CN111610522B - SA-ISAR imaging method for target with micro-motion component based on low-rank and sparse combined constraint - Google Patents

SA-ISAR imaging method for target with micro-motion component based on low-rank and sparse combined constraint Download PDF

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CN111610522B
CN111610522B CN202010501764.4A CN202010501764A CN111610522B CN 111610522 B CN111610522 B CN 111610522B CN 202010501764 A CN202010501764 A CN 202010501764A CN 111610522 B CN111610522 B CN 111610522B
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CN111610522A (en
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张双辉
刘永祥
黎湘
霍凯
姜卫东
田彪
张新禹
高勋章
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National University of Defense Technology
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Abstract

The invention belongs to the field of radar imaging, and relates to an SA-ISAR (synthetic aperture radar-inertial navigation radar) imaging method with a micro-motion component target based on low-rank and sparse joint constraint, which comprises the following steps of: s1, modeling a target one-dimensional range profile sequence with the micro-motion component after the target translation compensation; s2 modeling the sparse aperture ISAR imaging problem of the target with the micromotion component; s3 solves the ISAR imaging problem of the target sparse aperture of the micro-motion component by adopting linear ADMM. Has the advantages that: the ISAR imaging method can realize target sparse aperture ISAR imaging with the micro-motion component, can effectively separate one-dimensional range profile sequences of a target main body part and the micro-motion component under the sparse aperture condition, eliminates the m-D effect caused by the target micro-motion component, eliminates side lobe and grating lobe interference caused by the sparse aperture, further obtains ISAR images of the target main body part with good focusing effect, and has important engineering application value for radar imaging, micro-motion parameter estimation, feature extraction and target identification of the target with the micro-motion component under the data limitation condition.

Description

SA-ISAR imaging method for target with micro-motion component based on low-rank and sparse combined constraint
Technical Field
The invention belongs to the field of radar imaging, and particularly relates to a Sparse aperture inverse synthetic aperture radar (SA-ISAR) imaging method for a target with a micro-motion component based on low-rank and Sparse combined constraint.
Background
The Inverse Synthetic Aperture Radar (ISAR) imaging technology can acquire high-resolution radar images of moving targets, has the advantage of all weather in all days compared with an optical imaging means, and is widely applied to military and civil fields of space target monitoring, missile defense, shore-based warning, radar astronomy and the like.
The target with the micro-motion component refers to components which do rigid motion except for a target main body, and comprises other micro-motion components, such as helicopter rotor rotation, airplane propeller rotation, ship antenna rotation, wheel rotation and the like. When ISAR imaging is performed on such an object, the micro-motion components thereof will generate micro-Doppler (m-D) effect, resulting in defocusing of the ISAR image. At this time, generally, signal separation is performed on the radar echo, a target main body and a micro-motion component radar echo are separated from the radar echo, and then ISAR imaging processing is performed on the target main body radar echo to eliminate the influence of the m-D effect and realize high-resolution ISAR imaging of a target main body part.
Generally, incomplete radar echoes are called sparse aperture echoes, and the sparse aperture echoes are caused by 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. Under the condition of sparse aperture, the traditional method for separating signals of the main body and the micro-motion component is invalid, and the m-D effect cannot be effectively inhibited, so that the target ISAR image with the micro-motion component is defocused. In addition, non-uniform sampling of sparse aperture radar echoes also introduces a large amount of side lobe and grating lobe interference, further causing defocusing of the ISAR image. The ISAR imaging effect of the target with the micro-motion component under the sparse aperture condition is improved, and the method has important engineering application value.
Disclosure of Invention
The invention aims to solve the technical problem that under the condition of sparse aperture, ISAR imaging of a target with a micro-motion component is interfered by m-D effect, side lobe and grating lobe, so that ISAR images are defocused and the engineering application requirements are difficult to meet.
The invention provides an SA-ISAR imaging method for a target with a micro-motion component based on low-rank and sparse combined constraint, aiming at the problem of defocusing of the target ISAR image with the micro-motion component under the sparse aperture condition. According to the method, SA-ISAR imaging modeling of the target with the micro-motion component becomes an underdetermined problem based on triple constraints, wherein the triple constraints specifically comprise: the low-rank characteristic of the target main body part one-dimensional range profile sequence, the sparse characteristic of the target micro-motion component one-dimensional range profile sequence and the sparse characteristic of the target main body ISAR image. The triple-constrained underdetermined problem is further solved by adopting a linear alternating direction multiplier (ADMM) method so as to improve the operation efficiency. According to the method, through loop iteration, a one-dimensional range profile sequence of a target main body and a micro-motion component and an ISAR image of the target main body can be finally obtained.
The technical scheme adopted by the invention for solving the technical problems is as follows: a low-rank and sparse combined constraint-based SA-ISAR imaging method for targets with micro-motion components comprises the following steps:
s1, modeling a target one-dimensional range profile sequence after the translational compensation of the target with the micro-motion component:
the translation compensation is the first link of ISAR imaging, and the technical route is relatively mature after decades of development, so that the invention directly processes the one-dimensional distance image sequence after the translation compensation (shining protection, chenchenge, wangtong, radar imaging technology [ M ]. Beijing: electronics industry Press, 2005) on the assumption that the translation compensation is finished. For the target with the micro-motion component, the translation-compensated one-dimensional range profile sequence of the target can be modeled as follows:
Figure BDA0002524990490000021
wherein the content of the first and second substances,
Figure BDA0002524990490000022
representing a translation compensated one-dimensional range profile sequence of the target,
Figure BDA0002524990490000023
tmrespectively representing fast time and slow time, M is 1 and 2, M represents the number of pulses contained in the full aperture radar echo, fcB, c denote the center frequency, bandwidth and propagation speed of the radar signal, σpAnd Rp(tm) Respectively representing the reflection coefficient of the p-th scattering center of the target body part and the instantaneous rotation distance, sigma, of the relative radarqAnd Rq(tm) Respectively representing the reflection coefficient of the qth scattering center of the target micro-motion component and the instantaneous rotation distance relative to the radar, wherein P is 1,2, …, P, Q is 1,2, …, and Q, P represents that the target main body part contains P scattering centers, and Q represents that the target micro-motion component contains Q scattering centers; p for target body partA scattering center having an instantaneous rotation distance R with respect to the radarp(tm) Can be expressed as:
Rp(tm)＝xpsin(ωtm)+ypcos(ωtm)≈xpωtm+yp (2)
wherein (x)p,yp) Representing the coordinate of the p-th scattering center of the target main body part in a target specimen body coordinate system, and omega represents the rotating angular speed of the target main body part; because the ISAR imaging accumulation time is short, the rotation angle omega of the target relative to the radar in the imaging accumulation time is small, and therefore: sin (ω t)m)≈ωtm、cos(ωtm) 1 is approximately distributed; suppose that the scattering center of the target micro-motion component is around the point O' (x)O',yO') Rotating, and then for the qth scattering center of the target micro-motion component, the instantaneous rotating distance R of the target micro-motion component relative to the radarq(tm) Can be expressed as:
Figure BDA0002524990490000024
wherein (x)O',yO') Represents the coordinate of the qth scattering center of the target micro-motion component in the target specimen coordinate system, rqω', and θqRespectively representing the micromotion amplitude, the rotation angular speed and the initial phase of the qth scattering center of the target micromotion component. Respectively substituting the formula (2) and the formula (3) into the formula (1), and further carrying out slow time t on the formula (1)mPerforming Fourier transform to obtain a target ISAR image; comparing the formula (2) and the formula (3), the instantaneous rotation distance R of the qth scattering center of the target micro-motion component relative to the radar can be knownq(tm) Containing the cosine term rqcos(ω'tm+θq) During ISAR imaging, this term will produce an m-D effect, causing the ISAR image to defocus;
under sparse aperture conditions, the target one-dimensional range profile sequence shown in equation (1) can be expressed in the form of the following matrix:
H＝L+S (4)
wherein
Figure BDA0002524990490000025
And
Figure BDA0002524990490000026
respectively representing a one-dimensional range image sequence of the target, the target main body part and the target micro-motion component,
Figure BDA0002524990490000027
expressing a K multiplied by N complex matrix, wherein K and N respectively express the number of pulses and the number of distance units of a sparse aperture one-dimensional range profile sequence; for sparse aperture data, the number of pulses is less than that of pulses contained in full aperture radar echo, namely K < M, and the pulse sequence number set is a subset of full aperture pulse sequence numbers, namely
Figure BDA0002524990490000028
Wherein i represents a sparse aperture pulse sequence number set;
for a target with a micro-motion component, acquiring a clear ISAR image of a main body part of the target is a main target of ISAR imaging; under the ideal condition, the ISAR image of the target main body part and the one-dimensional range profile sequence L of the target main body part are mutually fourier transform pairs, that is:
L＝PX (5)
wherein
Figure BDA0002524990490000031
An ISAR image representing a subject body portion of the subject,
Figure BDA0002524990490000032
representing a partial Fourier matrix, assuming a complete Fourier matrix of
Figure BDA0002524990490000033
P is formed by extracting part of row vectors in X and combining, specifically, the sequence number set of the extracted row vectors is a sparse aperture pulse sequence number set i; since K is less than M, formula (5) is an underdetermined problem, and infinite groups of solutions exist. Thus, sparse aperture ISAR imaging with micro-motion component targets faces two major problems, first requiringSeparating a one-dimensional range profile sequence L of a target main body part from the target one-dimensional range profile sequence H, and then reconstructing an ISAR image X of the target main body part from the one-dimensional range profile sequence L of the target main body part, namely solving a signal separation problem shown in a formula (4) and an underdetermined problem shown in a formula (5);
s2 models the sparse aperture ISAR imaging problem of the target with the micromotion part:
the solution of the signal separation problem shown in equation (4) and the underdetermined problem shown in equation (5) is not unique, and a constraint condition needs to be added to obtain a unique solution: firstly, the column correlation of a one-dimensional range profile sequence L of a target main body part is strong, and the low-rank characteristic is achieved; secondly, energy of the one-dimensional range profile sequence S of the target micro-motion component is distributed in different range units, and the target micro-motion component has a sparse characteristic; thirdly, the ISAR image of the target main body generally consists of a few scattering centers and has strong sparse characteristic; therefore, this step limits equations (4) and (5) with the above three constraints to achieve sparse aperture ISAR imaging with the fine motion component target, specifically, the problem can be modeled as:
Figure BDA0002524990490000034
wherein | · | purple*(ii) counting & lt | & gt | & lt | & gt1Respectively representing the kernel norm and l of the matrix1Norm, which is respectively used for representing the rank and the sparsity of the matrix; λ and μ represent regularization parameters, which are used for adjusting weights of matrix decomposition and ISAR imaging, respectively, and λ ═ μ ═ 0.2;
s3, solving the ISAR imaging problem of the target sparse aperture of the micro-motion component by adopting linear ADMM:
solving the triple-constrained underdetermined problem shown in the formula (6) by using linear ADMM firstly needs to deduce the augmented Lagrangian function of the formula (6), as shown in the following formula:
Figure BDA0002524990490000035
wherein<·,·>Represents twoInner product of matrix, Y1、Y2Representing the Lagrange multiplier matrix, p1、ρ2Represents a penalty factor, | · |. non-woven phosphorFAn F norm representing a matrix; ADMM converts equation (6) into the following sub-problem solution:
Figure BDA0002524990490000041
wherein (·)(k)Representing the variable obtained by the kth iteration, and eta representing a rising factor for controlling the penalty factor rho1、ρ2η ═ 2; the method comprises the following steps:
s3.1, updating the one-dimensional range profile sequence L of the target main body part:
substituting formula (7) for formula (8), and omitting
Figure BDA0002524990490000042
Of (1), L independent terms, we can obtain:
Figure BDA0002524990490000043
equation (9) shows the problem of minimizing the nuclear norm, which can be solved by the singular value shrink operator (W.Qiu, J.Zhou, Q.Fu, "journal Using Low-Rank and space principles for space Inverse Synthetic Aperture radio Imaging," IEEE trans. image Process, vol.29, pp.100-115,2020):
Figure BDA0002524990490000044
wherein
Figure BDA0002524990490000045
The singular value contraction factor is expressed, and specifically, for an arbitrary matrix a and an arbitrary scalar γ, there are:
Figure BDA0002524990490000046
wherein A ═ Udiag (sigma) VHExpressing the singular value decomposition of A, U, V is a unitary matrix, sigma expresses the singular value vector of A, and diag (·) expresses a diagonal matrix formed by the vectors;
Figure BDA0002524990490000047
representing soft-threshold operators, for arbitrary scalars x, γ, having
Figure BDA0002524990490000048
Wherein sgn (·) represents a sign operator; for an arbitrary vector x, there are
Figure BDA0002524990490000049
Wherein xnRepresents the nth element of the vector x;
s3.2, updating the one-dimensional range profile sequence S of the target micro-motion component:
substituting formula (7) for formula (8), and omitting
Figure BDA00025249904900000410
S independent terms in (1), we can obtain:
Figure BDA0002524990490000051
formula (12) is represented by1Norm problem, which can be solved by a soft threshold operator (w.qiu, j.zhou, q.fu, "joint Using Low-Rank and Sparse colors for Sparse Inverse Synthetic Aperture Radar Imaging," IEEE trans. image process, vol.29, pp.100-115,2020):
Figure BDA0002524990490000052
s3.3, updating the ISAR image X of the target main body part:
substituting formula (7) for formula (8), and omitting
Figure BDA00025249904900000510
Of (1), the terms independent of X, can be:
Figure BDA0002524990490000053
equation (14) is not a standard minimization l due to the presence of a partial Fourier matrix P multiplied by X1The norm problem cannot be solved directly by a soft threshold operator; therefore, further solving for X using linear ADMM for the quadratic term in equation (14)
Figure BDA0002524990490000054
Carrying out linearization; in particular, for this quadratic term, X ═ X(k)The second order taylor expansion is performed by:
Figure BDA0002524990490000055
wherein G is(k)To represent
Figure BDA0002524990490000056
When X is equal to X(k)The gradient of (a):
Figure BDA0002524990490000057
wherein, PHRepresents the conjugate transpose of the partial fourier matrix P;
formula (15) may be substituted for formula (14):
Figure BDA0002524990490000058
formula (17) is1The norm problem can be solved by a soft threshold operator:
Figure BDA0002524990490000059
s3.4 updating Lagrange multiplier matrix Y1、Y2：
From equation (8), the Lagrange multiplier matrix Y1、Y2The updating expressions of (A) are respectively as follows:
Y1 (k+1)＝Y1 (k)+ρ1 (k)(H-L(k+1)-S(k+1)) (19)
Y2 (k+1)＝Y2 (k)+ρ2 (k)(L(k+1)-PX(k+1)) (20)
s3.3 updating penalty factor ρ1、ρ2：
As shown in the formula (8), the penalty factor ρ1、ρ2The updating expressions of (A) are respectively as follows:
ρ1 (k+1)＝ηρ1 (k) (21)
ρ2 (k+1)＝ηρ2 (k) (22)
s3.4, combining the iteration formulas (10), (13), (18) and (19) - (22) until the relative error (| X) of the ISAR images of the target main body part obtained by two adjacent iterations(k+1)-X(k)|/|X(k)Is less than a set threshold (e.g. 10)-4) And obtaining the ISAR image X of the target main body part with the micro-motion component under the sparse aperture condition.
The invention has the following beneficial effects: the invention can realize the ISAR imaging of the target with the micro-motion component, can effectively separate the one-dimensional range profile sequence of the main part of the target and the micro-motion component under the condition of the sparse aperture, eliminate the m-D effect caused by the micro-motion component of the target, and eliminate the interference of side lobes and grating lobes caused by the sparse aperture, thereby obtaining the ISAR image of the main part of the target with good focusing effect, and has important engineering application values for radar imaging, micro-motion parameter estimation, feature extraction and target identification of the target with the micro-motion component under the condition of limited data.
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FIG. 1 is a flow chart of an embodiment of the present invention;
FIG. 2, a prop-rotor aircraft An-26;
FIG. 3 full pore size conditions: (a) a target one-dimensional range profile sequence; (b) a target ISAR image;
fig. 4 under sparse aperture conditions with 50% sparsity: (a) a target one-dimensional range profile sequence; (b) obtaining a target ISAR image by a range-Doppler method; (c) obtaining an ISAR image;
fig. 5 sparse aperture condition with 25% sparsity: (a) a target one-dimensional range profile sequence; (b) obtaining a target ISAR image by a range-Doppler method; (c) obtaining an ISAR image;
fig. 6 shows the sparsity of 12.5% under the sparse aperture condition: (a) a target one-dimensional range profile sequence; (b) obtaining a target ISAR image by a range-Doppler method; (c) obtaining an ISAR image;
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 a low-rank and sparse combined constraint-based SA-ISAR imaging method for a target with a micro-motion component, which comprises the following steps of:
s1, modeling a target one-dimensional range profile sequence with the micro-motion component after the target translation compensation;
s2 modeling the sparse aperture ISAR imaging problem of the target with the micromotion component;
s3 solves the ISAR imaging problem of the target sparse aperture of the micro-motion component by adopting linear ADMM.
Fig. 2 shows a radar measured target: an-26, the airplane is a double-propeller airplane, and during the flying process, the propeller rotates around the bracket at a high speed and is a micro-motion component, and in addition, the propeller is An airplane body part. The radar emission signal parameters are as follows: the center frequency was 5.52GHz, the bandwidth was 400MHz, the pulse width was 25.6 mus, and the full aperture data contained 256 pulses, each containing 256 sampling points.
Fig. 3(a) and 3(b) are a target one-dimensional range image sequence and an ISAR image under the full aperture condition, respectively. As can be seen, the rotation of the airplane propeller produces a strong m-D effect, causing the two range bin ISAR images in which it is located to be out of focus.
128 pulses were randomly extracted from the full aperture data to simulate sparse aperture data with 50% sparsity, at which time the target one-dimensional range profile sequence is as shown in fig. 4 (a). The sparse aperture data is further subjected to ISAR imaging using a conventional range-Doppler (RD) method and the present invention, and the obtained ISAR images are respectively shown in FIG. 4(b) and FIG. 4 (c). As can be seen from FIG. 4(b), under the influence of the m-D effect generated by the target micro-motion component and the side lobe and grating lobe generated by the sparse aperture, the ISAR image obtained by the RD method is severely defocused. As can be seen from FIG. 4(c), the ISAR image obtained by the method has a good focusing effect, which shows that the ISAR image effectively inhibits the m-D effect introduced by the target micro-motion component and the side lobe and grating lobe interference introduced by the sparse aperture.
Further, 64 pulses were randomly extracted from the full aperture data to simulate sparse aperture data with a sparsity of 25%. Under these conditions, the target one-dimensional range image sequence, the ISAR image obtained by the RD method, and the ISAR image obtained by the present invention are shown in fig. 5(a), fig. 5(b), and fig. 5(c), respectively. Comparing fig. 5(b) and fig. 4(b), it can be seen that the ISAR image obtained by the RD method is more defocused as the sparsity of the sparse aperture data decreases. As can be seen from fig. 5(c), the method of the present invention can still obtain an ISAR image with a good focusing effect, which shows the effectiveness of the method in imaging the target sparse aperture ISAR of the micromotion component.
And finally, randomly extracting 32 pulses from the full aperture data, and simulating sparse aperture data with the sparsity of 12.5%. In this case, the target one-dimensional range image sequence, the ISAR image obtained by the RD method, and the ISAR image obtained by the present invention are shown in fig. 6(a), 6(b), and 6(c), respectively. Due to the low sparsity, the ISAR image obtained by the RD method is almost completely defocused, and the target main body part cannot be distinguished. The invention can still effectively obtain the ISAR image of the target main body part with good focusing effect, and further verifies the effectiveness of the ISAR image of the target with the micro-motion component under the condition of sparse aperture.
In conclusion, the invention can effectively eliminate the m-D effect introduced by the target micro-motion component and the interference of side lobes and grating lobes introduced by the sparse aperture, obtain a clear ISAR image of the main body part of the target with the micro-motion component under the sparse aperture condition, is still effective for sparse aperture data with the sparsity of less than 12.5 percent, and has higher engineering application value.

Claims (4)

1. A low-rank and sparse combined constraint-based SA-ISAR imaging method for targets with micro-motion parts is characterized by comprising the following steps:
s1, modeling a target one-dimensional range profile sequence after the translational compensation of the target with the micro-motion component:
for the target with the micro-motion component, the translation-compensated one-dimensional range profile sequence of the target can be modeled as follows:
Figure FDA0003332062520000011
wherein the content of the first and second substances,
Figure FDA0003332062520000012
representing a translation compensated one-dimensional range profile sequence of the target,
Figure FDA0003332062520000013
tmrespectively representing a fast time and a slow time, wherein M is 1,2, …, M represents the number of pulses contained in a full-aperture radar echo, fcB, c denote the center frequency, bandwidth and propagation speed of the radar signal, σpAnd Rp(tm) Respectively representing the reflection coefficient of the p-th scattering center of the target body part and the instantaneous rotation distance, sigma, of the relative radarqAnd Rq(tm) Respectively representing the reflection coefficient of the qth scattering center of the target micro-motion component and the instantaneous rotation distance relative to the radar, wherein P is 1,2, …, P, Q is 1,2, …, and Q, P represents that the target main body part contains P scattering centers, and Q represents that the target micro-motion component contains Q scattering centers; for the p-th scattering center of the target body part, the instantaneous rotation distance R relative to the radarp(tm) Can be expressed as:
Rp(tm)＝xpsin(ωtm)+ypcos(ωtm)≈xpωtm+yp (2)
wherein (x)p,yp) Representing the coordinate of the p-th scattering center of the target main body part in a target specimen body coordinate system, and omega represents the rotating angular speed of the target main body part; because the ISAR imaging accumulation time is short, the rotation angle omega of the target relative to the radar in the imaging accumulation time is small, and therefore: sin (ω t)m)≈ωtm、cos(ωtm) 1 is approximately distributed; suppose that the scattering center of the target micro-motion component is around the point O' (x)O',yO') Rotating, and then for the qth scattering center of the target micro-motion component, the instantaneous rotating distance R of the target micro-motion component relative to the radarq(tm) Can be expressed as:
Figure FDA0003332062520000014
wherein (x)O',yO') Represents the coordinate of the qth scattering center of the target micro-motion component in the target specimen coordinate system, rqω', and θqRespectively representing the micromotion amplitude, the rotation angular speed and the initial phase of the qth scattering center of the target micromotion component; respectively substituting the formula (2) and the formula (3) into the formula (1), and further carrying out slow time t on the formula (1)mPerforming Fourier transform to obtain a target ISAR image;
under sparse aperture conditions, the target one-dimensional range profile sequence shown in equation (1) can be expressed in the form of the following matrix:
H＝L+S (4)
wherein
Figure FDA0003332062520000015
And
Figure FDA0003332062520000016
respectively representing a one-dimensional range image sequence of the target, the target main body part and the target micro-motion component,
Figure FDA0003332062520000017
expressing a K multiplied by N complex matrix, wherein K and N respectively express the number of pulses and the number of distance units of a sparse aperture one-dimensional range profile sequence; for sparse aperture data, the number of pulses is less than that of pulses contained in full aperture radar echo, namely K < M, and the pulse sequence number set is a subset of full aperture pulse sequence numbers, namely
Figure FDA0003332062520000018
Wherein i represents a sparse aperture pulse sequence number set;
for a target with a micro-motion component, acquiring a clear ISAR image of a main body part of the target is a main target of ISAR imaging; under the ideal condition, the ISAR image of the target main body part and the one-dimensional range profile sequence L of the target main body part are mutually fourier transform pairs, that is:
L＝PX (5)
wherein
Figure FDA0003332062520000021
An ISAR image representing a subject body portion of the subject,
Figure FDA0003332062520000022
representing a partial Fourier matrix, assuming a complete Fourier matrix of
Figure FDA0003332062520000023
P is formed by extracting part of row vectors in X and combining, specifically, the sequence number set of the extracted row vectors is a sparse aperture pulse sequence number set i;
s2 models the sparse aperture ISAR imaging problem of the target with the micromotion part:
the solution of the signal separation problem shown in equation (4) and the underdetermined problem shown in equation (5) is not unique, and a constraint condition needs to be added to obtain a unique solution: firstly, the column correlation of a one-dimensional range profile sequence L of a target main body part is strong, and the low-rank characteristic is achieved; secondly, energy of the one-dimensional range profile sequence S of the target micro-motion component is distributed in different range units, and the target micro-motion component has a sparse characteristic; thirdly, the ISAR image of the target main body consists of a few scattering centers and has strong sparse characteristic; therefore, this step limits equations (4) and (5) with the above three constraints to achieve sparse aperture ISAR imaging with the fine motion component target, specifically, the problem can be modeled as:
Figure FDA0003332062520000024
wherein | · | purple*(ii) counting & lt | & gt | & lt | & gt1Respectively representing the kernel norm and l of the matrix1Norm, which is respectively used for representing the rank and the sparsity of the matrix; lambda and mu represent regularization parameters which are respectively used for adjusting the weights of matrix decomposition and ISAR imaging;
s3, solving the ISAR imaging problem of the target sparse aperture of the micro-motion component by adopting linear ADMM:
solving the triple-constrained underdetermined problem shown in the formula (6) by using linear ADMM firstly needs to deduce the augmented Lagrangian function of the formula (6), as shown in the following formula:
Figure FDA0003332062520000025
wherein<·,·>Representing the inner product of two matrices, Y1、Y2Representing the Lagrange multiplier matrix, p1、ρ2Represents a penalty factor, | · |. non-woven phosphorFAn F norm representing a matrix; ADMM converts equation (6) into the following sub-problem solution:
Figure FDA0003332062520000031
wherein (·)(k)Representing the variable obtained by the kth iteration, and eta representing a rising factor for controlling the penalty factor rho1、ρ2The rising trend of (2) is specifically divided into the following steps:
s3.1, updating the one-dimensional range profile sequence L of the target main body part:
substituting formula (7) for formula (8), and omitting
Figure FDA0003332062520000032
Of (1), L independent terms, we can obtain:
Figure FDA0003332062520000033
equation (9) shows the problem of minimizing the nuclear norm, which can be solved by the singular value contraction operator:
Figure FDA0003332062520000034
wherein
Figure FDA0003332062520000035
The singular value contraction factor is expressed, and specifically, for an arbitrary matrix a and an arbitrary scalar γ, there are:
Figure FDA0003332062520000036
wherein A ═ Udiag (sigma) VHExpressing the singular value decomposition of A, U, V is a unitary matrix, sigma expresses the singular value vector of A, and diag (·) expresses a diagonal matrix formed by the vectors;
Figure FDA0003332062520000037
representing soft-threshold operators, for arbitrary scalars x, γ, having
Figure FDA0003332062520000038
Wherein sgn (·) represents a sign operator; for an arbitrary vector x, there are
Figure FDA0003332062520000039
Wherein xnRepresents the nth element of the vector x;
s3.2, updating the one-dimensional range profile sequence S of the target micro-motion component:
substituting formula (7) for formula (8), and omitting
Figure FDA00033320625200000310
S independent terms in (1), we can obtain:
Figure FDA0003332062520000041
formula (12) is represented by1A norm problem that can be solved by a soft threshold operator:
Figure FDA0003332062520000042
s3.3, updating the ISAR image X of the target main body part:
substituting formula (7) for formula (8), and omitting
Figure FDA0003332062520000043
Of (1), the terms independent of X, can be:
Figure FDA0003332062520000044
equation (14) is not a standard minimization l due to the presence of a partial Fourier matrix P multiplied by X1The norm problem cannot be solved directly by a soft threshold operator; therefore, further solving for X using linear ADMM for the quadratic term in equation (14)
Figure FDA0003332062520000045
Carrying out linearization; in particular, for this quadratic term, X ═ X(k)The second order taylor expansion is performed by:
Figure FDA0003332062520000046
wherein G is(k)To represent
Figure FDA0003332062520000047
When X is equal to X(k)The gradient of (a):
Figure FDA0003332062520000048
wherein, PHRepresents the conjugate transpose of the partial fourier matrix P;
formula (15) may be substituted for formula (14):
Figure FDA0003332062520000049
formula (17) is1The norm problem can be solved by a soft threshold operator:
Figure FDA00033320625200000410
s3.4 updating Lagrange multiplier matrix Y1、Y2：
From equation (8), the Lagrange multiplier matrix Y1、Y2The updating expressions of (A) are respectively as follows:
Y1 (k+1)＝Y1 (k)+ρ1 (k)(H-L(k+1)-S(k+1)) (19)
Y2 (k+1)＝Y2 (k)+ρ2 (k)(L(k+1)-PX(k+1)) (20)
s3.3 updating penalty factor ρ1、ρ2：
As shown in the formula (8), the penalty factor ρ1、ρ2The updating expressions of (A) are respectively as follows:
ρ1 (k+1)＝ηρ1 (k) (21)
ρ2 (k+1)＝ηρ2 (k) (22)
s3.4, combining the iteration formulas (10), (13), (18) and (19) - (22) until the relative error (| X) of the ISAR images of the target main body part obtained by two adjacent iterations(k+1)-X(k)|/|X(k)And |)) is smaller than a set threshold, the ISAR image X of the target main body part with the micro-motion component under the condition of sparse aperture can be obtained.
2. The low-rank and sparse combined constraint-based SA-ISAR imaging method with micro-motion component targets according to claim 1, wherein the method comprises the following steps: in S2, the regularization parameter λ ═ μ ═ 0.2.
3. The low-rank and sparse combined constraint-based SA-ISAR imaging method with micro-motion component targets according to claim 1, wherein the method comprises the following steps: in S3, the increase factor η is 2.
4. The low-rank and sparse combined constraint-based SA-ISAR imaging method with micro-motion component targets according to claim 1, wherein the method comprises the following steps: in S3.4, the threshold is set to 10-4。
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