CN113466864A - Fast joint inverse-free sparse Bayesian learning super-resolution ISAR imaging algorithm - Google Patents

Fast joint inverse-free sparse Bayesian learning super-resolution ISAR imaging algorithm Download PDF

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CN113466864A
CN113466864A CN202110935986.1A CN202110935986A CN113466864A CN 113466864 A CN113466864 A CN 113466864A CN 202110935986 A CN202110935986 A CN 202110935986A CN 113466864 A CN113466864 A CN 113466864A
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CN113466864B (en
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何兴宇
任晓岳
刘桃
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Air Force Engineering University of PLA
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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
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Abstract

The invention relates to the technical field of signal processing, in particular to a fast combined inverse-free sparse Bayesian learning super-resolution ISAR imaging algorithm, which comprises the following steps: s1, initializing a parameter gamma, wherein gamma is a non-negative random initial value; s2, initializing the parameter Z, wherein
Figure DDA0003213161410000011
S3, according to
Figure DDA0003213161410000012
And
Figure DDA0003213161410000013
calculating the mean M and the variance Σ of the posterior probability distribution by
Figure DDA0003213161410000014
And
Figure DDA0003213161410000015
to calculate qα(alpha) and qγ(γ); s4, according to
Figure DDA0003213161410000016
Iteratively updating the parameter Z; s5, and circulating S3 and S4 until M(t)‑M(t‑1)||FDelta is less than or equal to delta, wherein delta is a preset threshold value.

Description

Fast joint inverse-free sparse Bayesian learning super-resolution ISAR imaging algorithm
Technical Field
The invention relates to the technical field of signal processing, in particular to a fast joint inverse-free sparse Bayesian learning super-resolution ISAR imaging algorithm.
Background
An Inverse Synthetic Aperture Radar (ISAR) imaging target is generally sparse in an observation scene, that is, a target image is sparse in the whole background domain, so that sparse reconstruction conditions are met, and imaging can be performed by a sparse reconstruction method. In general, it is difficult to satisfy the requirement of high resolution imaging for broadband and long-time continuous observation of a fixed scene, so the radar often faces the problem of sparse aperture imaging. Under the condition of sparse aperture, the traditional imaging method can cause strong side lobes and grating lobes to appear on the image, and the imaging effect is poor.
When a sparse reconstruction algorithm is used for imaging a moving target, the imaging effect of the algorithm capable of solving the most sparse solution is generally better. Tipping proposes that an original sparse signal is reconstructed through iterative optimization based on a Relevance Vector Machine (RVM) and by a sample learning method based on an SBL. The method is based on sparse probability learning, does not need additional prior information of signals, and is easy to obtain the most sparse solution of the signals, so that the SBL algorithm is widely applied to the fields of signal and image processing, pattern recognition and the like. The SBL-based super-resolution ISAR imaging is researched, a small amount of pulses are used for obtaining an ISAR image of a target, and the SBL-based imaging method has obvious advantages in the aspects of parameter estimation and selection, image reconstruction effect and the like compared with other CS-based imaging methods.
Most sparse signal reconstruction methods are directed to one-dimensional sparse signals, and these methods can be considered as Single Measurement Vector (SMV) reconstruction methods. When the methods are adopted to process two-dimensional signals such as images, the two-dimensional signals need to be vectorized into one-dimensional signals and then reconstructed, the algorithm efficiency can be reduced by the processing, and the reconstruction effect of the two-dimensional sparse signals is general. Most of the existing CS-based ISAR imaging methods perform vectorization operation on reconstructed signals and then complete reconstruction of the signals or perform column-by-column reconstruction on the signals. However, these methods only exploit one-dimensional sparsity of the target image and do not exploit two-dimensional joint sparsity of the image.
Disclosure of Invention
The invention aims to provide a fast combined inverse-free sparse Bayesian learning super-resolution ISAR imaging algorithm, which solves the problems of complexity and large calculation amount of the conventional imaging algorithm.
In order to solve the technical problems, the invention adopts the following technical scheme:
a fast joint inverse-free sparse Bayesian learning super-resolution ISAR imaging algorithm is characterized by comprising the following steps:
s1, initializing a parameter gamma, wherein gamma is a non-negative random initial value;
s2, initializing the parameter Z, wherein
Figure BDA0003213161390000021
S3, according to
Figure BDA0003213161390000022
And
Figure BDA0003213161390000023
calculating the mean M and the variance Σ of the posterior probability distribution by
Figure BDA0003213161390000024
And
Figure BDA0003213161390000025
to calculate qα(alpha) and qγ(γ)；
S4, according to
Figure BDA0003213161390000026
Iteratively updating the parameter Z;
s5, and circulating S3 and S4 until M(t)-M(t-1)||FDelta is less than or equal to delta, wherein delta is a preset threshold value。
A further technical solution is that, before initializing parameters, assuming that a radar transmits a linear radio frequency signal, a received signal can be represented as:
Figure BDA0003213161390000027
the distance compressed signal is represented as:
Figure BDA0003213161390000028
assuming that the number of pulses in the coherent accumulation time is M, the pulse repetition frequency is divided into N doppler cells, and x (τ, t) in the formula (2) is expressed as: x ═ Xnm]N×MApplying the sparse representation theory to the echo distance signal direction, the matrix form of the formula (1) is expressed as: y ═ Φ X + V (3).
A further technical solution is to reconstruct a signal X from the equation (3) by using a bayesian method, and first give X two layers of prior, where in the first layer, X is represented as a gaussian prior distribution characterized by a parameter α, that is:
Figure BDA0003213161390000029
in the second layer, it is assumed that the hyper-parameter α obeys the Gamma distribution, i.e.:
Figure BDA00032131613900000210
meanwhile, it is required to satisfy that the mean value of V noise in the formula (3) is zero, the covariance matrix is (1/γ) I, and in order to realize the estimation of γ through iterative learning, it is required to assume that V obeys Gamma distribution, that is:
p(γ)＝Gamma(γ|c,d)＝Γ(c)-1dcγc-1e-dγ (4)；
order to
Figure BDA0003213161390000031
The hidden variables of the hierarchical model in the formula (4) are expressed, and the variation distribution can be expressed as:
q(θ)＝qX(X)qα(α)qγ(γ)；
suppose q againXThe iterative update of (X) obeys a gaussian distribution function, a joint likelihood function and a prior distribution, and the jth column posterior probability density function of X can be expressed as:
Figure BDA0003213161390000032
the further technical scheme is that the formula (5) is solved by maximizing the unconstrained evidence lower bound, wherein the evidence lower bound form is as follows:
Figure BDA0003213161390000033
introducing a theorem: order to
Figure BDA0003213161390000034
Representing a continuous differentiable function, and having a Lipschitz constant and a Lipschitz continuous gradient, for arbitrary
Figure BDA0003213161390000035
And T ≧ T (f), the following inequality holds:
Figure BDA0003213161390000036
the equations (6) and (7) are combined to obtain the lower bound of unconstrained evidence, which is expressed as follows:
Figure BDA0003213161390000037
the lower bound on unconstrained evidence in equation (8) can be further expressed as:
Figure BDA0003213161390000038
then maximizing the unconstrained evidence lower bound by utilizing a variational expectation maximization algorithm
Figure BDA0003213161390000041
In E-step, the posterior distribution function for each hidden variable is calculated assuming the other variables are constants, and in M-step, q (θ) is made constant and maximized
Figure BDA0003213161390000042
Function with respect to Z.
According to a further technical scheme, the calculation process of the E-step comprises qXIterative update of (X), qαLoop iteration of (alpha) and qγLoop iteration of (γ).
According to a further technical scheme, the q isXIn the iterative update of (X), the posterior probability distribution qXThe iterative update of (X) is represented as:
Figure BDA0003213161390000043
wherein the content of the first and second substances,
Figure BDA0003213161390000044
<αn>denotes alphanWith respect to qα(α) expectation that q can be obtained from the formula (9)X(X) obeys a gaussian distribution with mean and variance:
Figure BDA0003213161390000045
and
Figure BDA0003213161390000046
according to a further technical scheme, the q isαIn the loop iteration of (a),posterior distribution qα(α) is represented by:
Figure BDA0003213161390000047
wherein the content of the first and second substances,
Figure BDA0003213161390000048
is composed of
Figure BDA0003213161390000049
With respect to qXExpectation of (X), XnlThe first element in the n-th row in X, i.e., α, satisfies the Gamma distribution in S3
Figure BDA00032131613900000410
Wherein
Figure BDA00032131613900000411
According to a further technical scheme, the q isγLoop iteration of (gamma) for qγVariation optimization of (γ) can result in:
Figure BDA0003213161390000051
that is, γ satisfies the Gamma distribution in S3
Figure BDA0003213161390000052
Wherein
Figure BDA0003213161390000053
The further technical proposal is that in the M-step, q (theta; Z)old) Bringing in
Figure BDA0003213161390000054
Then the optimization problem can be obtained to obtain S4
Figure BDA0003213161390000055
Let the gradient of the above equation for Z be zero, one obtains:
Figure BDA0003213161390000056
(10) in the formula, TI-2 phiTPhi is positive definite matrix and satisfies T>T(f)＝2λmax(ΦTΦ) where λmax(ΦTPhi) represents phiTMaximum eigenvalue of Φ.
Compared with the prior art, the invention has the beneficial effects that: in the process of iteratively updating the variance Σ in the scheme, although the inverse of the N × N matrix still needs to be calculated, the matrix inversion operation at this time is applied to the diagonal matrix, and the inverse can be quickly obtained.
Drawings
Fig. 1 is a reconstructed MSE graph of each algorithm under different sparsity K, where other parameters are set as follows: m-250, N-500, L-20.
Fig. 2 is a reconstructed MSE graph of each algorithm under different parameters M, where other parameters are set as follows: n-500, L-20, K-120.
Fig. 3 is a reconstructed MSE graph for each algorithm at different snr.
Fig. 4 is a graph of average operation time of each algorithm under different parameters N, and other parameters are set as follows: m is N/2, K is N/10, L is 20, SNR is 20 dB.
FIG. 5 is a diagram of the super-resolution ISAR imaging result of the B727 data by the PC-SBL algorithm.
FIG. 6 is a graph of the super-resolution ISAR imaging results of the M-FOCUSS algorithm on the B727 data.
FIG. 7 is a graph of the super-resolution ISAR imaging results of the M-SBL algorithm on B727 data.
FIG. 8 is a super-resolution ISAR imaging result graph of the B727 data by the algorithm of the present invention.
Fig. 9 is an MSE plot of super-resolution ISAR imaging of B727 with different algorithms.
FIG. 10 is a time contrast plot of super-resolution ISAR imaging for B727 with different algorithms.
FIG. 11 is a diagram of the result of super-resolution ISAR imaging of the Yak-42 by the PC-SBL algorithm.
FIG. 12 is a graph of the results of the M-FOCUSS algorithm on the super-resolution ISAR imaging of Yak-42.
FIG. 13 is a diagram of the result of super-resolution ISAR imaging of the Yak-42 by the M-SBL algorithm.
FIG. 14 is a graph of the super-resolution ISAR imaging results of the algorithm of the present invention on Yak-42.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
Example (b):
fig. 1 to 14 show a preferred embodiment of the fast joint inverse-free sparse bayesian learning super-resolution ISAR imaging algorithm of the present invention, and the fast joint inverse-free sparse bayesian learning super-resolution ISAR imaging algorithm in this embodiment specifically includes the following steps:
s1, initializing a parameter gamma, wherein gamma is a non-negative random initial value;
s2, initializing the parameter Z, wherein
Figure BDA0003213161390000061
S3, according to
Figure BDA0003213161390000062
And
Figure BDA0003213161390000063
calculating the mean M and the variance Σ of the posterior probability distribution by
Figure BDA0003213161390000064
And
Figure BDA0003213161390000065
to calculate qα(alpha) and qγ(γ)；
S4, according to
Figure BDA0003213161390000066
Iteratively updating the parameter Z;
s5, and circulating S3 and S4 until M(t)-M(t-1)||FDelta is less than or equal to delta, wherein delta is a preset threshold value.
Before initializing the parameters, assuming that the radar transmits a linear rf signal, the received signal can be expressed as:
Figure BDA0003213161390000067
the distance compressed signal is represented as:
Figure BDA0003213161390000071
where c represents the electromagnetic wave propagation velocity, ω0Angular velocity of rotation, R, being uniform rotation0Denotes the distance from the center of the rotating shaft to the radar, tau is the fast time, tmIs a slow time, TpDenotes the pulse width, fcFor the carrier frequency, μ represents the tuning frequency. B denotes the transmission signal bandwidth, AkDenotes the k-th scattering center Pk(xk,yk) Scattering coefficient of (1), TaRepresents the coherent integration time, and K represents the number of scattering points.
Assuming that the number of pulses in the coherent accumulation time is M, the pulse repetition frequency is divided into N doppler cells, and x (τ, t) in equation (2) is represented as: x ═ Xnm]N×MApplying the sparse representation theory to the echo distance signal direction, the matrix form of the formula (1) is expressed as: y ═ Φ X + V (3).
Wherein the content of the first and second substances,
Figure BDA0003213161390000072
is a matrix of y (tau, t) in the formula (1)In the form of a sheet of paper,
Figure BDA0003213161390000073
representing a noise matrix, sparse dictionary phiN×NCan be expressed as:
Figure BDA0003213161390000074
if it is not
Figure BDA0003213161390000075
Then
Figure BDA0003213161390000076
Representing a super-resolved range image.
Matrix XTCan be expressed in the following form: xT＝FA；
Wherein A ═ amn]To represent
Figure BDA0003213161390000077
The matrix is a two-dimensional super-resolution ISAR image of the target, and the element values in the matrix represent the scattering amplitude of scattering points. Parameter(s)
Figure BDA0003213161390000078
And
Figure BDA0003213161390000079
shows the super-resolution multiple and parameter of the range profile
Figure BDA00032131613900000710
And
Figure BDA00032131613900000711
the super-resolution multiple of the azimuth direction is shown.
Figure BDA00032131613900000712
Represents a partial fourier matrix, i.e.:
Figure BDA00032131613900000713
after the joint reconstruction of the range profile is completed, the X is solved by the traditional CS reconstruction methodTThe ISAR image of the target is acquired for the FA reconstruction problem. Suppose that the noise V obeys a mean of 0 and a variance of σ2Multivariate Gaussian distribution of I.
Reconstructing a signal X from the equation (3) by using a Bayesian method, firstly giving X two layers of prior, wherein in the first layer, X is expressed as a Gaussian prior distribution characterized by a parameter alpha, namely:
Figure BDA0003213161390000081
wherein the content of the first and second substances,
Figure BDA0003213161390000082
for non-negative superparameters controlling the prior variance of each line of X, X(n)And represents the nth row of X.
In the second layer, it is assumed that the hyper-parameter α obeys the Gamma distribution, i.e.:
Figure BDA0003213161390000083
wherein the content of the first and second substances,
Figure BDA0003213161390000084
representing a Gamma function. The parameter b is usually chosen to be a very small value, e.g. 10-4In contrast, the parameter a is usually selected to be a larger value, and a e [0,1 ] is generally selected]。
Meanwhile, it is required to satisfy that the mean value of V noise in the formula (3) is zero, the covariance matrix is (1/γ) I, and in order to realize the estimation of γ through iterative learning, it is required to assume that V obeys Gamma distribution, that is:
p(γ)＝Gamma(γ|c,d)＝Γ(c)-1dcγc-1e-dγ (4)；
order to
Figure BDA0003213161390000085
The hidden variables of the hierarchical model in the formula (4) are expressed, and the variation distribution can be expressed as:
q(θ)＝qX(X)qα(α)qγ(γ)；
suppose q againXThe iterative update of (X) obeys a gaussian distribution function, a joint likelihood function and a prior distribution, and the jth column posterior probability density function of X can be expressed as:
Figure BDA0003213161390000086
wherein the mean and variance can be expressed as follows:
Figure BDA0003213161390000087
and
Figure BDA0003213161390000088
wherein the content of the first and second substances,
Figure BDA0003213161390000089
Σt＝γI+ΦDΦTfrom this equation 2, it can be seen that in the iteration process of the posterior distribution based on the M-SBL algorithm, the inverse of the mxm matrix needs to be calculated each time. Thus, the computational complexity of the variational M-SBL algorithm is approximately O (M)3). Such high computational complexity may limit its application to many of the large data volume problems that need to be handled.
In the scheme, the formula (5) is solved by maximizing the unconstrained evidence lower bound, and the evidence lower bound form is as follows:
Figure BDA0003213161390000091
introducing a theorem: order to
Figure BDA0003213161390000092
Indicating continuous microminiaturizationFunction, and there are Lipschitz constants and Lipschitz continuous gradients, for arbitrary
Figure BDA0003213161390000093
And T ≧ T (f), the following inequality holds:
Figure BDA0003213161390000094
according to the above theorem, the lower bound of p (Y | X, γ) can be obtained by the following equation:
Figure BDA0003213161390000095
wherein the content of the first and second substances,
Figure BDA0003213161390000096
when the inequality relationship in the equation (a) holds for any Z, and when Z is X, the inequality in the equation becomes an equation.
The equations (6) and (7) are combined to obtain the lower bound of unconstrained evidence, which is expressed as follows:
Figure BDA0003213161390000097
wherein the content of the first and second substances,
Figure BDA0003213161390000098
the lower bound on unconstrained evidence in equation (8) can be further expressed as:
Figure BDA0003213161390000099
wherein the content of the first and second substances,
Figure BDA00032131613900000910
h (Z) can be represented as:
Figure BDA0003213161390000101
the unconstrained evidence lower bound is then maximized by utilizing the variance Expectation Maximization (EM) algorithm
Figure BDA0003213161390000102
In E-step, the posterior distribution function for each hidden variable is calculated assuming the other variables are constants, and in M-step, q (θ) is made constant and maximized
Figure BDA0003213161390000103
Function with respect to Z.
The calculation process of E-step includes qXIterative update of (X), qαLoop iteration of (alpha) and qγLoop iteration of (γ).
qXIn the iterative update of (X), the posterior probability distribution qXThe iterative update of (X) is represented as:
Figure BDA0003213161390000104
wherein the content of the first and second substances,
Figure BDA0003213161390000105
<αn>denotes alphanWith respect to qα(α) expectation that q can be obtained from the formula (9)X(X) obeys a gaussian distribution with mean and variance:
Figure BDA0003213161390000106
and
Figure BDA0003213161390000107
wherein < γ > represents the expectation of γ.
qαIn the loop iteration of (alpha), the posterior distribution qα(α) is represented by:
Figure BDA0003213161390000108
wherein the content of the first and second substances,
Figure BDA0003213161390000109
is composed of
Figure BDA00032131613900001010
With respect to qXExpectation of (X), XnlThe first element in the n-th row in X, i.e., α, satisfies the Gamma distribution in S3
Figure BDA00032131613900001011
Wherein
Figure BDA00032131613900001012
qγLoop iteration of (gamma) for qγVariation optimization of (γ) can result in:
Figure BDA0003213161390000111
that is, γ satisfies the Gamma distribution in S3
Figure BDA0003213161390000112
Wherein
Figure BDA0003213161390000113
In summary, E-step mainly implements iterative update of the posterior probability distribution of the hidden variables X, α, and γ. In the iterative update process, some parameters are given by:
Figure BDA0003213161390000114
Figure BDA0003213161390000115
wherein Tr (A) represents a trace of a square matrix A, Σn,nRepresenting the nth diagonal element of the matrix Σ.
In M-step, q (theta; Z)old) Bringing in
Figure BDA0003213161390000116
Then the optimization problem can be obtained to obtain S4
Figure BDA0003213161390000117
Let the gradient of the above equation for Z be zero, one obtains:
Figure BDA0003213161390000118
(10) the derivation in the formula is true because of TI-2 phiTPhi is positive definite matrix and satisfies T>T(f)＝2λmax(ΦTΦ) where λmax(ΦTPhi) represents phiTMaximum eigenvalue of Φ.
In summary, assume that the observed data is Y and the constructed sparse dictionary is Φ.
In the process of iteratively updating the variance Σ in the scheme, although the inverse of the N × N matrix still needs to be calculated, the matrix inversion operation at this time is applied to the diagonal matrix, and the inverse can be quickly obtained.
The advantages of the algorithm are demonstrated by several experiments as follows:
the first experimental example:
firstly, the effectiveness of the algorithm of the scheme is verified by using a simulation signal. Parameters a, b, u and v are all set to 10-6. Setting the parameter T to a value slightly larger than the Lipschitz constant, and making T ═ lambdamax(2ΦTΦ)+10-6. In the experiment, the recipe is comparedThe reconstruction performance of the algorithm of the scheme, a PC-SBL algorithm, a continuous block original dual active set with continuous activation algorithm, a GPDASC algorithm, an M-FOCUSS algorithm, an M-SBL algorithm and a T-MSBL algorithm. In the simulation experiment, L sparse vectors of the original data all contain K nonzero elements, and the positions of the nonzero elements of all the vectors are the same. Meanwhile, the non-zero bin positions are random, with amplitudes following a standard normal distribution. Matrix array
Figure BDA0003213161390000121
Is a Gaussian random matrix, and each vector element in the matrix is subjected to independent and identically distributed standard Gaussian distribution. The experiments are all 100 Monte Carlo simulation results.
Quantitative analysis of reconstruction performance of each algorithm is performed by using mean square error, which is defined as MSE | | | R-R0||F/||R0||FWherein R is0The full aperture ISAR image data matrix obtained by the RD algorithm is represented, and R represents the sparse aperture imaging data matrix obtained by other three different algorithms.
As can be seen from fig. 1-3, the PC-SBL algorithm and the GPDASC algorithm have relatively poor reconstruction effect, mainly because the two algorithms require that the signals are all block sparse in all directions and require knowledge of part of a priori information. The reconstruction effect of other algorithms is approximate, and the original signal can be reconstructed well. Here, the CPU average computation time is also used to measure the computational complexity of each algorithm. As can be seen from fig. 4, the algorithm of the present solution avoids the matrix inversion operation, so that the computation complexity is the lowest, and the computation time is the shortest. Therefore, the algorithm performance is best by comprehensively considering the algorithm reconstruction effect and the operation complexity.
Experiment example two:
the effectiveness of the algorithm of the scheme is verified by carrying out ISAR imaging experiments by utilizing the wave sound 727 simulation data, a radar transmits a linear frequency modulation signal, the carrier frequency is 9GHz, the signal bandwidth is 150MHz, and the pulse repetition frequency is 20 KHz. The values of the parameters a, b, u, v and T were set to be consistent with those in experiment one,
Figure BDA0003213161390000122
is a partial fourier matrix. Firstly, the ISAR super-resolution imaging is realized by using 128 pulses, as shown in FIGS. 5 to 8, since the T-MSBL algorithm has a large reconstruction error when reconstructing a complex signal, and the GPDASC algorithm mainly aims at the reconstruction problem of a block sparse signal, the results of reconstructing ISAR images of the two algorithms are not continuously given here, and are only compared with the other three algorithms in the first experimental example.
As shown in fig. 5-8, compared with other algorithms, the algorithm reconstruction in the present scheme has a higher resolution of the obtained ISAR image and a good focusing performance. In order to quantitatively analyze the imaging effect of each algorithm, the reconstructed mean square error of each algorithm is also analyzed and calculated. As can be seen from fig. 9-10 (all 200 monte carlo experimental results), the reconstructed ISAR image MSE of the algorithm of the present scheme is the minimum. In addition, the average operation time of each algorithm is compared, the operation time of the algorithm of the scheme is the shortest, and the calculation complexity of the algorithm of the scheme is proved to be smaller than that of other algorithms.
Experiment example three:
the validity of the algorithm is verified by using the Yak-42 measured data, the radar transmits linear frequency modulation signals as well, the center frequency is 10GHz, the signal bandwidth is 400MHz, the pulse repetition frequency is 100Hz, and 256 pulses are acquired within the coherent accumulation time of 2.56 s. The values of the parameters a, b, u, v and T are consistent with the settings of the parameters in experiment one.
Figure BDA0003213161390000131
Is a partial fourier matrix. As shown in fig. 11-14, ISAR high resolution imaging is achieved with 128 pulses.
Compared with other algorithms, the ISAR image reconstructed by the algorithm is high in resolution and good in focusing performance. The reconstructed mean square error of each algorithm is calculated through the same comparison analysis to quantitatively analyze the imaging effect of each algorithm, and as can be seen from fig. 11-14, the reconstructed ISAR image MSE of the algorithm of the scheme is minimum, the average operation time of each algorithm is compared, which is the same result of 200 monte carlo experiments, and it can be seen that the operation time required by the algorithm of the scheme is shortest compared with other bayesian learning algorithms, and the superiority of the algorithm of the scheme in the aspect of computational complexity is further proved.
Although the invention has been described herein with reference to a number of illustrative embodiments thereof, it should be understood that numerous other modifications and embodiments can be devised by those skilled in the art that will fall within the spirit and scope of the principles of this disclosure. More specifically, various variations and modifications are possible in the component parts and/or arrangements of the subject combination arrangement within the scope of the disclosure, the drawings and the appended claims. In addition to variations and modifications in the component parts and/or arrangements, other uses will also be apparent to those skilled in the art.

Claims (9)

1. A fast joint inverse-free sparse Bayesian learning super-resolution ISAR imaging algorithm is characterized by comprising the following steps:
s1, initializing a parameter gamma, wherein gamma is a non-negative random initial value;
s2, initializing the parameter Z, wherein
Figure FDA0003213161380000011
S3, according to
Figure FDA0003213161380000012
And
Figure FDA0003213161380000013
calculating the mean M and the variance Σ of the posterior probability distribution by
Figure FDA0003213161380000014
And
Figure FDA0003213161380000015
to calculate qα(alpha) and qγ(γ)；
S4, according to
Figure FDA0003213161380000016
Iteratively updating the parameter Z;
s5, and circulating S3 and S4 until M(t)-M(t-1)||FDelta is less than or equal to delta, wherein delta is a preset threshold value.
2. The fast joint inverse-free sparse Bayesian learning super-resolution ISAR imaging algorithm as recited in claim 1, wherein: before initializing the parameters, assuming that the radar transmits a linear rf signal, the received signal can be expressed as:
Figure FDA0003213161380000017
the distance compressed signal is represented as:
Figure FDA0003213161380000018
assuming that the number of pulses in the coherent accumulation time is M, the pulse repetition frequency is divided into N doppler cells, and x (τ, t) in the formula (2) is expressed as: x ═ Xnm]N×MApplying the sparse representation theory to the echo distance signal direction, the matrix form of the formula (1) is expressed as: y ═ Φ X + V (3).
3. The fast joint inverse-free sparse Bayesian learning super-resolution ISAR imaging algorithm as recited in claim 2, wherein: reconstructing a signal X from the equation (3) by using a Bayesian method, firstly giving X two layers of prior, wherein in the first layer, X is expressed as a Gaussian prior distribution characterized by a parameter alpha, namely:
Figure FDA0003213161380000019
in the second layer, it is assumed that the hyper-parameter α obeys the Gamma distribution, i.e.:
Figure FDA0003213161380000021
meanwhile, it is required to satisfy that the mean value of V noise in the formula (3) is zero, the covariance matrix is (1/γ) I, and in order to realize the estimation of γ through iterative learning, it is required to assume that V obeys Gamma distribution, that is:
p(γ)＝Gamma(γ|c,d)＝Γ(c)-1dcγc-1e-dγ (4)；
order to
Figure FDA0003213161380000022
The hidden variables of the hierarchical model in the formula (4) are expressed, and the variation distribution can be expressed as:
q(θ)＝qX(X)qα(α)qγ(γ)；
suppose q againXThe iterative update of (X) obeys a gaussian distribution function, a joint likelihood function and a prior distribution, and the jth column posterior probability density function of X can be expressed as:
Figure FDA0003213161380000023
4. the fast joint inverse-free sparse Bayesian learning super-resolution ISAR imaging algorithm as recited in claim 3, wherein: solving equation (5) by maximizing the unconstrained evidence lower bound, which is in the form:
Figure FDA0003213161380000024
introducing a theorem: order to
Figure FDA0003213161380000025
Representing a continuous differentiable function, and having a Lipschitz constant and a Lipschitz continuous gradient, for arbitrary
Figure FDA0003213161380000026
And T ≧ T (f), the following inequality holds:
Figure FDA0003213161380000027
the equations (6) and (7) are combined to obtain the lower bound of unconstrained evidence, which is expressed as follows:
Figure FDA0003213161380000028
the lower bound on unconstrained evidence in equation (8) can be further expressed as:
Figure FDA0003213161380000031
then maximizing the unconstrained evidence lower bound by utilizing a variational expectation maximization algorithm
Figure FDA0003213161380000032
In E-step, the posterior distribution function for each hidden variable is calculated assuming the other variables are constants, and in M-step, q (θ) is made constant and maximized
Figure FDA0003213161380000033
Function with respect to Z.
5. The fast joint inverse-free sparse Bayesian learning super-resolution ISAR imaging algorithm as recited in claim 4, wherein: the calculation process of the E-step comprises qXIterative update of (X), qαLoop iteration of (alpha) and qγLoop iteration of (γ).
6. The fast joint inverse-free sparse Bayesian learning super-resolution ISAR imaging algorithm as recited in claim 5, wherein the ISAR imaging algorithm is characterized in that: q is a number ofXIn the iterative update of (X), the posterior probability distribution qXThe iterative update of (X) is represented as:
Figure FDA0003213161380000034
wherein the content of the first and second substances,
Figure FDA0003213161380000035
<αn>denotes alphanWith respect to qα(α) expectation that q can be obtained from the formula (9)X(X) obeys a gaussian distribution with mean and variance:
Figure FDA0003213161380000036
and
Figure FDA0003213161380000037
7. the fast joint inverse-free sparse Bayesian learning super-resolution ISAR imaging algorithm as recited in claim 5, wherein: q is a number ofαIn the loop iteration of (alpha), the posterior distribution qα(α) is represented by:
Figure FDA0003213161380000038
wherein the content of the first and second substances,
Figure FDA0003213161380000039
is composed of
Figure FDA00032131613800000310
With respect to qXExpectation of (X), XnlThe first element in the n-th row in X, i.e., α, satisfies the Gamma distribution in S3
Figure FDA0003213161380000041
Wherein
Figure FDA0003213161380000042
8. The fast joint inverse-free sparse Bayesian learning super-resolution ISAR imaging algorithm as recited in claim 5, wherein: q is a number ofγLoop iteration of (gamma) for qγVariation optimization of (γ) can result in:
Figure FDA0003213161380000043
that is, γ satisfies the Gamma distribution in S3
Figure FDA0003213161380000044
Wherein
Figure FDA0003213161380000045
9. The fast joint inverse-free sparse Bayesian learning super-resolution ISAR imaging algorithm as recited in claim 4, wherein: in the M-step, q (theta; Z)old) Bringing in
Figure FDA0003213161380000046
Then the optimization problem can be obtained to obtain S4
Figure FDA0003213161380000047
Let the gradient of the above equation for Z be zero, one obtains:
Figure FDA0003213161380000048
(10) in the formula, TI-2 phiTPhi is positive definite matrix and satisfies T>T(f)＝2λmax(ΦTΦ) where λmax(ΦTPhi) represents phiTMaximum eigenvalue of Φ.
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