CN112099008B - SA-ISAR imaging and self-focusing method based on CV-ADMMN - Google Patents

SA-ISAR imaging and self-focusing method based on CV-ADMMN Download PDF

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CN112099008B
CN112099008B CN202010975711.6A CN202010975711A CN112099008B CN 112099008 B CN112099008 B CN 112099008B CN 202010975711 A CN202010975711 A CN 202010975711A CN 112099008 B CN112099008 B CN 112099008B
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张双辉
李瑞泽
刘永祥
霍凯
姜卫东
黎湘
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National University of Defense Technology
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S13/00—Systems using the reflection or reradiation of radio waves, e.g. radar systems; Analogous systems using reflection or reradiation of waves whose nature or wavelength is irrelevant or unspecified
    • G01S13/88—Radar or analogous systems specially adapted for specific applications
    • G01S13/89—Radar or analogous systems specially adapted for specific applications for mapping or imaging
    • G01S13/90—Radar or analogous systems specially adapted for specific applications for mapping or imaging using synthetic aperture techniques, e.g. synthetic aperture radar [SAR] techniques
    • G01S13/904—SAR modes
    • G01S13/9064—Inverse SAR [ISAR]

Abstract

The invention belongs to the field of radar imaging, and particularly relates to a SA-ISAR imaging and self-focusing method based on CV-ADMMN, which comprises the following steps: s1, modeling the moving target one-dimensional range profile sequence; s2 modeling the moving target sparse aperture ISAR imaging scene; s3, establishing an ADMM reconstruction model of the moving target sparse aperture ISAR imaging problem; s4, establishing a CV-ADMMN network structure model; s5, solving the sparse aperture ISAR imaging problem by using a CV-ADMMN network structure; the beneficial effects obtained by the invention are as follows: the invention can realize the sparse aperture ISAR imaging and self-focusing of the moving target, can quickly reconstruct the complete radar image under the sparse aperture condition and realize the phase error compensation. The algorithm performance has weak dependence on parameter selection, and further better reconstruction performance is obtained. The method has important engineering application value for sparse aperture ISAR imaging and self-focusing under the condition of data loss.

Description

SA-ISAR imaging and self-focusing method based on CV-ADMMN
Technical Field
The invention belongs to the field of radar imaging, and particularly relates to a target Sparse aperture inverse synthetic aperture radar (SA-ISAR) imaging and self-focusing method based on a Complex-domain alternating direction multiplier network (CV-ADMMN).
Background
The Inverse Synthetic Aperture Radar (ISAR) imaging technology can be used for high-resolution imaging of targets, has the all-weather characteristic all day long, and is widely applied to the civil and military fields.
SA-ISAR imaging refers to imaging a target using sparse aperture radar returns. Sparse aperture echo refers to an incomplete echo received by the radar. In general, environment and radar receiver noise, a 'wide-narrow' alternative mode of a multifunctional radar, a random sampling mode of a compressed sensing radar, a target switching mode of a multi-channel radar, and the like all result in sparse aperture echoes. Under sparse aperture conditions, the conventional Fast Fourier Transform (FFT) method cannot image the azimuth unit because the correlation between echoes is severely destroyed. At this time, the imaging result can be solved by sparsity prior iteration of the radar image by using a convex optimization method. The convex optimization method is sensitive to model parameter selection, and selection of different parameters greatly affects the algorithm effect. In practical application, parameters need to be finely adjusted manually, which brings inconvenience to engineering application.
The self-focusing can compensate the phase error generated by the translation of the moving object, thereby realizing fine translation compensation. Under the condition of sparse aperture, due to data loss, the traditional self-focusing method is difficult to obtain good effect, and image defocusing is caused. Therefore, under the condition of sparse aperture, the method has important engineering application value for efficiently imaging and self-focusing the moving target.
Disclosure of Invention
The invention aims to solve the technical problems that under the condition of sparse aperture, the traditional moving target ISAR imaging method has strong parameter sensitivity, the traditional self-focusing method has poor effect, and the engineering application requirements are difficult to meet.
The invention provides an SA-ISAR imaging and self-focusing method based on CV-ADMMN, aiming at the problems that an imaging algorithm has strong sensitivity on parameter selection and a traditional self-focusing method has poor effect under the condition of sparse aperture. The method is based on a deep learning network model, applies a traditional Alternating direction multiplier (ADMM) to the SA-ISAR problem by using a deep expansion mode, and models the SA-ISAR problem into the deep learning network model. The network is trained on the data set to adaptively adjust the algorithm parameters. In order to improve the self-focusing effect under the sparse aperture condition, the self-focusing module based on the minimum entropy is embedded in the existing network structure, so that a complete CV-ADMMN structure is formed. The structure can reconstruct an original radar image from a sparse aperture one-dimensional distance direction sequence through network forward propagation.
The technical scheme adopted by the invention for solving the technical problems is as follows: a SA-ISAR imaging and self-focusing method based on CV-ADMMN comprises the following steps:
s1, modeling the moving target one-dimensional range profile sequence:
translational compensation is the first link of ISAR imaging, and the technical route thereof is relatively mature (shining under Sharp, Meng, Wang Tong radar imaging technology [ M ]. Beijing: electronics industry Press, 2005) after decades of development, so the present invention assumes that translational compensation of a target has been completed. The radar transmits a Linear Frequency Modulation (LFM) signal, and the two-dimensional echo received for a moving target can be modeled as:
Figure GDA0003337171810000021
wherein the content of the first and second substances,
Figure GDA0003337171810000022
t represents fast time and full time, respectively, and
Figure GDA0003337171810000023
tmindicating a slow time. SigmaiAnd RiRespectively representing the reflection coefficient of the ith scattering center and the instantaneous rotation distance, f, relative to the radarcC and gamma respectively represent the center frequency of the radar signal, the vacuum light speed and the signal frequency modulation. Due to the short integration time of ISAR imaging, the motion of the target within one pulse time can be neglected when modeling the echo.
The signal expression obtained after the two-dimensional signal shown in the formula (1) is subjected to line-off frequency modulation is as follows:
Figure GDA0003337171810000024
under sparse aperture conditions, the fast time echo pulse waveform remains unchanged, so the signal shown in equation (2) is in fast time
Figure GDA00033371718100000219
Performing FFT to obtain a target one-dimensional range profile sequence;
s2, modeling the moving target sparse aperture ISAR imaging scene:
under the condition of sparse aperture, the observation of the radar system to the moving target can be represented by the following down-sampling model:
y＝Φx+n＝dfx+n (3)
wherein the content of the first and second substances,
Figure GDA0003337171810000025
representing a radar image column vector formed by an image matrix
Figure GDA0003337171810000026
The method is obtained by rearranging along the columns,
Figure GDA0003337171810000027
representing the MN-dimensional complex column vector,
Figure GDA0003337171810000028
representing an M multiplied by N dimensional complex matrix, wherein M represents the number of azimuth units of the radar image, and N represents the number of range units;
Figure GDA0003337171810000029
representing a received radar one-dimensional range profile vector formed by a one-dimensional range profile matrix
Figure GDA00033371718100000210
Is obtained by rearrangement along the column, L represents the number of the one-dimensional range profile after the down sampling, L<<M；
Figure GDA00033371718100000211
Represents a down-sampling matrix that is down-sampled,
Figure GDA00033371718100000212
representing a gaussian white noise vector rearranged along a column;
Figure GDA00033371718100000213
represents a block Fourier transform matrix, which can be expressed as
Figure GDA00033371718100000214
Wherein INRepresents an N × N dimensional identity matrix, and F represents an M × M dimensional fourier transform matrix.
Figure GDA00033371718100000215
Represents a block down-sampling matrix, which can be expressed as
Figure GDA00033371718100000216
Where D represents an L × M dimensional down-sampling matrix, the elements consisting of 0 and 1. Let V represent the distance image index being sampled, then
Figure GDA00033371718100000217
For the i row and m column elements D in the matrix Dl,mWhen the l-th element V of the vector V islWhen m, there is Dl,m＝1，l＝1,2,…,L，m＝1,2,…,M。
The down-sampling model given by equation (3) models sparse aperture imaging as the solution of a linear underdetermined inverse problem, which can be solved by using a Compressive Sensing (CS) method (d.l.donoho, "Compressive sensing," IEEE Transactions on Information Theory, vol.52, No.4, pp.1289-1306,2006.);
s3, establishing an ADMM reconstruction model of the moving target sparse aperture ISAR imaging problem:
for the down-sampling model given by equation (3), it is solved using conventional ADMM:
s3.1, constructing an optimization model as follows:
Figure GDA00033371718100000218
where z is an introduced intermediate variable and λ represents a regularization parameter.
S3.2, aiming at the optimization model of the formula (4), obtaining an augmented Lagrange function:
Figure GDA0003337171810000031
in the formula (5), α represents a lagrangian multiplier, ρ represents a penalty factor, | · | | luminance2Represents a vector l2Norm, | · | luminance1Representing a vector or matrix of l1And (4) norm.
S3.3, converting the optimization problem of the formula (4) into the following subproblems by using the formula (5) to carry out iterative solution:
Figure GDA0003337171810000032
k represents the number of iterations; by substituting formula (5) for formula (6), x can be obtained(k)And z(k)And finally obtaining the complete iteration steps as follows:
Figure GDA0003337171810000033
wherein the content of the first and second substances,
Figure GDA0003337171810000034
representing soft threshold operators, with any complex scalar x and real threshold t
Figure GDA0003337171810000035
For any complex vector x and real threshold t, there are
Figure GDA0003337171810000036
Wherein xiRepresents the ith element of the complex phasor x;
s4, establishing a CV-ADMMN network structure model:
each iteration of equation (7) includes x(k)、z(k)、α(k)Three calculation steps, which correspond to three different network layers: x is the number of(k)Referred to as the k-th reconstruction layer, z(k)Called the kth noise reduction layer, α(k)Is called asThe kth Lagrangian multiplier updates the layer. X is to be(k)、z(k)、α(k)And connecting the structures in sequence to obtain a kth-level structure, and repeatedly cascading the structures to obtain the CV-ADMMN model.
In order to reduce the operation amount in practical application, the vector expression in the expression (7) is rearranged into a matrix form, and the following CV-ADMMN forward propagation expression can be obtained:
Figure GDA0003337171810000037
wherein the content of the first and second substances,
Figure GDA0003337171810000038
respectively representing a penalty factor of a kth denoising layer, a regularization parameter of the kth denoising layer, a penalty factor of a kth Lagrange multiplier updating layer and a penalty factor of a kth reconstruction layer. The above parameters are all independently adjustable parameters. Mask represents an M × N sparse sampling Mask matrix, elements of the M × N sparse sampling Mask matrix are composed of 0 and 1, positions of original signals which are sampled and reserved are 1, and otherwise, the positions are 0. 1M×NRepresenting an all 1 matrix of size M × N. Z(k)And A(k)Respectively represent the intermediate variable z(k)And lagrange multiplier alpha(k)The matrix obtained after the rearrangement is performed,
Figure GDA0003337171810000041
a fourier transform matrix is represented which is,
Figure GDA0003337171810000042
representing a down-sampled one-dimensional range profile matrix, each row of the matrix representing a set of one-dimensional range profiles. And Y is network input.
S5, solving the sparse aperture ISAR imaging problem by using CV-ADMMN:
s5.1 training CV-ADMMN:
s5.1.1, constructing a data set similar to an actual application scene. The data set comprises a plurality of groups of distance image-label data pairs
Figure GDA0003337171810000043
Wherein
Figure GDA0003337171810000044
Representing the q-th set of sparse aperture one-dimensional range profile matrices,
Figure GDA0003337171810000045
representing the q-th group of image tags. And sequentially inputting the data in the data set into the CV-ADMMN model generated by S4, and training the CV-ADMMN.
S5.1.2 define two loss functions as follows:
Figure GDA0003337171810000046
Figure GDA0003337171810000047
wherein the content of the first and second substances,
Figure GDA0003337171810000048
represents the input YqThe reconstructed image obtained by time network output, xi represents a penalty coefficient, | | · | | non calculationFAnd representing the F norm of the matrix, Q representing the total number of data contained in the data set, and abs (-) representing the matrix or vector obtained by modulo the matrix or vector element by element. L is1The loss function represents the Root Mean Square Error (RMSE), L, of the label image and the reconstructed image2Loss function representation of RMSE superposition l of label image and reconstructed image1Norm regularization term. Loss function L2Better results are often obtained under low signal-to-noise ratio conditions, and the loss function L1Is often suitable for high signal-to-noise ratio conditions.
S5.1.3 use Complex field Back Propagation (BP) and gradient descent algorithm (G.M. Georgiou and C.Koutsouugeras, "Complex domain Back propagation," in IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, vol.39, No.5, pp.330-334, May 1992) to update network parameters. The complex derivative employed follows the definition:
Figure GDA0003337171810000049
where O represents a real number and O represents a complex matrix or vector. Re {. and Im {. can represent the real and imaginary parts of the complex phasor, respectively.
The back propagation process requires solving the partial derivatives of the loss function for each network layer and its parameters. For convenience of representation, the formula derivation in the part adopts a vector form which is not rearranged into a matrix, and in practical application, CV-ADMMN is still realized through the matrix form.
S5.1.3.1 define the vector form of the network output
Figure GDA00033371718100000410
Is composed of
Figure GDA00033371718100000411
Computing pairs of loss functions along the vectors obtained by the rearrangement of the columns
Figure GDA00033371718100000412
Derivative of (a):
Figure GDA0003337171810000051
wherein the content of the first and second substances,
Figure GDA0003337171810000056
representing label image vectors, by a matrix of label images
Figure GDA0003337171810000053
Resulting in a rearrangement along the columns, the symbol |, indicates a matrix or vector element-by-element multiplication,
Figure GDA0003337171810000054
representing a matrix or vector divided element by element;
S5.1.3.2CV-ADMMN the partial derivatives of the loss function for each layer can be represented by the backward corresponding network layer derivatives. After the partial derivative of the output layer is obtained, the partial derivative of each layer can be further solved through a chain rule, and the specific expression of the chain rule is as follows:
Figure GDA0003337171810000055
wherein L denotes a loss function L1Or L2。
S5.1.3.3 obtaining the loss function for the (k +1) th reconstruction layer x using equation (12)(k+1)(k +1) th noise reduction layer z(k+1)(k +1) th Lagrange multiplier update layer alpha(k+1)After partial derivative, the gradient of the parameter to be solved in each layer can be further calculated, and the specific expression is expressed in a matrix form as follows:
Figure GDA0003337171810000061
where sum (-) denotes summing all elements of the matrix.
S5.1.3.4 the parameters are updated by gradient descent method during training. For the network parameters in the kth level structure body, the updating expression is as follows
Figure GDA0003337171810000062
Wherein, the first and the second end of the pipe are connected with each other,
Figure GDA0003337171810000063
respectively represent the parameters after the current structural body is updated, and eta represents the learning rate of parameter updating.
S5.1.3.5 when the parameter is updated to a gradient of approximately 0, the training is stopped, and a CV-ADMMN model with fixed parameters is obtained.
S5.2 embedding a self-focusing module based on minimum entropy:
s5.2.1, constructing a sparse aperture observation scene containing a phase error:
y＝edfx+n (15)
wherein the content of the first and second substances,
Figure GDA0003337171810000064
E＝diag[exp(jφ1),exp(jφ2),...,exp(jφL)]representing the phase error in the one-dimensional range profile, philIndicating the phase error in the l-th one-dimensional distance direction.
S5.2.2 in the above model, E is unknown phase error matrix, and in order to realize the self-focusing function, the invention estimates E by the minimum entropy method. For any reconstructed layer output X in equation (8)(k)The estimation result of the phase error matrix E is given by:
Figure GDA0003337171810000065
wherein phi is [ phi ]1,φ2,...,φL]，e(X(k)(phi)) represents matrix X(k)Entropy of (φ), the expression is as follows:
Figure GDA0003337171810000071
wherein the content of the first and second substances,
Figure GDA0003337171810000072
representative matrix X(k)The elements of row i and column j,
Figure GDA0003337171810000073
representing the total energy of the matrix. And the phase error value that minimizes the entropy is obtained by solving the following equation:
Figure GDA0003337171810000074
where L1, 2.., L, the value of vector phi may ultimately be solved for.
X in the formula (8)(k)The analytical expression is substituted into formula (18) to obtain philThe analytical expression of (c):
Figure GDA0003337171810000075
wherein Y is.lRepresenting the l-th column of the matrix Y. 0 represents a matrix of all 0 elements. The self-focusing function can be realized by using the phase error obtained by estimation.
S5.2.3 embedding the self-focusing module into the CV-ADMMN structure by using formula (19) analysis expression, obtaining the CV-ADMMN forward propagation expression with the self-focusing function:
Figure GDA0003337171810000076
by using the formula (20), a CV-ADMMN model with a self-focusing function can be constructed. Unknown parameters in the formula (20)
Figure GDA0003337171810000077
Figure GDA0003337171810000078
Obtained by the training step of S5.1. Compared with the model of the formula (8), the model constructed by the formula (20) can adaptively compensate the initial phase in the radar signal, and has wider application scenarios.
S5.3 sparse aperture imaging and self-focusing by using CV-ADMMN embedded into self-focusing module
S5.3.1, acquiring actual observation sparse aperture echo, and obtaining a sparse aperture one-dimensional range profile sequence through fast time FFT. And carrying out translation coarse compensation on the one-dimensional range profile sequence by using a cross-correlation method. (have luck, Chen Meng, Wang Tong. Radar imaging technology [ M ]. Beijing: electronic industry Press, 2005)
S5.3.2, inputting the roughly compensated one-dimensional range image sequence Y into CV-ADMMN, and carrying out forward propagation through the network to obtain a high-quality ISAR image X.
The invention has the following beneficial effects: the invention can realize the sparse aperture ISAR imaging and self-focusing of the moving target, can quickly reconstruct the complete radar image under the sparse aperture condition and realize the phase error compensation. The algorithm performance has weak dependence on parameter selection, and further better reconstruction performance is obtained. The method has important engineering application value for sparse aperture ISAR imaging and self-focusing under the condition of data loss.
Drawings
FIG. 1 is a flow chart of an embodiment of the present invention;
FIG. 2 is a schematic diagram of CV-ADMMN construction;
FIG. 3 is a diagram of a CV-ADMMN architecture embedded in a self-focusing module;
FIG. 4 full pore size conditions: (a) a target one-dimensional range profile sequence; (b) a target ISAR image;
fig. 5 shows the sparsity of 25% under the sparse aperture condition and in the presence of phase error: (a) a target one-dimensional range profile sequence; (b) obtaining a target ISAR image by a range-Doppler method; (c) the invention utilizes L1Training the obtained ISAR image by using a loss function; (d) the invention utilizes L2Training the obtained ISAR image by using a loss function;
fig. 6 considers raw data containing phase errors and randomly extracts 64 pulses from them to simulate sparse aperture data with 25% sparsity: (a) one-dimensional distance direction of the target; (b) imaging results of the traditional RD method; (c) from L1A CV-ADMMN imaging result of the embedded self-focusing module trained by the loss function; (d) from L2And (4) losing the CV-ADMMN imaging result of the function training embedded self-focusing module.
Detailed Description
The invention is further illustrated with reference to the accompanying drawings:
FIG. 1 is a flow chart of the present invention.
Fig. 2 and 3 show the CV-ADMMN structure and the CV-ADMMN structure embedded in the self-focusing module, respectively. The invention provides a complex domain ADMM-Net based target SA-ISAR imaging and self-focusing method, which comprises the following steps:
s1, modeling the moving target one-dimensional range profile sequence;
s2 modeling the moving target sparse aperture ISAR imaging scene;
s3, establishing an ADMM reconstruction model of the moving target sparse aperture ISAR imaging problem;
s4, establishing a CV-ADMMN network structure model;
s5, solving the sparse aperture ISAR imaging problem by using a CV-ADMMN network structure;
fig. 4(a) and 4(b) show a target one-dimensional range profile sequence and an ISAR image under the full aperture condition of a target actually measured by a radar. The radar emission 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 data contains 256 pulses, each pulse containing 256 sample points.
64 pulses were randomly extracted from the full aperture data without phase error to simulate sparse aperture data with 25% sparsity. At this time, the target one-dimensional distance direction is as shown in fig. 5 (a). The sparse aperture data is further imaged using the conventional Range-Doppler (RD) method and the present invention. The obtained ISAR images are shown in fig. 5(b), (c), and (d), respectively. Wherein, the invention adopts two different loss functions to train, and obtains two different network structures, and the graphs (c) and (d) are respectively corresponding to L1、L2And (5) obtaining an imaging result after the loss function training. As can be seen from fig. 5(b), due to the sparse aperture effect, the correlation between pulses is seriously destroyed, and it is difficult for the RD algorithm to obtain an image with good focusing effect. As can be seen from fig. 5(c) and (d), the ISAR image obtained by the present invention has a good focusing effect.
Further, consider raw data containing phase errors and randomly draw 64 pulses from it to simulate sparse aperture data with 25% sparsity. At this time, the target one-dimensional distance direction is as shown in fig. 6 (a). The conventional no-self-focusing RD method is used for comparison with the present invention. The imaging result of the conventional RD method is shown in fig. 6 (b). At this time, not only the correlation between pulses is destroyed, but also a phase error is superimposed on the echo data, and the conventional RD method cannot adaptively compensate the phase error, so that imaging cannot be completed. FIGS. 6(c) and (d) show the general formula L1、L2And (4) losing the CV-ADMMN imaging result of the function training embedded self-focusing module. As shown in FIGS. 6(c) and (d), the present inventionThe method overcomes the defect of correlation between echo pulses, and can correctly compensate phase errors.
In conclusion, the invention can effectively realize the functions of imaging and self-focusing of the moving target under the condition of sparse aperture, has good effect on the sparse aperture data with the sparsity of 25 percent, and has higher engineering application value.

Claims (1)

1. A SA-ISAR imaging and self-focusing method based on CV-ADMMN is characterized by comprising the following steps:
s1, modeling the moving target one-dimensional range profile sequence:
the radar transmits a chirp signal, and the two-dimensional echo received for a moving target can be modeled as:
Figure FDA0003337171800000011
wherein the content of the first and second substances,
Figure FDA0003337171800000012
t represents fast time and full time, respectively, and
Figure FDA0003337171800000013
tmrepresents a slow time; sigmaiAnd RiRespectively representing the reflection coefficient of the ith scattering center and the instantaneous rotation distance, f, relative to the radarcC and gamma respectively represent the center frequency of a radar signal, the vacuum light speed and the signal frequency modulation; due to the short integration time of ISAR imaging, the movement of the target in one pulse time can be ignored when the echo is modeled;
the signal expression obtained after the two-dimensional signal shown in the formula (1) is subjected to line-splitting frequency modulation is as follows:
Figure FDA0003337171800000014
under sparse aperture conditionsSince the fast echo pulse waveform remains unchanged, the signal shown in the formula (2) is applied in a fast time
Figure FDA00033371718000000119
Performing FFT to obtain a target one-dimensional range profile sequence;
s2, modeling the moving target sparse aperture ISAR imaging scene:
under the condition of sparse aperture, the observation of the radar system to the moving target can be represented by the following down-sampling model:
y＝Φx+n＝dfx+n (3)
wherein the content of the first and second substances,
Figure FDA0003337171800000015
representing a radar image column vector formed by an image matrix
Figure FDA0003337171800000016
The method is obtained by rearranging along the columns,
Figure FDA0003337171800000017
representing the MN-dimensional complex column vector,
Figure FDA0003337171800000018
representing an M multiplied by N dimensional complex matrix, wherein M represents the number of azimuth units of the radar image, and N represents the number of range units;
Figure FDA0003337171800000019
representing a received radar one-dimensional range profile vector formed from a one-dimensional range profile matrix
Figure FDA00033371718000000110
Obtained by rearrangement along the column, L represents the number of the one-dimensional range profile after down sampling, L<<M；
Figure FDA00033371718000000111
Represents a down-sampling matrix that is down-sampled,
Figure FDA00033371718000000112
representing a gaussian white noise vector rearranged along a column;
Figure FDA00033371718000000113
represents a block Fourier transform matrix, which can be expressed as
Figure FDA00033371718000000114
Wherein INRepresenting an N × N dimensional identity matrix, and F representing an M × M dimensional Fourier transform matrix;
Figure FDA00033371718000000115
represents a block down-sampling matrix, which can be expressed as
Figure FDA00033371718000000116
Wherein D represents an L × M dimensional down-sampling matrix, the elements of which are composed of 0 and 1; let V represent the distance image index being sampled, then
Figure FDA00033371718000000117
For the i row and m column elements D in the matrix Dl,mWhen the l-th element V of the vector V islWhen m, there is Dl,m＝1，l＝1,2,…,L，m＝1,2,…,M；
The down-sampling model given by the formula (3) models the sparse aperture imaging into the solution of the linear underdetermined inverse problem, and the solution can be carried out by using a compressed sensing method;
s3, establishing an ADMM reconstruction model of the moving target sparse aperture ISAR imaging problem:
for the down-sampling model given by equation (3), it is solved using conventional ADMM:
s3.1, constructing an optimization model as follows:
Figure FDA00033371718000000118
wherein z is an introduced intermediate variable, and λ represents a regularization parameter;
s3.2, aiming at the optimization model of the formula (4), obtaining an augmented Lagrange function:
Figure FDA0003337171800000021
in the formula (5), α represents a lagrangian multiplier, ρ represents a penalty factor, | · | | luminance2Represents a vector l2Norm, | · | luminance1Representing a vector or matrix of l1A norm;
s3.3, converting the optimization problem of the formula (4) into the following subproblems by using the formula (5) to carry out iterative solution:
Figure FDA0003337171800000022
k represents the number of iterations; by substituting formula (5) for formula (6), x can be obtained(k)And z(k)And finally obtaining the complete iteration steps as follows:
Figure FDA0003337171800000023
wherein the content of the first and second substances,
Figure FDA0003337171800000024
representing soft threshold operators, with any complex scalar x and real threshold t
Figure FDA0003337171800000025
For any complex vector x and real threshold t, there are
Figure FDA0003337171800000026
Wherein xiRepresents the ith element of the complex phasor x;
s4, establishing a CV-ADMMN network structure model:
each iteration of equation (7) includes x(k)、z(k)、α(k)Three calculation steps, which correspond to three different network layers: x is the number of(k)Referred to as the k-th reconstruction layer, z(k)Referred to as the kth noise reduction layer, α(k)Referred to as the kth lagrangian multiplier update layer; x is to be(k)、z(k)、α(k)Sequentially connecting to obtain a kth-level structural body, and repeatedly cascading the structural bodies to obtain a CV-ADMMN model;
in order to reduce the operation amount in practical application, the vector expression in the expression (7) is rearranged into a matrix form, and the following CV-ADMMN forward propagation expression can be obtained:
Figure FDA0003337171800000027
wherein the content of the first and second substances,
Figure FDA0003337171800000028
respectively representing a penalty factor of a kth denoising layer, a regularization parameter of the kth denoising layer, a penalty factor of a kth Lagrange multiplier updating layer and a penalty factor of a kth reconstruction layer, wherein the parameters are independently adjustable parameters; mask represents an M multiplied by N sparse sampling Mask matrix, elements of the M multiplied by N sparse sampling Mask matrix are composed of 0 and 1, the position of the original signal which is sampled and reserved is 1, and otherwise, the position is 0; 1M×NRepresents a full 1 matrix of size mxn; z(k)And A(k)Respectively represent the intermediate variable z(k)And lagrange multiplier alpha(k)The matrix obtained after the rearrangement is performed,
Figure FDA0003337171800000031
a fourier transform matrix is represented which is,
Figure FDA0003337171800000032
representing a one-dimensional range profile matrix after down sampling, wherein each row of the matrix represents a group of one-dimensional range profiles, and Y is network input;
s5, solving the sparse aperture ISAR imaging problem by using CV-ADMMN:
s5.1 training CV-ADMMN:
s5.1.1, constructing a data set similar to an actual application scene: the data set comprises a plurality of groups of distance image-label data pairs
Figure FDA0003337171800000033
Wherein
Figure FDA0003337171800000034
Representing a q-th set of sparse aperture one-dimensional range-image matrices,
Figure FDA0003337171800000035
representing the q-th group of image tags; sequentially inputting the data in the data set into a CV-ADMMN model generated by S4, and training the CV-ADMMN;
s5.1.2 define two loss functions as follows:
Figure FDA0003337171800000036
Figure FDA0003337171800000037
wherein the content of the first and second substances,
Figure FDA0003337171800000038
represents the input YqThe reconstructed image obtained by time network output, xi represents a penalty coefficient, | | · | | non calculationFRepresenting the F norm of the matrix, Q representing the total number of data contained in the data set, abs (-) representing the matrix or vector obtained by modulo the matrix or vector element by element; l is1The loss function represents the root mean square error, L, of the label image and the reconstructed image2Loss function representation of RMSE superposition l of label image and reconstructed image1A norm regularization term; loss function L2Better results are often obtained under low signal-to-noise ratio conditions, and the loss function L1Is often suitable for high signal-to-noise ratio conditions;
s5.1.3 updating network parameters using complex domain back propagation and gradient descent algorithms, the complex derivatives used follow the following definitions:
Figure FDA0003337171800000039
wherein O represents a real number, O represents a complex matrix or vector, and Re {. cndot.and Im {. cndot.represent the real and imaginary parts of the complex vector, respectively;
in the back propagation process, the partial derivative of the loss function to each network layer and the parameters thereof needs to be solved; for convenience of representation, the derivation of the formulas in this part adopts a vector form which is not rearranged into a matrix, and in practical application, CV-ADMMN is still realized through the matrix form:
s5.1.3.1 define the vector form of the network output
Figure FDA00033371718000000310
Is composed of
Figure FDA00033371718000000311
Computing pairs of loss functions along the vectors obtained by the rearrangement of the columns
Figure FDA00033371718000000312
Derivative of (a):
Figure FDA0003337171800000041
wherein the content of the first and second substances,
Figure FDA0003337171800000042
representing a label image vector, the symbol |, representing a matrix or vector element-by-element multiplication,
Figure FDA0003337171800000043
representing matrix or vector elementsPerforming element division;
S5.1.3.2CV-ADMMN, the partial derivative of each layer of the loss function can be expressed by backward corresponding network layer derivatives, after the partial derivative of the output layer is obtained, the partial derivative of each layer can be further solved by a chain rule, and the specific chain rule expression is as follows:
Figure FDA0003337171800000044
wherein L denotes a loss function L1Or L2；
S5.1.3.3 obtaining the loss function for the (k +1) th reconstruction layer x using equation (12)(k+1)(k +1) th noise reduction layer z(k+1)(k +1) th Lagrange multiplier update layer alpha(k+1)After partial derivative, the gradient of the parameter to be solved in each layer can be further calculated, and the specific expression is expressed in a matrix form as follows:
Figure FDA0003337171800000051
where sum (-) denotes summing all elements of the matrix;
s5.1.3.4, updating the parameters by using a gradient descent method in the training process; for the network parameters in the kth level structure body, the updating expression is as follows
Figure FDA0003337171800000052
Wherein the content of the first and second substances,
Figure FDA0003337171800000053
respectively representing the updated parameters of the current structural body, wherein eta represents the learning rate of parameter updating;
s5.1.3.5 when the parameter is updated to the gradient of 0, stopping training to obtain a CV-ADMMN model with fixed parameters;
s5.2 embedding a self-focusing module based on minimum entropy:
s5.2.1, constructing a sparse aperture observation scene containing a phase error:
y＝edfx+n (15)
wherein the content of the first and second substances,
Figure FDA0003337171800000054
E＝diag[exp(jφ1),exp(jφ2),...,exp(jφL)]representing the phase error in the one-dimensional range profile, philRepresenting the phase error in the l one-dimensional distance direction;
s5.2.2 in the above model, E is unknown phase error matrix, in order to realize self-focusing function, the invention estimates E by minimum entropy method; for any reconstructed layer output X in equation (8)(k)The estimation result of the phase error matrix E is given by:
Figure FDA0003337171800000055
wherein phi is [ phi ]1,φ2,...,φL]，e(X(k)(phi)) represents matrix X(k)Entropy of (φ), the expression is as follows:
Figure FDA0003337171800000061
wherein the content of the first and second substances,
Figure FDA0003337171800000062
representative matrix X(k)The elements of row i and column j,
Figure FDA0003337171800000063
representing the total energy of the matrix; and the phase error value that minimizes the entropy is obtained by solving the following equation:
Figure FDA0003337171800000064
where L1, 2.., L, the value of the vector Φ may eventually be solved for;
x in the formula (8)(k)The analytical expression is substituted into formula (18) to obtain philThe analytical expression of (1):
Figure FDA0003337171800000065
wherein, Y.lThe l column of the matrix Y is represented, 0 represents a matrix with all 0 elements, and the self-focusing function can be realized by utilizing the phase error obtained by estimation;
s5.2.3 embedding the self-focusing module into the CV-ADMMN structure by using formula (19) analysis expression, obtaining the CV-ADMMN forward propagation expression with the self-focusing function:
Figure FDA0003337171800000066
by using the formula (20), a CV-ADMMN model with a self-focusing function can be constructed; unknown parameters in the formula (20)
Figure FDA0003337171800000067
Figure FDA0003337171800000068
Obtained through the training step of S5.1;
s5.3, performing sparse aperture imaging and self-focusing by using the CV-ADMMN embedded in the self-focusing module:
s5.3.1, acquiring actual observation sparse aperture echo, obtaining a sparse aperture one-dimensional range profile sequence through fast time FFT, and performing translational coarse compensation on the one-dimensional range profile sequence by using a cross-correlation method;
s5.3.2, inputting the roughly compensated one-dimensional range image sequence Y into CV-ADMMN, and carrying out forward propagation through the network to obtain a high-quality ISAR image X.
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