986590000044
is set to a three-dimensional tensor in which all elements are 0.
4. The MD-ADMM-based MIMO-ISAR three-dimensional imaging method according to claim 1, wherein: the value of the penalty coefficient ρ is set to 1.
5. The MD-ADMM-based MIMO-ISAR three-dimensional imaging method according to claim 1, wherein: the value range of the regular coefficient lambda is [2,6 ].
6. The MD-ADMM-based MIMO-ISAR three-dimensional imaging method according to claim 5, wherein: the value of the regularization coefficient λ is 5.
7. A kind of baseThe MD-ADMM based MIMO-ISAR three-dimensional imaging method of claim 1, wherein: in S3.3, the threshold is set to 10-4。
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