CN112162280B - SF ISAR one-dimensional high-resolution distance imaging method based on atomic norm minimization - Google Patents

SF ISAR one-dimensional high-resolution distance imaging method based on atomic norm minimization Download PDF

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CN112162280B
CN112162280B CN202010860243.8A CN202010860243A CN112162280B CN 112162280 B CN112162280 B CN 112162280B CN 202010860243 A CN202010860243 A CN 202010860243A CN 112162280 B CN112162280 B CN 112162280B
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CN112162280A (en
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吕明久
徐芳
马建朝
赵欣
陈文峰
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Radar Sergeant School Of Chinese People's Liberation Army Air Force Early Warning Academy
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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]
    • 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
    • G01S13/9004—SAR image acquisition techniques

Abstract

The invention relates to an SF ISAR one-dimensional high-resolution distance imaging method based on atomic norm minimization, which comprises the following steps: step 1: the SF ISAR transmits an SF signal for detecting a target; step 2: the receiving end carries out distance image synthesis to realize high distance resolution; by converting the distance imaging problem into the atomic norm minimization problem, one-dimensional high-resolution distance imaging is realized on a continuous domain, and rapid sparse reconstruction is realized by using an ADMM method. Compared with the traditional discretization method, the method has the advantages of good reconstruction performance under the conditions of grid mismatch and low measurement data, and high distance resolution capability is kept.

Description

SF ISAR one-dimensional high-resolution distance imaging method based on atomic norm minimization
Technical Field
The invention relates to the technical field of radar signal processing, in particular to an SF ISAR one-dimensional high-resolution distance imaging method based on atomic norm minimization.
Background
An Inverse Synthetic Aperture Radar (ISAR) system detects a target by using a transmission signal with a large bandwidth, and high range resolution can provide finer target characteristics. While a Stepped Frequency (SF) waveform realizes a large synthesized wideband at a receiving end by transmitting a narrowband sub-pulse with continuously hopping carrier frequencies. Therefore, the combination of the stepping frequency waveform and the modern ISAR can realize high one-dimensional distance resolution capability, and has great significance for reducing the system complexity and realizing a radar multi-mode working mode, so that numerous scholars widely research SF ISAR high-resolution distance imaging methods.
Currently, the distance synthesis method for SF signals mainly includes an "IFFT" method, a target extraction algorithm, a wideband synthesis method, and the like. Among them, it is most common to directly perform "IFFT" transformation on the received sampled signal. However, the method has the defects of low resolution, poor imaging effect under the conditions of signal loss, interference and the like. By combining sparse representation, particularly Compressive Sensing (CS), with SF and utilizing the sparse property of ISAR observation targets, one-dimensional high-resolution imaging under signal deficiency conditions can be realized, which has become an important means for SF range imaging research. However, most of the existing CS-based SF high-resolution imaging methods are discretized sparse representation models, and are all based on an ideal assumption that target scattering points are accurately located at preset grid points. In fact, the grid mismatch problem always exists due to the randomness of the distribution of the target scattering points and the distance walking phenomenon in the ISAR imaging process, so that the improvement of the sparse reconstruction performance is severely restricted. To solve this problem, there are two main solutions: one is to increase the meshing density so that the target is most likely to fall on the grid points. However, not only may this approach lead to a drastic increase in processing complexity, but the corresponding reconstruction performance may also be severely degraded by the improved dictionary dependency. And secondly, reducing the influence of grid mismatch by a dictionary optimization method. In the prior art, a grid mismatch optimization method for an adaptive dictionary is provided, and an adaptive grid is constructed in a resolution unit through frequency shift, so that the influence caused by mismatch is reduced. In another prior art, a lattice mismatch is regarded as a model error, and an off-grid sparse Bayesian inference (OGSBI) algorithm is proposed based on a bayes theory. However, these algorithms are based on the premise of grid dispersion, and approach to the real scattering point position by means of grid further refinement or search optimization, and the like, so that the influence of grid mismatch is not completely eliminated. Another prior art proposes a Continuous Compressed Sensing (CCS) theory based on Atomic Norm Minimization (ANM), which avoids mesh discretization by sparse modeling in a continuous domain, and realizes accurate estimation of sinusoidal signal frequency under mesh mismatch. Applying this theory to the DOA estimation field in another prior art results in better estimation performance than the traditional discretized CS method. If the atomic norm theory can be applied to the field of ISAR imaging, the method has very important significance and value for solving SF ISAR one-dimensional high-resolution distance imaging under the grid mismatch condition.
In conclusion, how to apply the atomic norm to the field of SF ISAR imaging to solve the problem of grid mismatch caused by the distance walking phenomenon in the ISAR imaging process becomes very important.
Disclosure of Invention
Therefore, the invention provides an SF ISAR one-dimensional high-resolution range imaging method based on atomic norm minimization, which is used for overcoming the problem of grid mismatch caused by range walk phenomenon in the prior art.
In order to achieve the above object, the present invention provides a SF ISAR one-dimensional high-resolution range imaging method based on atomic norm minimization, comprising:
step 1: the SF ISAR transmits an SF signal for detecting a target;
step 2: the receiving end carries out distance image synthesis to realize high distance resolution;
in step 1, the SF ISAR observes K scattering point targets, and corresponding received echo signals are represented as:
Figure BDA0002647828830000021
where n denotes the nth subpulse in each train of pulses, n a N-th indicating azimuth a The pulse is divided into a group of pulses,
Figure BDA0002647828830000022
denotes the n-th a The intensity of the K scattering point in each pulse group; f. of c Is the carrier frequency; Δ f is the carrier frequency step; e represents noise; (x) k ，y k ) Is the coordinate of the scattering point in the target coordinate system, delta theta is the approximate rotation step length, n is1. 2, 3 … n, c is the speed of light.
Further, when the azimuth information and the constant term are changed into an additional phase term, the nth a The group range profile information is represented as:
Figure BDA0002647828830000031
where n denotes the nth subpulse in each train of pulses, n a N-th indicating azimuth a The pulse is divided into a group of pulse,
Figure BDA0002647828830000032
denotes the n-th a The intensity of the K-th scatter point in each pulse group,
Figure BDA0002647828830000033
a phase term associated with the azimuthal imaging is represented,
Figure BDA00026478288300000314
denotes the n-th a Position information of the Kth scattering point of each pulse group, wherein delta f is carrier frequency stepping amount; e represents noise, n is 1, 2, 3.. n, and c is the speed of light.
Further, according to the SF signal characteristics, N sub-pulses with carrier frequency step quantity Δ f are transmitted, the size of a synthesized bandwidth obtained by broadband synthesis is N Δ f, the corresponding distance resolution is Δ R ═ c/2N Δ f, and the unambiguous distance resolution range is R u C/2 Δ f, then n a The group range profile information is represented as:
Figure BDA0002647828830000034
while
Figure BDA0002647828830000035
Expressed by formula (4) is:
Figure BDA0002647828830000036
e is a noise matrix;
Figure BDA0002647828830000037
can be expressed as:
Figure BDA0002647828830000038
wherein the content of the first and second substances,
Figure BDA0002647828830000039
is a constant.
Further, the one-dimensional distance image of the SF ISAR is synthesized and converted into a coefficient
Figure BDA00026478288300000310
And normalized frequency
Figure BDA00026478288300000315
The n th a Solving and converting the group range image information into K forms selected from the convex hulls conv (A)
Figure BDA00026478288300000311
Wherein A represents an atom
Figure BDA00026478288300000312
A corresponding set of atoms.
Further, when the atomic norm of the one-dimensional distance image transformation of the SF ISAR is minimized, then the nth a The group range profile information is represented as:
Figure BDA00026478288300000313
wherein the content of the first and second substances,
Figure BDA0002647828830000041
and
Figure BDA0002647828830000042
are all made ofThe constant coefficient is a constant coefficient of the linear motor,
Figure BDA0002647828830000043
denotes the n-th a The group distance is like the corresponding kth atom,
Figure BDA0002647828830000044
denotes the n-th a The atomic norm of the group echo signals.
Furthermore, the preset distance direction is a sparse measurement mode, the number of transmitted sub-pulses is M, M is less than or equal to N, and the subset formed by the M sub-pulse sequences is represented as omega, then
Figure BDA0002647828830000045
And M ═ Ω | ≦ N, then ANM under sparse sampling conditions is expressed as:
Figure BDA0002647828830000046
where τ is related to the noise level, assuming that the noise obeys N (0, σ) 2 I N ) Then, then
Figure BDA0002647828830000047
s Ω (n a ) Randomly measuring data for the range profile;
Figure BDA0002647828830000048
for metrology data to be recovered, A (Ω) represents a set of atoms with a subset of Ω.
Further, the ANM under the sparse sampling condition is converted into a semi-positive plan, which is expressed as:
Figure BDA0002647828830000049
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA00026478288300000410
represented as the full data that is recovered,
Figure BDA00026478288300000411
is a constant number u 1 Is a vector
Figure BDA00026478288300000412
The first of the elements in (1) is,
Figure BDA00026478288300000413
representing the conjugate transpose of the data to be recovered.
Further, the SF ISAR converts the ANM into a semi-positive definite plan by using an ADMM algorithm, wherein the ADMM method mainly constructs a lagrangian function optimized by the semi-positive definite, and the lagrangian function corresponding to the ANM converted into the semi-positive definite plan is expressed as:
Figure BDA00026478288300000414
wherein, the Lambda Lagrange multiplier is defined, rho is a penalty parameter, the matrixes Z and Lambda are Hermitian matrixes,<·，·>is the inner product sign, u 1 Is a vector
Figure BDA00026478288300000415
The first of the elements in (1) is,
Figure BDA00026478288300000416
the squared value of the 2 norm representing the observation minus the recovered data, and the structural partition introduced into the matrices Z, Λ is expressed as:
Figure BDA0002647828830000051
wherein the matrices Z, Λ are Hermirian matrices, and
Figure BDA0002647828830000052
Figure BDA0002647828830000053
further, the ADMM updates a plurality of variables separately, for a selected variable to be updated, the other variables are kept unchanged, and the selected variable is iterated according to the fixed variables, where the iteration is expressed as follows:
Figure BDA0002647828830000054
where l denotes the first iteration, t l+1 Represents the iteration value, u, of the parameter t at iteration l +1 l+1 Represents the iterative value, Z, of the time vector u at the (l +1) th time l+1 Representing the iteration value, Λ, of the matrix Z at the 1 st iteration l The iteration values of the matrix Λ at the ith iteration are represented. Z ' represents the iteration value of the matrix Z ' at the time of the i-th iteration, u ' represents the iteration value of the vector u at the time of the i-th iteration,
Figure BDA0002647828830000055
is the estimated data.
Figure BDA0002647828830000056
Data for l +1 iterations
Figure BDA0002647828830000057
The value of the iteration of (a) is,
based on the iterative process, the iterative computation expression of the parameters is obtained by increasing Lagrange function partial derivatives as follows:
Figure BDA0002647828830000058
wherein omega c Denotes the complement of the set Ω, which is also the set of positions of the sparse echo signals, and e is the first element 1, W ═ diag [1/N,1/(2(N-1)),]is a diagonal matrix, other elements being N-dimensional vectors, s, of 0 Ω In order to measure the data, the measurement data,
Figure BDA0002647828830000059
representing the matrix Z in the first iteration 1 Corresponding to the measured data.
Figure BDA00026478288300000510
Representing the matrix Z obtained in the first iteration 1 。
Figure BDA0002647828830000061
Representing the matrix Λ in the ith iteration 1 Updating the matrix Z according to the corresponding measurement data:
Figure BDA0002647828830000062
further, in the updating iteration of ADMM, the estimation error or the maximum iteration number is set as a stagnation condition, and the final t, toe (u) and
Figure BDA0002647828830000063
if the rank of the matrix K ═ rank (toe (u)) < N, toe (u)) can be decomposed exclusively as Vandermonde:
Toep(u)＝APA H (16)
wherein A is Vandermonde matrix, and each column is composed of atom a (f) k ，φ k ) Is composed of, i.e.
Figure BDA0002647828830000064
P is a K x K dimensional positive diagonal matrix, and
Figure BDA0002647828830000065
it can therefore be concluded that the rank K of the toep (u) matrix at this time represents the sparsity of the sparse signal.
Compared with the prior art, the SF ISAR high-resolution one-dimensional range profile synthesis method based on atomic norm minimization has the advantages that firstly, a meshless step frequency ISAR range sparse representation model based on atomic norm is constructed, and the one-dimensional range imaging problem is converted into the problems of atomic coefficients and frequency estimation. And then, converting the atomic norm minimization problem into a semi-definite programming problem by utilizing the semi-definite property of the atomic norm, and realizing quick solution based on an alternating direction multiplier method. And finally, obtaining a final one-dimensional high-resolution distance imaging result by utilizing Vandermonde decomposition.
In particular, the SF ISAR converts the range imaging problem into the atomic norm minimization problem, thereby realizing one-dimensional high-resolution range imaging in a continuous domain and realizing rapid sparse reconstruction by using the ADMM method. Compared with the traditional discretization method, the method has the advantages of good reconstruction performance under the conditions of grid mismatch and low measurement data, and high distance resolution capability is kept. However, the method of the present invention has the disadvantage of large computation amount, which is also a problem to be researched and solved in the next step.
Further, in the SF ISAR processing process, since grid discretization processing is avoided, high-resolution distance imaging under the conditions of grid mismatch and low measurement value can be realized, and high distance resolution capability is maintained.
Further, the SF ISAR imaging method includes the steps that firstly, an SF ISAR distance sparse representation model is built in an atomic norm domain, and a one-dimensional distance imaging problem is converted into an atomic norm minimization problem. And then, converting the minimization of the atomic norm into a semi-definite programming problem, and quickly solving by using an alternating direction multiplier method. And finally, realizing final high-resolution one-dimensional distance imaging by utilizing Vandermonde decomposition. The SF ISAR imaging method directly carries out the sparse modeling of the range profile in the continuous domain, avoids the problem of grid mismatching caused by grid discretization operation, has the advantage of accurate estimation performance under the condition of grid mismatching, and keeps better reconstruction performance under the condition of low measurement value.
Drawings
FIG. 1 is a schematic flow chart of a method for high resolution range imaging based on atomic norms according to an embodiment of the present invention;
FIG. 2a is a schematic diagram of a reconstruction result under different algorithm one-dimensional range profile network matching conditions in the embodiment of the present invention;
FIG. 2b is a schematic diagram of a reconstruction result under a mismatch condition of one-dimensional distance image networks of different algorithms in the embodiment of the present invention;
FIG. 3a is a schematic diagram illustrating comparison of estimation errors of an algorithm support set under different measurement values according to an embodiment of the present invention;
FIG. 3b is a schematic diagram illustrating comparison of algorithm estimation errors under different measurement values according to an embodiment of the present invention;
FIG. 4a is a graph illustrating error comparison of algorithm support set reconstruction accuracy estimates under different SNR conditions according to the present invention;
FIG. 4b is a graph illustrating error comparison of algorithm reconstruction accuracy under different SNR conditions according to an embodiment of the present invention;
FIG. 5 is a diagram illustrating a resolution analysis result of algorithm reconstruction according to an embodiment of the present invention;
FIG. 6a is a distance image reconstruction result of the OMP algorithm according to the embodiment of the present invention;
FIG. 6b is an ISAR imaging result of the OMP algorithm according to the embodiment of the present invention;
FIG. 6c is a distance image reconstruction result of the OGSBI algorithm according to the embodiment of the present invention;
FIG. 6d is the ISAR imaging result of the OGSBI algorithm according to the embodiment of the present invention;
FIG. 6e is a diagram illustrating the reconstructed distance image of the ANM-SDPT3 algorithm according to the embodiment of the present invention;
FIG. 6f is the ISAR imaging results of the ANM-SDPT3 algorithm in accordance with the exemplary embodiment of the present invention;
FIG. 6g is a diagram illustrating a distance image reconstruction result of the ANM-ADMM algorithm according to the embodiment of the present invention;
FIG. 6h is a diagram illustrating a distance image reconstruction result of the ANM-ADMM algorithm according to the embodiment of the present invention;
FIG. 7a is a diagram of entropy comparison results of different algorithm reconstructions in the embodiment of the present invention;
fig. 7b is a diagram of contrast results reconstructed by different algorithms in the embodiment of the present invention.
Detailed Description
In order that the objects and advantages of the invention will be more clearly understood, the invention is further described below with reference to examples; it should be understood that the specific embodiments described herein are merely illustrative of the invention and do not delimit the invention.
Preferred embodiments of the present invention are described below with reference to the accompanying drawings. It should be understood by those skilled in the art that these embodiments are only for explaining the technical principle of the present invention, and do not limit the scope of the present invention.
It should be noted that in the description of the present invention, the terms of direction or positional relationship indicated by the terms "upper", "lower", "left", "right", "inner", "outer", etc. are based on the directions or positional relationships shown in the drawings, which are only for convenience of description, and do not indicate or imply that the device or element must have a specific orientation, be constructed and operated in a specific orientation, and thus, should not be construed as limiting the present invention.
Furthermore, it should be noted that, in the description of the present invention, unless otherwise explicitly specified or limited, the terms "mounted," "connected," and "connected" are to be construed broadly, and may be, for example, fixedly connected, detachably connected, or integrally connected; can be mechanically or electrically connected; they may be connected directly or indirectly through intervening media, or they may be interconnected between two elements. The specific meanings of the above terms in the present invention can be understood by those skilled in the art according to specific situations.
First, technical terms used in the embodiments of the present invention are explained,
ISAR: the Chinese characters are all called as: inverse synthetic aperture radar, known in english as: inverse Synthetic Aperture Radar can perform high-resolution imaging on a long-distance target, a large-bandwidth transmitting signal is provided to detect the target, and high distance resolution can provide more precise target characteristics.
SF: chinese full name: step frequency, english is called: and (3) a stepped frequency waveform realizes a large synthesized broadband at a receiving end by transmitting a narrow-band sub-pulse with continuously hopping carrier frequency.
CS: chinese characters are fully called: compressed sensing, english is commonly referred to as: compressive Sensing, the Compressive Sensing method discards redundant information in the current signal sample. The CS transforms the compressed samples directly from the continuous-time signal and then processes the compressed samples in digital signal processing using an optimization method.
OGSBI: chinese characters are fully called: sparse Bayesian reconstruction of lattice mismatch, English is called as: off-grid spark Bayesian inference, an algorithm based on Bayesian theory.
ANM: chinese characters are fully called: atomic norm minimization, english is called: atom norm minimization.
CCS: chinese full name: the theory of continuous compressed sensing, english is called: continuous compressed sensing.
ADMM: chinese full name: the alternating direction multiplier method, English is called as: an alternating direct method of multipliers.
Referring to fig. 1, the present invention provides a SF ISAR one-dimensional high-resolution range imaging method based on atomic norm minimization, which includes
Step 1: the SF ISAR transmits an SF signal for detecting a target;
step 2: the receiving end carries out distance image synthesis to realize high distance resolution.
In the step 1, in the SF signals transmitted by the SF ISAR, the SF ISAR observes K scattering point targets, and a corresponding received echo signal formula (1) is represented as:
Figure BDA0002647828830000091
where n denotes the nth sub-pulse in each train of pulses, n a N-th indicating azimuth a The pulse is divided into a group of pulses,
Figure BDA0002647828830000092
denotes the n-th a The intensity of the K scattering point in each pulse group; f. of c Is the carrier frequency; Δ f is the carrier frequency step; e represents noise; (x) k ，y k ) Is the coordinate of the scattering point in the target coordinate system, delta theta is the approximate rotation step length, n is 1, 2,3 … n, c is the speed of light.
Changing the azimuth information and constant term into additional phase term, at this time, n a The group range profile information can be expressed as equation (2):
Figure BDA0002647828830000093
where n denotes the nth sub-pulse in each train of pulses, n a N-th indicating azimuth a The pulse is divided into a group of pulse,
Figure BDA0002647828830000094
denotes the n-th a The intensity of the K-th scatter point in each pulse group,
Figure BDA0002647828830000095
a phase term associated with the azimuthal imaging is represented,
Figure BDA0002647828830000096
denotes the n-th a Position information of the Kth scattering point of each pulse group, wherein delta f is carrier frequency stepping amount; e represents noise, and n is 1, 2, 3 … n.
Specifically, in the embodiment of the present invention, according to the SF signal characteristics, N sub-pulses with carrier frequency step amount Δ f are transmitted, the synthesized bandwidth obtained by wideband synthesis is N Δ f, the corresponding range resolution is Δ R ═ c/2N Δ f, and the unambiguous range resolution range is R u When c/2 Δ f, the n-th group of the formula (2) a The group range profile information can be expressed as equation (3):
Figure BDA0002647828830000101
and then
Figure BDA0002647828830000102
Expressed by formula (4) is:
Figure BDA0002647828830000103
e is a noise matrix;
Figure BDA0002647828830000104
can be expressed as:
Figure BDA0002647828830000105
wherein the content of the first and second substances,
Figure BDA0002647828830000106
is a constant.
At this time, the one-dimensional range profile synthesis of the SF ISAR has been converted into coefficients
Figure BDA0002647828830000107
And normalizing the frequency
Figure BDA0002647828830000108
To (3) is described. According to the atomic norm theory, the same principle is applied to the nth a Solving the group range image information can be converted into selecting K types of range image information in the convex hull conv (A)
Figure BDA0002647828830000109
Wherein A represents an atom
Figure BDA00026478288300001010
A corresponding set of atoms. Therefore, the one-dimensional range profile estimation problem of the SF ISAR can be converted into an atomic norm minimization problem to be solved.
Specifically, in the embodiment of the present invention, the atomic norm theory is a form of norm describing continuous parameters, which covers various commonly used norms, such as l 1 Norm, nuclear norm, etc. Assume atom set a, whose corresponding convex hull is denoted conv (a), and conv (a) is a central symmetric compact set containing the origin. Thus, the atomic norm is the norm form defined by the scaling function of the convex hull conv (a), and is expressed by the following equation (5):
Figure BDA00026478288300001011
wherein | · | purple sweet A Denotes the atomic norm, a (f) k ，φ k ) Is the atom in convex hull conv (A), t is constant coefficient, k is sparsity, C k Is an atomic coefficient. Y is echo data.
From this, the atomic norm is given by the coefficient c k By selecting the least atoms a (f) in the set A as the lower bound k ，φ k ) To characterize the signal Y, the atomic norm can be considered to add a sparse constraint to the set of atoms a. Under the constraint, the set A is regarded as an infinite set with continuous atoms, so that the problems of atom mismatching and the like caused by introducing grid discretization parameters are avoided. In addition, if a (f) k ，φ k ) Is a base vector in Euclidean space, and the atomic norm is l of a general sparse vector 1 A norm; in the same way, if a (f) k ，φ k ) The matrix is a matrix with a rank of 1, and the atomic norm is a nuclear norm form in the matrix filling theory at this time. The atomic norm has the property of semi-definite programming, so that the problem of minimizing the atomic norm can be converted into the problem of semi-definite programming of the atomic norm, namely, the semi-definite programming of the atomic norm is expressed by the following formula (6):
Figure BDA0002647828830000111
wherein N is the signal length; trace (·) represents the trace of the matrix,
Figure BDA0002647828830000112
is a Toeplitz matrix, t is a normal number, and can be represented by formula (7):
Figure BDA0002647828830000113
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002647828830000119
representing the first row element of the Toep (u) matrix, (. cndot.) H Representing a conjugate transpose.
In the process of solving the problem of converting the one-dimensional distance image of the SF ISAR into the atomic norm minimization a The group range profile information may be expressed by equation (8):
Figure BDA0002647828830000115
wherein the content of the first and second substances,
Figure BDA0002647828830000116
and
Figure BDA0002647828830000117
are all constant coefficients of the light source,
Figure BDA0002647828830000118
denotes the n-th a Group distance image corresponding kth atom, | s (n) a )|| A Denotes the n-th a The atomic norm of the group echo signals.
Specifically, in the embodiment of the present invention, assuming that the distance direction is a sparse measurement mode, the number of transmitted sub-pulses is M, M is less than or equal to N, and the subset formed by M sub-pulse sequences is represented as Ω, then
Figure BDA0002647828830000121
And M | ≦ N, the ANM problem under sparse sampling conditions may be represented by equation (9):
Figure BDA0002647828830000122
where τ is related to the noise level, assuming that the noise obeys N (0, σ) 2 I N ) Then, then
Figure BDA0002647828830000123
s Ω (n a ) Is composed ofRandom measurement data of the range profile;
Figure BDA0002647828830000124
for metrology data to be recovered, a (Ω) represents a set of atoms with a subset of Ω.
If equation (9) is converted into a semi-positive plan, equation (10) can be expressed as:
Figure BDA0002647828830000125
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002647828830000126
represented as the full data that is recovered,
Figure BDA0002647828830000127
is a constant number u 1 Is a vector
Figure BDA0002647828830000128
In the first of the elements of (a) to (b),
Figure BDA0002647828830000129
representing the conjugate transpose of the data to be recovered.
According to the semi-definite programming problem, an optimization algorithm based on an interior point method, such as SDPT3 and SeDuMi, is conventionally adopted for solving, but the method has the problems of large operation amount and low calculation efficiency. The invention omits the subscript symbol n of the labeled pulse group a . At this time, the augmented lagrangian function corresponding to the conversion of the ANM into the semi-definite programming according to equation (10) can be expressed as equation (11):
Figure BDA00026478288300001210
wherein, the Lamlagrangian multiplier is defined, rho is a penalty parameter, the matrixes Z and Lambda are Hermitian matrixes,<·，·>is the inner product sign, u 1 Is a vector
Figure BDA00026478288300001211
The first of the elements in (1) is,
Figure BDA00026478288300001212
representing the observation minus the 2-norm squared value of the recovered data.
In the above equation (11), the structural division equation (12) in which the matrix Z, Λ is introduced is:
Figure BDA0002647828830000131
wherein the matrix Z, Λ are Hermirian matrices, and
Figure BDA0002647828830000132
Figure BDA0002647828830000133
ADMM is optimizing the above problem as: updating a plurality of variables separately, namely for the selected variable needing to be updated, firstly keeping other variables unchanged, and iterating the selected variable according to the fixed variables, wherein the specific process is shown as the following formula (13):
Figure BDA0002647828830000134
where l denotes the first iteration, t l+1 Represents the iteration value, u, of the parameter t at iteration number l +1 l+1 Represents the iterative value, Z, of the time vector u at the (l +1) th time l+1 Representing the iteration value, Λ, of the matrix Z at the 1 st iteration l Representing the matrix Λ at the ith iteration l The iteration value of (2). Z' denotes the iteration value of the matrix Z at the first iteration, u l Represents the iteration value of the time vector u at the i-th time,
Figure BDA0002647828830000138
in order to be able to estimate the data,
Figure BDA0002647828830000135
data for l +1 iterations
Figure BDA0002647828830000136
The iteration value of (2).
Based on the above updating process, by solving the augmented lagrangian function partial derivative, the iterative calculation of the above parameters can be obtained as expressed by equation (14):
Figure BDA0002647828830000137
wherein omega c Denotes the complement of the set Ω, which is also the set of positions of the sparse echo signals, and e is 1, W ═ diag [1/N,1/(2(N-1)),.., 1/2]Is a diagonal matrix, other elements being N-dimensional vectors, s, of 0 Ω In order to measure the data, the measurement data,
Figure BDA0002647828830000141
representing the matrix Z in the first iteration 1 Corresponding metrology data.
Figure BDA0002647828830000142
Representing the matrix Z obtained in the first iteration 1 。
Figure BDA0002647828830000143
Representing the matrix Λ in the first iteration 1 Corresponding metrology data.
The matrix Z is updated according to the following equation (15):
Figure BDA0002647828830000144
in the above update iteration, the estimation error or the maximum number of iterations may be set as a stall barThe final t, Toep (u) and
Figure BDA0002647828830000145
after the signal to be recovered is obtained, the distance image synthesis can be performed by using a traditional "IFFT" synthesis method or a dictionary discretization CS method, but the "mesh-free" advantage brought by solving the atomic norm is lost.
According to the cartheidory theorem, for any N × N dimensional semi-positive definite Toeplitz matrix toep (u), if the rank K of the matrix K ═ rank (toep (u)) < N, toep (u) can be uniquely decomposed into Vandermonde as shown in formula (16):
Toep(u)＝APA H (16)
wherein A is a Vandermonde matrix and each column thereof is composed of atoms a (f) k ，φ k ) Is composed of, i.e.
Figure BDA0002647828830000146
P is a K x K dimensional positive diagonal matrix, and
Figure BDA0002647828830000147
it can therefore be concluded that the rank K of the toep (u) matrix at this time represents the sparsity of the sparse signal.
Specifically, in the embodiment of the present invention, after the Toeplitz matrix toep (u) is obtained by the SF ISAR system, coefficients can be obtained by performing Vandermonde decomposition on the toep (u)
Figure BDA00026478288300001411
And normalizing the frequency
Figure BDA00026478288300001412
Thereby realizing one-dimensional high resolution. According to the embodiment of the invention, the conventional grid dictionary is replaced by the continuous dictionary through the atomic norm minimized high-resolution one-dimensional distance imaging method, the problem of distance image mismatch caused by grid division is avoided, and better reconstruction performance is achieved.
Specifically, in the embodiment of the present invention, the preset pair frequency
Figure BDA0002647828830000148
Distribution interval is [0, 1 ]]Carrying out N-point discretization processing to obtain an N-point discrete frequency sequence
Figure BDA0002647828830000149
When it corresponds to a set of atoms as
Figure BDA00026478288300001410
The corresponding atomic norm for any set of echo signals is expressed as equation (17) below:
Figure BDA0002647828830000151
wherein the content of the first and second substances,
Figure BDA00026478288300001510
Figure BDA0002647828830000152
is a corresponding discrete metrology dictionary.
When the set of atoms is finite, the atomic norm can be viewed as l of a sparse vector 1 Norm, when N → + ∞ discrete set of atoms A N (omega) infinitely approaches a continuous set of atoms A (omega), also
Figure BDA0002647828830000153
At this time, the atomic norm and l under discrete conditions 1 The norm relationship is expressed by the following formula (18):
Figure BDA0002647828830000154
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002647828830000155
denotes s Ω The discrete atomic norm of (a) of (b),
Figure BDA0002647828830000156
denotes s Ω M is the number of measured pulses, and N is the number of pulses.
Specifically, in the embodiment of the present invention, the method using the atomic norm is compared with the method based on the discrete dictionary 1 Norm and atomic norm have better reconstruction performance. When the semi-positive timing optimization is adopted, the results are mainly influenced by the random measurement number M and the signal sparsity K, and the minimum frequency interval delta is shown in the prior art f ≥1/[N/4]Wherein
Figure BDA0002647828830000157
Representing normalized frequency f k Has a constant C so that the random observed number M satisfies the following expression (19):
Figure BDA0002647828830000158
wherein K is sparsity. By solving the semi-positive definite optimization, the original signal s can be reconstructed with high probability of 1-delta, and delta is f ≥1/[N/4]This condition is not critical, at Δ f 1/N or even Delta f Under the condition of < 1/N, frequency separation can still be successfully realized, and the high resolution capability of the signal is maintained by the overall optimization method based on atomic norm minimization. In addition, in the ANM theory, the sparsity K of the signal to be recovered is generally required to satisfy the following condition (20):
Figure BDA0002647828830000159
the sparse [ A (omega) ] represents the least number of irrelevant atoms used for representing the sparse signals in the atom set A (omega), and the sparse [ A (omega) | is not less than 2 and not more than M +1, so that the sparse signal reconstruction method based on the atomic norm can be obtained, and the accurate reconstruction condition is similar to that of the traditional discrete CS. To ensure that the reconstruction sparsity K with a high probability satisfies the conditions of the above equations (19) and (20), it is necessary to decompose Vandermonde of the equation (16) and to make K ═ rank (toe (u)) < N.
Specifically, in the embodiment of the present invention, for the sparse reconstruction method based on the ANM, the computation complexity is mainly related to the dimension of the matrix Z, and the computation complexity of the optimization method based on the interior point method can be represented as o ((N +1) 6 ) When the data amount is large, the operation amount is increased sharply. And the ADMM-based rapid optimization method can reduce the computational complexity to O ((N +1) through decomposition of the optimization problem and iterative solution in turn 3 ) Compared with an interior point method, the method has the advantages that the operation complexity is greatly reduced, and the method is more suitable for quick reconstruction under the condition of high data volume. In the prior art, the complexity of the traditional discrete grid-based compressed sensing method mainly comes from matrix inversion, and the computational complexity is about o (kmn); the complexity of the OGSBI algorithm of the grid mismatch optimization method is about o (M) 2 N+M 3 ) Since K is more than M and less than or equal to N, no obvious hint can be given on the operation complexity in the embodiment of the invention.
Specifically, in the embodiment of the invention, during the verification process of the ANM-based SF ISAR distance high-resolution imaging method through simulation, firstly, the reconstruction error calculation mode is
Figure BDA0002647828830000161
Where x is the original signal and x ^ is the estimated signal, carrier frequency f of the SF signal used for simulation c The carrier frequency step quantity delta f is 5MHz, the number N of sub-pulses is 50, and the synthesis bandwidth is 250MHz at 10 GHz. The signal sampling rate is set to a-M/N. All simulations are based on Matlab R2012b simulation platform, and the computer processor Intel core i7-6700HQ, main frequency 2.6GHz and internal memory 8GB are used. In the embodiment of the invention, the reconstruction performance of the verification algorithm is mainly analyzed from three aspects of measurement value size, signal-to-noise ratio and resolution capability.
Referring to fig. 2a and fig. 2b, for simulation comparison of the reconstruction accuracy of the SF ISAR, the determination is performed by using one-dimensional high-resolution range imaging maps obtained by different reconstruction methods, and first, the reconstruction performance of the SF ISAR under the conditions that the target scattering point matches with the range grid and the target scattering point is mismatched is determined. 4 scattering point targets are generated through simulation, the positions and the amplitudes of the scattering point targets are generated randomly, the positions of the grid matching targets are set on grid points, and the signal-to-noise ratio is set to be 20 dB.
Referring to fig. 3a and 3b, for comparison under different measurement value conditions of the SF ISAR, the sparse reconstruction results are determined by different methods under different sampling rate a conditions under different measurement value conditions, different signal measurement values are set under the same parameter conditions, the signal-to-noise ratio is set to 20dB, and the analysis is mainly performed from two aspects of support set estimation accuracy and range image amplitude estimation error, where the support set is a correct range image position set. Under different measurement value conditions, the OMP algorithm has poor estimation accuracy, because the support set estimation error always exists under the grid mismatch condition, and the OGSBI algorithm has the grid mismatch correction function, so the estimation accuracy is greatly improved. The method based on the ANM in the embodiment of the invention has the best support set estimation precision because the reconstruction is directly carried out in the continuous domain. In comparison, the performance of the ANM reconstruction method based on SDPT3 is slightly better than that of the reconstruction result based on the ADMM method under the condition of low measurement values, but the result difference is small, and the reconstruction performance based on the SDPT3 method and the ADMM method are also closer, so that the best estimation accuracy is achieved.
Referring to fig. 4a and 4b, for the reconstruction performance of the SF ISAR under the condition of different signal-to-noise ratios, different signal-to-noise ratios are set under the same parameter condition, and the measurement value is set to 40, so that it can be obtained that under the condition of different signal-to-noise ratios, the method based on the ANM of the present invention also has the best estimation accuracy of the support set, and has higher estimation accuracy no matter based on the SDPT3 method or based on the ADMM method, which shows stronger robustness.
Referring to fig. 5, for the reconstruction resolution analysis of the SF ISAR, the difference of the processing time of different methods is further compared, and the processing time of different methods is compared under different conditions of transmitting sub-pulses by taking the signal parameters as an example and keeping the algorithm setting parameters unchanged. It can be seen that the conventional OMP algorithm has the least computation time, and the computation amount thereof increases slightly with the increase of the number M of the transmitted sub-pulses, because the computation complexity thereof is related to the number of the transmitted sub-pulses. In addition, the OGSBI algorithm also has a calculation time related to the number of transmission sub-pulses, so the calculation time gradually increases as the number of transmission sub-pulses increases. The ANM method based on SDPT3 has the longest operation time because the operation amount is related to the power of the exponent of the number N of sub-pulses, and the operation amount is substantially the same for different numbers M of transmitted sub-pulses. The ADMM-based ANM method has the advantages that the calculation amount is remarkably reduced, and the calculation complexity is only related to N, so that the same calculation time is realized under different emission sub-pulse M conditions. Compared with the four methods, the OMP algorithm has the least computation amount, the ANM reconstruction method based on the ADMM is approximately in the same computation order as the OGSBI algorithm, and the simulation result also verifies the correctness of the theoretical analysis conclusion.
Referring to fig. 6a to 6h, validity of the SF ISAR based on the ANM algorithm is verified by using MIG-25 data, a data transmission signal is a stepped frequency waveform, a carrier frequency of the data transmission signal is 9GHz, 64 sub-pulses are transmitted altogether, a synthesis bandwidth is 512MHz, a corresponding sub-pulse bandwidth is 8MHz, an azimuth direction has 512 groups of sub-pulses altogether, a pulse repetition frequency is 15KHz, a target rotation angular velocity is approximately 10o/s within an imaging time, and motion compensation has been performed. Since the correct distance image result cannot be known in advance, a proper index for measuring the distance imaging quality cannot be found. However, in ISAR imaging, the quality of range image imaging can have an effect on the final two-dimensional imaging result. For this reason, the range image synthesis result and the final two-dimensional ISAR imaging result are given in the experiment, and the Entropy (Entropy) and Contrast (Contrast) for measuring the ISAR imaging quality are used to compare the reconstruction performance of different algorithms. The distance image synthesis adopts 4 different reconstruction algorithms of the invention, and the azimuth is processed by using an FFT method. In a simulation experiment, the former 128 th group of sub-pulses are selected to carry out two-dimensional imaging result reconstruction, the sampling rate is set to be 0.5, namely 32 azimuth sub-pulses are randomly selected to be processed, and the distance synthetic result and the final ISAR imaging result obtained by different methods are utilized, so that the distance image synthetic result obtained by utilizing the OMP algorithm has more false reconstructions. The OGSBI algorithm is slightly better than the OMP algorithm in distance image synthesis result. In comparison, the distance image synthesis result of the method has fewer false reconstruction points, so that the performance is the best, and the effectiveness of the method is further verified. From the ISAR imaging result, the final ISAR imaging quality is influenced by the range image synthesis result, so that the two-dimensional imaging quality of the OMP algorithm and the OGSBI algorithm has more false scattering points and poorer imaging quality. The ISAR imaging result obtained by the sparse reconstruction method based on ANM (sparse reconstruction method based on SDPT3 and ADMM) is best in focusing, and false scattering points are least.
Referring to fig. 7a-7b, entropy and contrast curves of two-dimensional ISAR imaging results under different range-wise sampling rates are shown. It can be seen that the imaging entropy values of the method under the condition of low range sampling rate are lower than those of the other two methods, and the imaging contrast is higher than those of the other two methods, so that the reconstruction performance of the method under the condition of low sampling rate is also favorably verified. In addition, it can be seen that both the SDPT3 method and the ADMM method based ANM sparse reconstruction algorithm have similar imaging results and better imaging quality. Under the condition of the signal parameters, the total operation time for realizing the distance image synthesis by different algorithms is compared statistically, based on the simulation platform, the method for realizing 128 groups of distance images by using the ANM based on the SDPT3 needs about 343.93s, while under the same condition, the method for realizing the distance image synthesis by using the ANM based on the ADMM needs about 104.20s, the OGSBI method needs about 108.79s, and the OMP algorithm only needs about 2.86s, so that although the method for quickly solving the ADMM is used, the operation efficiency still has a larger improvement space. However, with the rapid increase of the operation performance of modern supercomputers, the complexity is not the primary factor to be considered.
So far, the technical solutions of the present invention have been described in connection with the preferred embodiments shown in the drawings, but it is easily understood by those skilled in the art that the scope of the present invention is obviously not limited to these specific embodiments. Equivalent changes or substitutions of related technical features can be made by those skilled in the art without departing from the principle of the invention, and the technical scheme after the changes or substitutions can fall into the protection scope of the invention.

Claims (10)

1. An SF ISAR one-dimensional high-resolution range imaging method based on atomic norm minimization, which is characterized by comprising the following steps:
step 1: the SF ISAR transmits an SF signal for detecting a target;
and 2, step: the receiving end carries out distance image synthesis to realize high distance resolution;
in step 1, the SF ISAR observes K scattering point targets, and corresponding received echo signals are represented as:
Figure FDA0003739055630000011
where n denotes the nth sub-pulse in each train of pulses, n a N-th indicating azimuth a The pulse is divided into a group of pulses,
Figure FDA0003739055630000012
denotes the n-th a The intensity of the K scattering point in each pulse group; f. of c Is the carrier frequency; Δ f is the carrier frequency step; e represents noise; (x) k ，y k ) The coordinate of the scattering point in the target coordinate system is shown, delta theta is approximate rotation step length, n is 1, 2 and 3 … n, and c is light speed;
in the step 1, firstly, an SFISAR distance direction sparse representation model is constructed in an atomic norm domain, a one-dimensional distance imaging problem is converted into an atomic norm minimization problem, then, the atomic norm minimization is converted into a semi-definite programming problem, a fast solution is carried out by using an alternative direction multiplier method, and finally, the Vandermonde decomposition is used for realizing the final high-resolution one-dimensional distance imaging.
2. The SFISAR one-dimensional high-resolution range imaging method based on atomic norm minimization as claimed in claim 1, wherein when the azimuth information and constant term are changed into additional phase term, n-th a The group range profile information is represented as:
Figure FDA0003739055630000013
where n denotes the nth subpulse in each train of pulses, n a N-th indicating azimuth a The pulse is divided into a group of pulse,
Figure FDA0003739055630000014
denotes the n-th a The intensity of the kth scattering point in each pulse group,
Figure FDA0003739055630000015
a phase term associated with the azimuthal imaging is represented,
Figure FDA0003739055630000016
denotes the n-th a Position information of the Kth scattering point of each pulse group, wherein delta f is carrier frequency stepping amount; e represents noise, n is 1, 2, 3 … n, and c is the speed of light.
3. The SF ISAR one-dimensional high-resolution range imaging method based on atomic norm minimization of claim 2, wherein N sub-pulses with carrier frequency step quantity Δ f are transmitted according to SF signal characteristics, a synthesized bandwidth obtained after wideband synthesis is N Δ f, a corresponding range resolution is Δ R ═ c/2N Δ f, and an unambiguous range resolution range is R u C/2 Δ f, then n a The group range profile information is represented as:
Figure FDA0003739055630000021
while
Figure FDA0003739055630000022
Expressed by formula (4) is:
Figure FDA0003739055630000023
e is a noise matrix;
Figure FDA0003739055630000024
can be expressed as:
Figure FDA0003739055630000025
wherein the content of the first and second substances,
Figure FDA0003739055630000026
Figure FDA0003739055630000027
is a constant.
4. The SFISAR one-dimensional high-resolution range imaging method based on atomic norm minimization of claim 3, wherein the SF ISAR one-dimensional range image synthesis is converted into coefficients
Figure FDA0003739055630000028
And normalizing the frequency
Figure FDA0003739055630000029
The n th a Solving and converting the group of distance image information into K forms of distance image information selected from convex hulls conv (A)
Figure FDA00037390556300000210
Wherein A represents an atom
Figure FDA00037390556300000211
A corresponding set of atoms.
5. The SFISAR one-dimensional high-resolution range imaging method based on atomic norm minimization of claim 4, wherein the SF ISAR one-dimensional range image is transformed into the originalWhen the sub-norm is minimized, then n a The group range profile information is represented as:
Figure FDA00037390556300000212
wherein the content of the first and second substances,
Figure FDA00037390556300000213
and with
Figure FDA00037390556300000214
Are all constant coefficients of the number of the magnetic poles,
Figure FDA00037390556300000215
denotes the n-th a The group distance is like the corresponding kth atom,
Figure FDA00037390556300000216
denotes the n-th a The atomic norm of the group echo signals.
6. The SFISAR one-dimensional high-resolution range imaging method based on atomic norm minimization as claimed in claim 5, wherein the preset range direction is a sparse measurement mode, the number of transmitted sub-pulses is M, M is less than or equal to N, the subset consisting of M sub-pulse sequences is represented as Ω, and then
Figure FDA0003739055630000031
And M | Ω | ≦ N, the ANM under sparse sampling conditions is represented as:
Figure FDA0003739055630000032
where τ is related to the noise level, assuming that the noise obeys N (0, σ) 2 I N ) Then, then
Figure FDA0003739055630000033
s Ω (n a ) Randomly measuring data for the range profile;
Figure FDA0003739055630000034
in order to recover the metrology data to be recovered,
Figure FDA0003739055630000035
representing a set of atoms with a subset Ω.
7. The SF ISAR one-dimensional high resolution range imaging method based on atomic norm minimization of claim 6, wherein converting the ANM under sparse sampling condition to semi-positive definite programming is expressed as:
Figure FDA0003739055630000036
wherein the content of the first and second substances,
Figure FDA0003739055630000037
represented as the full data that is recovered,
Figure FDA0003739055630000038
is a constant number u 1 Is a vector
Figure FDA0003739055630000039
The first of the elements in (1) is,
Figure FDA00037390556300000310
representing the conjugate transpose of the data to be recovered.
8. The SF ISAR one-dimensional high-resolution range imaging method based on atomic norm minimization according to claim 7, wherein the SF ISAR adopts an ADMM algorithm to convert ANM into semi-definite programming, wherein the ADMM method is mainly implemented by constructing a lagrange function optimized by semi-definite, and the lagrange function corresponding to the conversion of ANM into semi-definite programming is expressed as:
Figure FDA00037390556300000311
wherein, the Lambda Lagrange multiplier is defined, rho is a penalty parameter, the matrixes Z and Lambda are Hermitian matrixes,<·，·>is the inner product sign, u 1 Is a vector
Figure FDA00037390556300000312
In the first of the elements of (a) to (b),
Figure FDA00037390556300000313
the square value of the 2 norm of the observation data minus the recovery data is represented, and the structural division formula of the introduced matrix Z and Lambda is represented as follows:
Figure FDA00037390556300000314
wherein the matrices Z, Λ are Hermirian matrices, and
Figure FDA0003739055630000041
Figure FDA0003739055630000042
9. the SF ISAR one-dimensional high-resolution range imaging method based on atomic norm minimization of claim 8, wherein ADMM updates several variables separately, for a selected variable to be updated, the other variables are kept unchanged first, and the selected variable is iterated according to the fixed variables, wherein the iteration is represented as follows:
Figure FDA0003739055630000043
wherein, l represents the l iteration process, t l+1 Represents the iteration value, u, of the parameter t at iteration l +1 l+1 Representing the iteration value, Z, of the time vector u at the (l +1) th time l+1 Representing the iteration value, Λ, of the matrix Z at the 1 st iteration l Representing the matrix Λ at the ith iteration l Iteration value of, Z l Representing the iteration value, u, of the matrix Z at the ith iteration l Represents the iteration value of the time vector u at the i-th time,
Figure FDA0003739055630000044
in order to be able to estimate the data,
Figure FDA0003739055630000045
data at l +1 iterations
Figure FDA0003739055630000046
The value of the iteration of (c) is,
based on the iterative process, the iterative computation expression of the parameters is obtained by increasing Lagrange function partial derivatives as follows:
Figure FDA0003739055630000047
wherein omega c Denotes the complement of the set Ω, which is also the set of positions of the sparse echo signals, and e is the first element 1, W ═ diag [1/N,1/(2(N-1)),]is a diagonal matrix, other elements being N-dimensional vectors, s, of 0 Ω In order to measure the data, the measurement data,
Figure FDA0003739055630000048
representing the matrix Z in the first iteration 1 The corresponding measured data of (1) is obtained,
Figure FDA0003739055630000049
representing the matrix Z obtained in the first iteration 1 ，
Figure FDA00037390556300000410
Representing the matrix Λ in the ith iteration 1 Updating the matrix Z according to the corresponding measurement data:
Figure FDA0003739055630000051
10. the SF ISAR one-dimensional high-resolution range imaging method based on atomic norm minimization of claim 9, wherein ADMM sets estimation error or maximum number of iterations as a stagnation condition in update iteration, and obtains the final t, toe (u) and
Figure FDA0003739055630000052
if the rank of the matrix K ═ rank (toe (u)) < N, toe (u)) can be decomposed exclusively as Vandermonde:
Toep(u)＝APA H (16)
wherein A is a Vandermonde matrix and each column thereof is composed of atoms a (f) k ，φ k ) Is composed of, i.e.
Figure FDA0003739055630000053
P is a K x K dimensional positive diagonal matrix, and
Figure FDA0003739055630000054
it can therefore be concluded that the rank K of the toep (u) matrix at this time represents the sparsity of the sparse signal.
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