Regular Articles

Radar coincidence imaging with phase error using Bayesian hierarchical prior modeling

[+] Author Affiliations
Xiaoli Zhou, Hongqiang Wang, Yongqiang Cheng, Yuliang Qin

National University of Defense Technology, Research Institute of Space Electronics Technology, College of Electronic Science and Engineering, 157 Deya Road, Changsha 410073, China

J. Electron. Imaging. 25(1), 013018 (Feb 01, 2016). doi:10.1117/1.JEI.25.1.013018
History: Received July 27, 2015; Accepted January 7, 2016
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Abstract.  Radar coincidence imaging (RCI) is a high-resolution imaging technique without the limitation of relative motion between target and radar. In sparsity-driven RCI, the prior knowledge of imaging model requires to be known accurately. However, the phase error generally exists as a model error, which may cause inaccuracies of the model and defocus the image. The problem is formulated using Bayesian hierarchical prior modeling, and the self-calibration variational message passing (SC-VMP) algorithm is proposed to improve the performance of RCI with phase error. The algorithm determines the phase error as part of the imaging process. The scattering coefficient and phase error are iteratively estimated using VMP and Newton’s method, respectively. Simulation results show that the proposed algorithm can estimate the phase error accurately and improve the imaging quality significantly.

Figures in this Article

Radar coincidence imaging (RCI), motivated by classical coincidence imaging in optical systems, is a novel staring imaging technique.13 RCI can obtain focused high-resolution image without the limitation of target relative motion and operate under the observing geometry of forward-looking/staring, with significant potentials for resolution enhancement, interference, and jamming suppression. In RCI, the temporal-spatial stochastic waveforms are transmitted, thus the spatial variety of wavefront is increased, so the super-resolution within a beam emerges.

In RCI, sparse recovery is widely used as the scatterers of target, which are often distributed sparsely in many radar imaging applications. Solving the linear inverse problem with a sparsity constraint depends on the perfect prior knowledge of the system. However, phase error among the transmitter–receiver pairs exists generally since perfect synchronization is impossible for a multitransmitter configuration in RCI,4 because of plenty of factors, such as isolated local oscillators. Phase error results in the dictionary mismatch and induces the performance to degrade considerably, as the imaging performance depends on presetting an appropriate sparsifying dictionary based on an accurate prior known model.

Various studies have been presented on phase error. The Cramer–Rao bound for MIMO radar target localization with phase errors has been derived.5,6 To compensate the phase error, several eigenstructure-based methods are proposed.711 These methods are less sensitive to phase error but lack adaptation to demanding scenarios with low signal-to-noise ratio (SNR), limited snapshots, and spatially adjacent sources.12 Recently, sparse recovery and compressive sensing13 are introduced into signal processing by exploiting the sparsity. A sparsity-driven iterative method for joint synthetic aperture radar (SAR) imaging and phase error correction in a nonquadratic regularization-based framework is proposed in Ref. 14. The method cycles through steps of sparse recovery and phase error compensation, and the idea is widely used in SAR/inverse synthetic aperture radar (ISAR) imaging and MIMO radar imaging with phase error. Inspired by the idea of sparsity-driven iterative method introduced in Ref. 14, a sparse autocalibration imaging (SACI) method is presented to solve the RCI with gain-phase error.3 From the Bayesian statistics perspective, sparse imaging via expectation maximization algorithm and sparse self-calibration method via iterative minimization (SSCIM) algorithm are proposed, respectively, to alleviate the influence of phase synchronization mismatch by exploiting the maximum a posterior (MAP) criterion.15,16 Likewise, MAP estimator is also used into ISAR imaging by exploiting the sparseness prior of ISAR image,1719 while the phase error is corrected via modified quasi-Newton algorithm. To realize array calibration and direction-of-arrival estimation, a unified framework based on sparse Bayesian learning (SBL) is formulated and a sparse Bayesian array calibration method is then proposed in Ref. 12. Using variational Bayesian inference (VBI), an array autocalibration SBL algorithm in the full conjugate Bayesian framework is proposed to achieve DOA estimation with gain/phase errors in Ref. 20. In Ref. 21, an autocalibration algorithm via variational Bayesian expectation maximization is presented to solve the problem of multiplicative perturbation.

In this paper, we focus on the sparsity-driven RCI with phase error and propose a self-calibration variational message passing (SC-VMP) algorithm in SBL framework. The merit of SBL is its flexibility in modeling sparse signals that can not only promote the sparsity but also exploit the possible structure of the signal to be recovered.22 As exact Bayesian inference is typically intractable, approximations are needed, such as evidence procedure23 and VMP.24 Using Bayesian hierarchical prior, the scattering coefficient is assigned an appropriate prior, i.e., three-level (3-L) hierarchical Gaussian-Gamma-Gamma (G-Ga-Ga) prior. Then, we propose a self-calibration imaging algorithm for joint imaging and phase error calibration in SBL framework. The algorithm involves an iterative method, which cycles through steps of target reconstruction and phase error estimation, where VMP and Newton’s method are adopted, respectively. Numerical simulations show that the algorithm realizes the imaging robustly and achieves both high resolution and outstanding imaging quality in the presence of phase error and is also simple to implement without changing the algorithm parameters.

The rest of the paper is organized as follows. In Sec. 2, the RCI model with phase error in the range-azimuth space is presented. Section 3 presents the SC-VMP algorithm in detail. In Sec. 4, the performance of the proposed method is verified by numerical examples. Finally, Sec. 5 concludes the paper.

In this paper, RCI can be realized by a multitransmitter configuration to transmit time-independent and group-orthogonal waveforms.1 Consider a RCI system with M transmitters and one receiver, each transmitter emits an independent stochastic waveform. The RCI geometry is illustrated in Fig. 1. The imaging plane is a range-azimuth space. In sparsity-driven RCI, the continuous imaging plane is discretized to generate K small grid-cells with uniform size and shape. Denote βk by the scattering coefficient of the scattering center exactly located at the prediscretized k’th grid-cell center, i.e., rk and βk=0 for the grid-cell without scattering center.

As the backscattering of a radar target in the high-frequency region can be approximated as coming from a few dominant scattering centers,25 the target is assumed to be composed of a very limited amount of strong scattering centers, the number of which is much smaller than that of grid cells in the image plane.19 Thus, the RCI image is spatially sparse, which can be exploited to achieve super-resolution, denoising, and feature extraction.26

The echo is a linear combination of all the scatterers’ reflected waveforms from all the transmitters. Considering the phase error, the echo at the receiver can be expressed as Display Formula

y(t)=kSm=1MβkejϕmStm(tτmk)+w(t),(1)
where Stm(t) is the signal emitted by the m’th transmitter, ϕm is the phase error between the m’th transmitter and the receiver pair. w(t), an independent complex Gaussian random process, denotes the noise at the receiver. τmk is the propagation delay corresponding to the m’th transmitter and receiver with respect to the k’th scatterer. In addition, the RCI formula needs a detecting signal,1 which is simply structured as Display Formula
S(t,rk)=m=1MejϕmStm(tτmk).(2)

Thus, the echo can be expressed as the superposition of the detecting signals, i.e., y(t)=kSβkS(t,rk). After sampling the echo, the imaging equation can be given as follows: Display Formula

y=S·β+w[y(t1)y(t2)y(tN)]=[S(t1,r1)S(t1,r2)S(t1,rK)S(t2,r1)S(t2,r2)S(t2,rK)S(tN,r1)S(tN,r2)S(tN,rK)]·[β1β2βK]+[w(t1)w(t2)w(tN)],(3)
where N is the number of samples, S is the dictionary in sparse recovery framework, y, w and β are the echo, noise, and unknown scattering coefficient vector, respectively.

Under the assumption of sparse prior of the target, the imaging model shown in Eq. (3) is a typical linear model used in most sparse recovery applications. However, the dictionary S obtained from practical applications is inevitably disturbed by perturbations, which cannot be known accurately in practice, e.g., phase error. Then, S can be rewritten as S(ϕ) involving the phase error ϕ, where ϕ=[ϕ1,,ϕM]T. Then, Eq. (3) can be rewritten as Display Formula

y=S(ϕ)·β+w.(4)

Involving the generally unknown parameter ϕ, S(ϕ) is a parameterized unknown dictionary and then β could not be reconstructed directly based on the conventional sparse recovery algorithms. Therefore, a self-calibration imaging algorithm, which is presented in the following section, is proposed to solve the problem.

Conventional sparsity-driven radar imaging methods assume that the model contains no error and the dictionary is precisely known, which is not realizable in most application scenarios. In the presence of phase error, the structure of the dictionary is destroyed, which leads to the direct use of sparse recovery methods failure. In this section, the SC-VMP algorithm calibrating the phase error is presented. This algorithm is formulated in a Bayesian hierarchical prior model, and works by jointly reconstructing the target and estimating the phase error, where VMP and Newton’s method are adopted, respectively.

Hierarchical Prior Model

The hierarchical representation of sparsity-inducing prior allows us to choose simple and analytically tractable probability density functions (PDFs), and results in the construction of efficient yet computationally tractable iterative inference algorithms with analytical derivation of the inference expressions.27

The graphical representation of RCI model priors is shown in Fig. 2. To recover β in Eq. (4), we model it as hidden (unobserved) variable and assign it a 3-L hierarchical G-Ga-Ga prior typically to induce sparsity: Display Formula

p(β|α)=CN(β|0,Λ)=1πKdet(Λ)exp{βHΛ1β},(5)
Display Formula
p(α|η;ϵ)=k=1KΓ(αk|ϵ,η),(6)
Display Formula
p(η;a,b)=Γ(η|a,b),(7)
where CN(·|μ,Σ) denotes the complex Gaussian distribution with mean vector μ and covariance matrix Σ. α=[α1,α2,,αK]T, Λ=diag(α), αk is the prior variance of βk and Γ(αk|a,b) is the gamma distribution with parameters a and b. Then, 3-L prior model involves three free parameters, i.e., ϵ, a, and b. Here, we set ϵ=0, a=1, and b=106, which strongly promotes a sparse estimate and make the prior for η noninformative.27

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Fig. 2
F2 :

A graphical representation of the model priors.

Compared with the two-layer (2-L) prior model, the 3-L prior model considers η as random variable specified by a distribution and incorporates it into the inference framework. This leads to additional degrees of freedom in controlling the sparsity property of the resulting inference scheme.27

In addition, the measurement noise is assumed to be complex Gaussian with zero-mean and variance α01, to allow conjugate-exponential analysis. Then, the following 2-L hierarchical Gaussian-Gamma (G-Ga) prior is assigned: Display Formula

p(w|α0)=CN(w|0,α01I)=(πα01)Kexp{α0w22},(8)
Display Formula
p(α0;c,d)=Γ(α0|c,d).(9)

Encoded by the graphical model shown in Fig. 2, the joint PDF of the model [Eq. (4)] is of the form: Display Formula

p(y,β,α0,α,η;ϕ)=p(y|β,α0;ϕ)p(α0)p(β|α)p(α|η)p(η).(10)

Variational Message Passing

In this part, we present the VMP algorithm to estimate β. Denote Ω={β,α,η,α0} as the set of hidden variables. In Bayesian inference framework, the exact posterior distribution p(Ω|y;ϕ)=p(y,Ω;ϕ)/p(y) is intractable since p(y) cannot be expressed explicitly. VBI28 could be used to find a tractable distribution q(Ω) that closely approximates the true posterior distribution p(Ω|y;ϕ) by minimizing the Kullback–Leibler divergence (KLD) between them.29 A structured mean field approximation over p(Ω|y;ϕ) is further assumed as Display Formula

q(Ω)=q(β)q(α)q(η)q(α0).(11)
VBI is carried out using VMP24 in this paper. VMP is an iterative scheme that uses a message passing procedure on a graphical model and attempts to compute the auxiliary PDF. The variational distributions {q(β),q(α),q(η),q(α0)} are iteratively updated in VMP procedure to monotonically decrease the KLD and thus, the convergence is guaranteed.

The updates of {q(β),q(α),q(η),q(α0)} are similar to those in Ref. 27, the derivation detail is shown in Appendix 1. It can be concluded that the posterior distributions of β and αk are complex Gaussian distribution and generalized inverse Gaussian (GIG) distribution, respectively, while both η and α0 obey a Gamma distribution.

Based on the derivation of {q(β),q(α),q(η),q(α0)}, the optimal approximated distribution q(Ω) can be obtained by iteratively calculating Eqs. (19), (20), (23), (25), and (26) until convergence.

Phase Error Estimation

In the presence of phase error, the convergence of SC-VMP is not a direct result of VMP. Consider the phase error ϕ as an unknown deterministic parameter as it is not varying generally during the entire coherent processing interval.6 Then, the resulting algorithm can be interpreted as a variational EM algorithm.28

In VMP framework, the phase error can be estimated from the maximum likelihood: Display Formula

ϕ=argmaxϕlnp(y,Ω;ϕ)q(β)q(α0)q(α)q(η),(12)
where p(x)q(x) denotes the expectation of p(x) with respect to a density q(x). After simplification, Eq. (12) can be rewritten as ϕ=argminϕ{yS(ϕ)β22} and then changed to Display Formula
ϕ=argminϕ{yS(ϕ)μ22+tr([S(ϕ)]HS(ϕ)Σ)}.(13)
Define f(ϕ)=yS(ϕ)μ22+tr{[S(ϕ)]HS(ϕ)Σ} as the objective function. Clearly, Eq. (13) is a nonlinear least-squares problem, which is not tractable to obtain the closed-form expression. To solve the problem, we resort to Newton’s method to update the parameter,30 which reveals the behavior of phase error with no approximation being required. The updated ϕ estimate is Display Formula
ϕ=ϕ[2f(ϕ)]1[f(ϕ)],(14)
where f(ϕ) and 2f(ϕ) represent the gradient and Hessian with respect to the phase error, respectively. The derivations of f(ϕ) and 2f(ϕ) are shown in Appendix 2.

Algorithm Description

In SC-VMP procedure, each iteration consists of matrix inversion and matrix-vector multiplication in Eqs. (19) and (20), which leads to expensive computational burden in case of large number of grid cells. Besides, VMP suffers from low rate of convergence. Hence, a fast implementing approach is developed.

In conventional SBL, many of the prior precisions of coefficients, i.e., αk1q(α), typically take on quite large values upon convergence, which implies that the corresponding coefficients are quite small and in turn the contributions of the corresponding bases could be negligible. Thus, the corresponding bases could be removed from the model to realize sparsity when αk1q(α) exceeds a certain large threshold αth. Then, the computational complexity of SC-VMP can be drastically reduced. Therefore, to speed up the algorithm, the pruning of the current basis set Θi can be achieved via Display Formula

Θi+1={k|αk1q(α)<αth,kΘi},(15)
where Θi+1 denotes the basis set after pruning. Because of basis pruning, the computational burden is reduced, the convergence speed is improved, and the sparsity of the reconstruction results is enhanced.

In addition, we terminate the algorithm if μi+1μi2/μi2<γ or the maximum number of iterations Imax is reached, where γ is a user-defined tolerance and the superscript i refers to the iteration index.

Based on the above discussions, the procedure of SC-VMP algorithm is given in Algorithm 1.

Table Grahic Jump Location
Algorithm 1SC-VMP algorithm.
Discussion

After investigating the SC-VMP algorithm in depth, we provide some discussions to gain insight into the algorithm.

  1. Compared with related work: Similar with most of the sparsity-based methods for radar imaging with phase error presented in Refs. 3, 12, 1415161718192021, the proposed SC-VMP algorithm also employs the alternating strategy to update the sparse scattering coefficients and phase error iteratively. The methods listed in Refs. 3, 141516171819 can be summarized as l1-based regularization method, where a sparsity-inducing Laplace prior is directly imposed on the signal, then the sparse solution is exploited from the MAP estimation, which corresponds to the point estimate of the sparse scattering coefficient. Comparably, the SC-VMP utilizes the hierarchical modeling procedure to encode signal sparsity and achieve better sparse solutions, which could obtain the approximate posterior distribution and is regarded as a full Bayesian method. During the procedure, the statistical information is used to enhance the estimation performance and avoid converging to a shallow local minimum, due to the utilization of higher order statistical information (such as the estimation covariance matrix) in the full Bayesian framework. Besides, the proposed algorithm can properly utilize uncertainty information during iterations to ameliorate the error propagation problem, which means the estimation error of the sparse signal would degrade the estimation accuracy of phase error during iterations. Inevitably, the error propagation problem exists generally in these methods as the point estimate of the sparse scattering coefficient and phase error are updated alternately. Furthermore, different from the methods in Refs. 14, 20, and 21, we resort to Newton’s method to update the phase error, as it is not tractable to obtain the closed-from expression for updating the parameter when solving the nonlinear least-squares problem expressed in Eq. (13).
  2. Convergence: Since the SC-VMP algorithm can be interpreted as a variational EM algorithm, the update of β and ϕ will monotonically decrease the KLD and the negative expected log-likelihood function, respectively, until convergence. Thus, the (marginal) likelihood monotonically increases during the iterations and the convergence is guaranteed.28 Moreover, the possibility of converging to a local minimum is reduced due to the utilization of higher order statistical information.
  3. Computational complexity: The main computational burden at each iteration of SC-VMP procedure comes from the update of β in VMP procedure and the update of ϕ in Newton’s method. Two main time-consuming operations are the matrix inversion and matrix-vector multiplication whose computational complexity are o(K3) and o(K2), respectively. This can incur a high computational cost when K is large. Fortunately, the grid pruning could reduce the computational burden and improve the convergence speed, as the remaining grid number after pruning decreases fast. Additionally, the computational cost can be also controlled by restricting the maximum number of iteration.

In this section, simulations are carried out to verify the SC-VMP algorithm. An X-band RCI radar system with carrier frequency of 10 GHz is considered. The transmitters are configured as a uniform linear array with M=16 and the interelement spacing d=0.5  m. The transmitters emit independent frequency-hopping (FH) waveforms with the bandwidth of 500 MHz. A range-azimuth imaging plane, covering 40  m×40  m, is discretized to 40×40 grid cells. Further, there are supposed to be 17 point scatterers in the imaging plane.

Illustrative Example

In this part, the proposed SC-VMP algorithm is used to reconstruct the target image. For comparison, we also give the results obtained by four sparse-based imaging methods, i.e., VMP-3L, SSCIM,16 sparsity-driven autofocus (SDA) method14 and SACI3 method. VMP-3L is presented in Sec. 3. During the numerical experiments, the phase error randomly varied at [45  deg,45  deg], and the SNR is fixed at 15 dB. The experiments are performed on a computer with Intel Core CPU i3-4130 at 3.4 GHz and 4 GB of memory.

Figure 3 shows the true image and target images reconstructed by VMP-3L, SSCIM, SDA, SACI, and SC-VMP. It can be concluded that our proposed SC-VMP algorithm achieves superior imaging performance over the other four algorithms tested above. As shown in Fig. 6(f), the target is reconstructed perfectly without any spurious scatterers. Comparably, the other four algorithms suffer from imperfect reconstruction performance, and the reconstructed images are defocused with some spurious scatterers and the strength of scattering centers is not reconstructed exactly, although the target profiles are clear, which makes the images recognizable. Comparing the list algorithms, the difference is mainly focused on the method to reconstruct the scattering coefficients (see Sec. 3.5 for details). For SSCIM, SDA, and SACI, they suffer from the effect of phase error and the signal energy spills over the imaging plane because of the dictionary mismatch caused by phase error, as shown in Figs. 3(c)3(e), while our proposed SC-VMP utilizes the hierarchical modeling procedure and is considered as a full Bayesian method. More importantly, the method is less sensitive to the phase error, as shown in Fig. 3(b). In addition, the phase error is estimated and compensated perfectly by the SC-VMP algorithm, as shown in Fig. 4. Then, we could conclude that exploiting the phase error improves the performance of SC-VMP. Consequently, benefiting from full utilization of the sparsity prior and phase error calibration, the image quality shown in Fig. 3(f) is improved significantly, compared with the images reconstructed by other algorithms.

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Fig. 3
F3 :

RCI with phase error. (a) True image; (b) VMP-3L; (c) SSCIM; (d) SDA; (e) SACI; (f) SC-VMP.

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Fig. 4
F4 :

True and estimated phase error.

To evaluate the numerical complexity of the proposed algorithm, we record the runtime of the five algorithms during the simulations, and the result is shown in Table 1. We can see that SSCIM is time consuming for their heavy computational complexity, because the scattering coefficients are reconstructed by basis pursuit denoising algorithm. As orthogonal matching pursuit is employed in the iterations, SACI shows the best computational efficiency. The other three algorithms, i.e., VMP-3L, SDA, and SC-VMP, are all Bayesian-based algorithms and their computational complexity lies about the same level. Thus, it can be observed that our proposed algorithm does not show superior performance over other algorithms from the perspective of computational complexity. However, the SC-VMP algorithm can be improved to make it more computationally effective, which will be investigated in our future work.

Table Grahic Jump Location
Table 1Runtime of the five algorithms.
Performance of Self-Calibration Variational Message Passing Algorithm

To evaluate the performance of the proposed algorithm, we introduce two criterions, i.e., the relative imaging error (RIE) and probability of successful imaging (PSI). RIE is defined as 20log10(ββ2/β2), where β and β are the true and estimated scattering coefficient vector, respectively. To define PSI, a metric Δ=min(β)Λ/max(β)Λ31 is given first, where (β)Λ contains the values that β carries at the correct basis set Λ and (β¯)Λ takes 0 at Λ and takes the same values as β at every other indices. Δ>1 can guarantee exact estimation of the scatter’s position, which means a successful imaging trial. Hence, PSI is defined as the percentage of successful imaging trials.

First, we quantitatively analyze the performance of the aforementioned five algorithms versus SNR, and the result is shown in Fig. 5. It can be seen that the RIEs decrease quickly with the increase in SNR, which implies that the image quality is improved as the SNR increases and the algorithms are sensitive to noise, especially in low SNR regimes. Comparing with other algorithms, the RIE obtained by SC-VMP is much lower when SNR0  dB. Besides, it is worth noting that in Fig. 5(b), the SC-VMP outperforms other reported algorithms with a higher PSI, especially SNR>5  dB can guarantee the target reconstruction for SC-VMP (PSI96% in this case). Thus, SC-VMP exhibits significant performance gains over other algorithms in terms of both metrics, by compensating the phase error, whereas its performance is sensitive to SNR. More precisely, both the scattering coefficients estimation and phase error estimation are seriously affected by noise. The scattering coefficients are reconstructed by VMP-3L whose performance is shown in Fig. 5, while the performance of phase error estimation is given by Fig. 6, which shows the normalized mean square error (NMSE) of phase error, NMSE is defined as 20log10(ϕ2/ϕ2). We can see that low SNR results in high estimation error, which means that the phase error cannot be compensated accurately, and the target reconstruction would be degraded accordingly.

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Fig. 5
F5 :

Comparison of metrics versus SNR. (a) RIE and (b) PSI.

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Fig. 6
F6 :

NMSE of phase error versus SNR.

Next, we investigate the robustness of SC-VMP against the hopping frequency number and the scope of phase error. In this paper, the sampling interval is setup as the FH duration, which means hopping frequency number is equal to the number of samples N. The scope of phase error is [ϕmax,ϕmax], where ϕmax=[15k]°(k=1,,9) is the maximal value of phase error. For each scope of phase error, 100 independent trials are conducted and ϕm is randomly distributed within the scope.

As is clear from Fig. 7, the performance improves as the hopping frequency number increases, for both high imaging quality and high PSI. More independent hopping frequencies could provide more information of resolvability for RCI, which would also lead to the robustness of imaging. Besides, the algorithm is sensitive to the scope of phase error. When the phase error increases, the convergence of SC-VMP degrades, then the phase error could not be estimated exactly and calibrated perfectly. Thus, the imaging performance degrades severely, especially when ϕmax90  deg.

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Fig. 7
F7 :

Comparison of metrics versus the hopping frequency number and the scope of phase error: (a) RIE and (b) PSI.

RCI is a high-resolution imaging technique, while its performance degrades seriously in the presence of phase error. The SC-VMP algorithm is proposed in this paper to realize the target reconstruction and phase error calibration simultaneously in SBL framework. Bayesian hierarchical prior is used in the proposed algorithm, where the scattering coefficient is modeled statistically and estimated iteratively to achieve sparsity. As an iterative algorithm, SC-VMP cycles through steps of target reconstruction and phase error estimation, where VMP and Newton’s method are adopted, respectively. The proposed algorithm can estimate the phase error accurately and improve the reconstruction performance significantly. Besides, it requires no prior information and provides superior and robust imaging performance for various imaging scenes and system parameters, which shows the potential for the algorithm to be applied in practical RCI or other imaging radar systems.

Implementation of Variational Message Passing Procedure

During the VMP procedure, the hidden variables Ω={β,α,η,α0} can be updated individually.

  1. Update of q(β). It is obvious to conclude from Eq. (8) that the echo y obeys a complex Gaussian distribution and the likelihood function of the observation can be formulated as Display Formula
    p[p(y|β,αo;ϕ)]=CN(y|Sβ,α01I).(16)
    According to Ref. 27, the approximated posterior of β can be expressed as Display Formula
    q(β)exp{lnp(y|β,α0;ϕ)q(α0)p(β|α)q(α)}.(17)
    Substituting Eqs. (5) and (16) into Eq. (17), we can conclude that β obeys a complex Gaussian distribution: Display Formula
    q(β)CN(β|μ,Σ),(18)
    where the mean μ and covariance Σ are given by Display Formula
    Σ=(α0q(α0)SHS+Λ1q(α))1,(19)
    Display Formula
    μ=α0q(α0)ΣSHy.(20)
  2. Update of q(α). The approximated posterior of α can be given by Display Formula
    q(α)exp{p(β|α)q(β)lnp(α|η)q(η)}.(21)
    Then, we substitute Eqs. (5) and (6) into Eq. (21) and obtain q(α) as Display Formula
    q(α)k=1Kαkϵ2exp(αk1|βk|2q(β)αkηq(η)),(22)
    where |βk|2q(β)=|μk|2+Σkk. Thus, q(α) is the product of GIG PDFs with order p=ϵ1. The moments of GIG distribution are given in closed form for any nR: Display Formula
    αknq(α)=(|βk|2q(β)ηq(η))n2κp+n(2ηq(η)|βk|2q(β))κp(2ηq(η)|βk|2q(β)),(23)
    where κυ(·) is the modified Bessel function of the second kind with order υ. The case of n=1 in Eq. (23) gives the evaluation of Λ1q(α) used in Eq. (19), and the case of n=1 gives the calculation of αkq(α) used in Eq. (25).
  3. Update of q(η). The approximated posterior distribution of η is obtained by Display Formula
    q(α)exp{p(α|η)q(η)p(η;a,b)}.(24)
    It can be seen that the posterior distribution of η is a Gamma distribution: q(η)=Γ(η|Kϵ+a,k=1Kαkq(α)+b), while the mean of η is Display Formula
    ηq(η)=Kϵ+ak=1Kαkq(α)+b.(25)
  4. Update of q(α0). It can be shown that q(α0)=Γ(α0|N+c,yS·β22q(β)+d). The first moment of α0 is Display Formula
    α0q(α0)=N+cyS·β22q(β)+d.(26)

Derivation of Estimating Phase Error

In this part, we derive the gradient f(ϕ) and Hessian 2f(ϕ), which are used in Eq. (14) to estimating the phase error ϕ. For convenience of derivation, f(ϕ) is decomposed by two parts, i.e., f(ϕ)=f1(ϕ)+f2(ϕ), where f1(ϕ)=yS(ϕ)μ22 and f2(ϕ)=tr{[S(ϕ)]HS(ϕ)Σ}. Then, we have Display Formula

f(ϕ)=f1(ϕ)+f2(ϕ),(27)
Display Formula
2f(ϕ)=2f1(ϕ)+2f2(ϕ).(28)

After derivation and simplification, the gradient and Hessian matrix can be computed as Display Formula

[f1(ϕ)]m=2Im{[Stm(ϕm)μ]Hw},(29)
Display Formula
[f2(ϕ)]m=tr(j{[S(ϕ)]HStm(ϕm)[Stm(ϕm)]HS(ϕ)}Σ),(30)
Display Formula
[2f1(ϕ)]mm=2Re{[Stm(ϕm)·μ]Hw}+2Stm(ϕm)·μ2,(31)
Display Formula
[2f1(ϕ)]ml=2Re{[Stm(ϕm)·μ]H[Stl(ϕl)·μ]},ml,(32)
Display Formula
[2f2(ϕ)]mm=tr({2[Stm(ϕm)]H[Stm(ϕm)][Stm(ϕm)]HS(ϕ)[S(ϕ)]H[Stm(ϕm)]}Σ),(33)
Display Formula
[2f2(ϕ)]ml=tr({[Stm(ϕm)]H[Stl(ϕl)]+[Stl(ϕl)]H[Stm(ϕm)]}Σ),ml,(34)
where Re(·) and Im(·) denote the real and imaginary parts, respectively. tr(·) is the trace operator. [a]m is the m’th element of a, [A]ml denotes the (m,l)-th element of A. Stm(ϕm)=[Stm(r1,ϕm),,Stm(rK,ϕm)], Stm(rk,ϕm)=ejϕm[Stm(t1τmk),,Stm(tNτmk)]T, and w=yS(ϕ)μ.

This work is supported by the National Natural Science Foundation of China (No. 61302149 and 61302142) and Research Fund for the Doctoral Program of Higher Education of China (20124307110013). The authors would like to thank the editors and reviewers for their insightful comments.

Li  D.  et al., “Radar coincidence imaging: an instantaneous imaging technique with stochastic signals,” IEEE Trans. Geosci. Remote Sens.. 52, (4 ), 2261 –2277 (2014). 0196-2892 CrossRef
Li  D.  et al., “Radar coincidence imaging in the presence of target-motion-induced error,” J. Electron. Imaging. 23, (2 ), 023014  (2014). 1017-9909 CrossRef
Zhou  X.  et al., “Sparse auto-calibration for radar coincidence imaging with gain-phase errors,” Sensors. 15, , 27611 –27624 (2015). 0746-9462 CrossRef
Martorella  M., “Analysis of the robustness of bistatic inverse synthetic aperture radar in the presence of phase synchronisation errors,” IEEE Trans. Aerosp. Electron. Syst.. 47, (4 ), 2673 –2689 (2011). 0018-9251 CrossRef
Godrich  H., and Haimovich  A. M., “Localization performance of coherent MIMO radar systems subject to phase synchronization errors,” in  4th Int. Symp. on Communications, Control and Signal Processing ,  Limassol, Cyprus  (2010).
He  Q., and Blum  R., “Cramer-Rao bound for MIMO radar target localization with phase errors,” IEEE Signal Process. Lett.. 17, (1 ), 83 –86 (2010). 1070-9908 CrossRef
Liu  A.  et al., “An eigenstructure method for estimating DOA and sensor gain-phase errors,” IEEE Trans. Signal Process.. 59, (12 ), 5944 –5956 (2011). 1053-587X CrossRef
Cao  S.  et al., “A Hadamard product based method for DOA estimation and gain-phase error calibration,” IEEE Trans. Aerosp. Electron. Syst.. 49, (2 ), 1224 –1233 (2013). 0018-9251 CrossRef
Li  J.  et al., “Reduced-dimension MUSIC for angle and array gain-phase error estimation in bstatic MIMO radar,” IEEE Commun. Lett.. 17, (3 ), 443 –1482 (2013). 1089-7798 CrossRef
Li  J., , Zhang  X., and Gao  X., “A joint scheme for angle and array gain-phase error estimation in bistatic MIMO radar,” IEEE Geosci. Remote Sens. Lett.. 10, (6 ), 1478 –1482 (2013).CrossRef
Li  J.  et al., “Transmit and receive array gain-phase error estimation in bistatic MIMO radar,” IEEE Antennas Wireless Propag. Lett.. 14, , 32 –35 (2015). 1536-1225 CrossRef
Liu  Z., and Zhou  Y., “A unified framework and sparse Bayesian perspective for direction-of-arrival estimation in the presence of array imperfections,” IEEE Trans. Signal Process.. 61, (15 ), 3786 –3798 (2013). 1053-587X CrossRef
Donoho  D. V., “Compressed sensing,” IEEE Trans. Inf. Theory. 52, (4 ), 1289 –1306 (2006). 0018-9448 CrossRef
Özben Önhon  N., and Çetin  M., “A sparsity-driven approach for joint SAR imaging and phase error correction,” IEEE Trans. Image Process.. 21, (4 ), 2075 –2088 (2012). 1057-7149 CrossRef
Ding  L., and Chen  W., “MIMO radar sparse imaging with phase mismatch,” IEEE Geosci. Remote Sens. Lett.. 12, (4 ), 816 –820 (2015).CrossRef
Ding  L.  et al., “Sparse self-calibration via iterative minimization against phase synchronization mismatch for MIMO radar imaging,” in  Proc. IEEE Radar Conf. ,  Ottawa, Ontario, Canada , pp. 1 –4 (2013).
Xu  G.  et al., “Bayesian inverse synthetic aperture radar imaging,” IEEE Geosci. Remote Sens. Lett.. 8, (6 ), 1150 –1154 (2011).CrossRef
Zhang  L.  et al., “High-resolution ISAR imaging by exploiting sparse apertures,” IEEE Trans. Antennas Propag.. 60, (2 ), 997 –1008 (2012). 0018-926X CrossRef
Zhang  L.  et al., “High-resolution ISAR imaging with sparse stepped-frequency waveforms,” IEEE Trans. Geosci. Remote Sens.. 49, (11 ), 4630 –4651 (2011). 0196-2892 CrossRef
Zhao  L.  et al., “Phase/gain error compensation in sensor array via variational Bayesian inference,” in  IEEE 9th Conf. on Industrial Electronics and Applications , pp. 443 –446,  IEEE ,  Hangzhou, China  (2014).CrossRef
Zhao  L.  et al., “An improved auto-calibration algorithm based on sparse Bayesian learning framework,” IEEE Signal Process. Lett.. 20, (9 ), 889 –892 (2013). 1070-9908 CrossRef
Yang  Z., , Xie  L., and Zhang  C., “Off-grid direction of arrival estimation using sparse Bayesian inference,” IEEE Trans. Signal Process.. 61, (1 ), 38 –43 (2013). 1053-587X CrossRef
MacKay  D., “Bayesian interpolation,” Neural Comput.. 4, (3 ), 415 –447 (1992). 0899-7667 CrossRef
Winn  J., and Bishop  C. M., “Variational message passing,” J. Mach. Learn. Res.. 6, , 661 –694 (2005).
Potter  L. C.  et al., “A GTD-based parametric model for radar scattering,” IEEE Trans. Antennas Propag.. 43, (10 ), 1058 –1067 (1995). 0018-926X CrossRef
Çetin  M., and Karl  W. C., “Feature-enhanced synthetic aperture radar image formation based on nonquadratic regularization,” IEEE Trans. Image Process.. 10, (4 ), 623 –631 (2001). 1057-7149 CrossRef
Pedersen  N.  et al., “Sparse estimation using Bayesian hierarchical prior modeling for real and complex models,” http://arxiv.org/pdf/1108 (2011).
Tzikas  D., , Likas  A., and Galatsanos  N., “The variational approximation for Bayesian inference,” IEEE Signal Process Mag.. 25, (6 ), 131 –146 (2008). 1053-5888 CrossRef
Yang  Z., , Xie  L., and Zhang  C., “Variational Bayesian algorithm for quantized compressed sensing,” IEEE Trans. Signal Process.. 61, (11 ), 2815 –2824 (2013). 1053-587X CrossRef
Boyd  S., and Vandenberghe  L., Convex Optimization. ,  Cambridge University Press ,  Cambridge, United Kingdom  (2004).
Gogineni  S., and Nehorai  A., “Target estimation using sparse modeling for distributed MIMO radar,” IEEE Trans. Signal Process.. 59, (11 ), 5315 –5325 (2011). 1053-587X CrossRef

Xiaoli Zhou received his BS and MS degrees in instrument science and technology from the National University of Defense Technology, Changsha, China, in 2010 and 2012, respectively. He is currently pursuing his PhD degree with the National University of Defense Technology. His current research interests include remote sensing signal processing and coincidence imaging, as well as advanced signal processing with application to radar target imaging and identification.

Hongqiang Wang received his BS, MS, and PhD degrees from the National University of Defense Technology, Changsha, China, in 1993, 1999, and 2002, respectively. He is currently a professor with the School of Electronic Science and Engineering, National University of Defense Technology. He has been involved in modern radar signal processing research and development since 1996. His current research interests include automatic target recognition, radar imaging, and target tracking.

Yongqiang Cheng received his BS, MS, and PhD degrees in information and communication engineering from the National University of Defense Technology, Changsha, China, in 2005, 2007, and 2012, respectively. He is currently an associate professor with the National University of Defense Technology. From September 2009 to November 2010, he was a visiting research student with Melbourne Systems Laboratory, Department of Electrical and Electronic Engineering, University of Melbourne, Melbourne, Australia. His current research interests include statistical signal processing and information geometry.

Yuliang Qin received his BS, MS, and PhD degrees in information and communication engineering from the National University of Defense Technology, Changsha, China, in 2002, 2004, and 2008, respectively. He is currently an associate professor with the School of Electronic Science and Engineering, National University of Defense Technology. His current research interests include SAR imaging and radar signal processing.

© The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

Citation

Xiaoli Zhou ; Hongqiang Wang ; Yongqiang Cheng and Yuliang Qin
"Radar coincidence imaging with phase error using Bayesian hierarchical prior modeling", J. Electron. Imaging. 25(1), 013018 (Feb 01, 2016). ; http://dx.doi.org/10.1117/1.JEI.25.1.013018


Figures

Graphic Jump Location
Fig. 2
F2 :

A graphical representation of the model priors.

Graphic Jump Location
Fig. 3
F3 :

RCI with phase error. (a) True image; (b) VMP-3L; (c) SSCIM; (d) SDA; (e) SACI; (f) SC-VMP.

Graphic Jump Location
Fig. 4
F4 :

True and estimated phase error.

Graphic Jump Location
Fig. 5
F5 :

Comparison of metrics versus SNR. (a) RIE and (b) PSI.

Graphic Jump Location
Fig. 6
F6 :

NMSE of phase error versus SNR.

Graphic Jump Location
Fig. 7
F7 :

Comparison of metrics versus the hopping frequency number and the scope of phase error: (a) RIE and (b) PSI.

Tables

Table Grahic Jump Location
Algorithm 1SC-VMP algorithm.
Table Grahic Jump Location
Table 1Runtime of the five algorithms.

References

Li  D.  et al., “Radar coincidence imaging: an instantaneous imaging technique with stochastic signals,” IEEE Trans. Geosci. Remote Sens.. 52, (4 ), 2261 –2277 (2014). 0196-2892 CrossRef
Li  D.  et al., “Radar coincidence imaging in the presence of target-motion-induced error,” J. Electron. Imaging. 23, (2 ), 023014  (2014). 1017-9909 CrossRef
Zhou  X.  et al., “Sparse auto-calibration for radar coincidence imaging with gain-phase errors,” Sensors. 15, , 27611 –27624 (2015). 0746-9462 CrossRef
Martorella  M., “Analysis of the robustness of bistatic inverse synthetic aperture radar in the presence of phase synchronisation errors,” IEEE Trans. Aerosp. Electron. Syst.. 47, (4 ), 2673 –2689 (2011). 0018-9251 CrossRef
Godrich  H., and Haimovich  A. M., “Localization performance of coherent MIMO radar systems subject to phase synchronization errors,” in  4th Int. Symp. on Communications, Control and Signal Processing ,  Limassol, Cyprus  (2010).
He  Q., and Blum  R., “Cramer-Rao bound for MIMO radar target localization with phase errors,” IEEE Signal Process. Lett.. 17, (1 ), 83 –86 (2010). 1070-9908 CrossRef
Liu  A.  et al., “An eigenstructure method for estimating DOA and sensor gain-phase errors,” IEEE Trans. Signal Process.. 59, (12 ), 5944 –5956 (2011). 1053-587X CrossRef
Cao  S.  et al., “A Hadamard product based method for DOA estimation and gain-phase error calibration,” IEEE Trans. Aerosp. Electron. Syst.. 49, (2 ), 1224 –1233 (2013). 0018-9251 CrossRef
Li  J.  et al., “Reduced-dimension MUSIC for angle and array gain-phase error estimation in bstatic MIMO radar,” IEEE Commun. Lett.. 17, (3 ), 443 –1482 (2013). 1089-7798 CrossRef
Li  J., , Zhang  X., and Gao  X., “A joint scheme for angle and array gain-phase error estimation in bistatic MIMO radar,” IEEE Geosci. Remote Sens. Lett.. 10, (6 ), 1478 –1482 (2013).CrossRef
Li  J.  et al., “Transmit and receive array gain-phase error estimation in bistatic MIMO radar,” IEEE Antennas Wireless Propag. Lett.. 14, , 32 –35 (2015). 1536-1225 CrossRef
Liu  Z., and Zhou  Y., “A unified framework and sparse Bayesian perspective for direction-of-arrival estimation in the presence of array imperfections,” IEEE Trans. Signal Process.. 61, (15 ), 3786 –3798 (2013). 1053-587X CrossRef
Donoho  D. V., “Compressed sensing,” IEEE Trans. Inf. Theory. 52, (4 ), 1289 –1306 (2006). 0018-9448 CrossRef
Özben Önhon  N., and Çetin  M., “A sparsity-driven approach for joint SAR imaging and phase error correction,” IEEE Trans. Image Process.. 21, (4 ), 2075 –2088 (2012). 1057-7149 CrossRef
Ding  L., and Chen  W., “MIMO radar sparse imaging with phase mismatch,” IEEE Geosci. Remote Sens. Lett.. 12, (4 ), 816 –820 (2015).CrossRef
Ding  L.  et al., “Sparse self-calibration via iterative minimization against phase synchronization mismatch for MIMO radar imaging,” in  Proc. IEEE Radar Conf. ,  Ottawa, Ontario, Canada , pp. 1 –4 (2013).
Xu  G.  et al., “Bayesian inverse synthetic aperture radar imaging,” IEEE Geosci. Remote Sens. Lett.. 8, (6 ), 1150 –1154 (2011).CrossRef
Zhang  L.  et al., “High-resolution ISAR imaging by exploiting sparse apertures,” IEEE Trans. Antennas Propag.. 60, (2 ), 997 –1008 (2012). 0018-926X CrossRef
Zhang  L.  et al., “High-resolution ISAR imaging with sparse stepped-frequency waveforms,” IEEE Trans. Geosci. Remote Sens.. 49, (11 ), 4630 –4651 (2011). 0196-2892 CrossRef
Zhao  L.  et al., “Phase/gain error compensation in sensor array via variational Bayesian inference,” in  IEEE 9th Conf. on Industrial Electronics and Applications , pp. 443 –446,  IEEE ,  Hangzhou, China  (2014).CrossRef
Zhao  L.  et al., “An improved auto-calibration algorithm based on sparse Bayesian learning framework,” IEEE Signal Process. Lett.. 20, (9 ), 889 –892 (2013). 1070-9908 CrossRef
Yang  Z., , Xie  L., and Zhang  C., “Off-grid direction of arrival estimation using sparse Bayesian inference,” IEEE Trans. Signal Process.. 61, (1 ), 38 –43 (2013). 1053-587X CrossRef
MacKay  D., “Bayesian interpolation,” Neural Comput.. 4, (3 ), 415 –447 (1992). 0899-7667 CrossRef
Winn  J., and Bishop  C. M., “Variational message passing,” J. Mach. Learn. Res.. 6, , 661 –694 (2005).
Potter  L. C.  et al., “A GTD-based parametric model for radar scattering,” IEEE Trans. Antennas Propag.. 43, (10 ), 1058 –1067 (1995). 0018-926X CrossRef
Çetin  M., and Karl  W. C., “Feature-enhanced synthetic aperture radar image formation based on nonquadratic regularization,” IEEE Trans. Image Process.. 10, (4 ), 623 –631 (2001). 1057-7149 CrossRef
Pedersen  N.  et al., “Sparse estimation using Bayesian hierarchical prior modeling for real and complex models,” http://arxiv.org/pdf/1108 (2011).
Tzikas  D., , Likas  A., and Galatsanos  N., “The variational approximation for Bayesian inference,” IEEE Signal Process Mag.. 25, (6 ), 131 –146 (2008). 1053-5888 CrossRef
Yang  Z., , Xie  L., and Zhang  C., “Variational Bayesian algorithm for quantized compressed sensing,” IEEE Trans. Signal Process.. 61, (11 ), 2815 –2824 (2013). 1053-587X CrossRef
Boyd  S., and Vandenberghe  L., Convex Optimization. ,  Cambridge University Press ,  Cambridge, United Kingdom  (2004).
Gogineni  S., and Nehorai  A., “Target estimation using sparse modeling for distributed MIMO radar,” IEEE Trans. Signal Process.. 59, (11 ), 5315 –5325 (2011). 1053-587X CrossRef

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