Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 1782178, 12 pages

http://dx.doi.org/10.1155/2016/1782178

## Off-Grid Radar Coincidence Imaging Based on Variational Sparse Bayesian Learning

School of Electronic Science and Engineering, National University of Defense Technology, Changsha 410073, China

Received 24 October 2015; Revised 15 March 2016; Accepted 31 March 2016

Academic Editor: Erik Cuevas

Copyright © 2016 Xiaoli Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Radar coincidence imaging (RCI) is a high-resolution staring imaging technique motivated by classical optical coincidence imaging. In RCI, sparse reconstruction methods are commonly used to achieve better imaging result, while the performance guarantee is based on the general assumption that the scatterers are located at the prediscretized grid-cell centers. However, the widely existing off-grid problem degrades the RCI performance considerably. In this paper, an algorithm based on variational sparse Bayesian learning (VSBL) is developed to solve the off-grid RCI. Applying Taylor expansion, the unknown true dictionary is approximated accurately to a linear model. Then target reconstruction is reformulated as a joint sparse recovery problem that recovers three groups of sparse coefficients over three known dictionaries with the constraint of the common support shared by the groups. VSBL is then applied to solve the problem by assigning appropriate priors to the three groups of coefficients. Results of numerical experiments demonstrate that the algorithm can achieve outstanding reconstruction performance and yield superior performance both in suppressing noise and in adapting to off-grid error.

#### 1. Introduction

Motivated by classical coincidence imaging, radar coincidence imaging (RCI) is a novel high-resolution imaging technique which has been realized in optical systems [1–3]. The RCI can realize staring imaging without the requirement of the target relative motion and operate under the nonideal observing geometry of forward-looking/staring and shorten the imaging time to even a single pulse width, with significant potentials for resolution enhancement, interference, and jamming suppression. The basic principle of RCI is to excite time-space independent signals in the imaging area. Thus the spatial variety of wavefront is increased, and the scatterers within a beam then reflect independent signals associated with their respective positions, so the super-resolution within a beam emerges.

To achieve better imaging performance, sparse reconstruction is commonly used, and the continuous space needs to be discretized to a fine grid and target-scattering centers are assumed to be exactly located at the center of these prediscretized grids [3]. Based on the model, the independent* detecting signals* at different grid-cell centers can be formed as the* atoms* of sparse representation dictionary. Meanwhile, the scatterers of target are often distributed sparsely in most radar imaging applications; thus sparse reconstruction and compressive sensing (CS) [4, 5] are suitable for RCI by exploiting the sparsity of target. In sparse reconstruction theory, signal reconstruction depends on presetting an appropriate sparsifying dictionary which is supported on the prediscretized grids and defines the signal sparsity. However, as the scatterers distributed in a continuous scene are generally located off the grid-cell centers, the* off-grid problem* emerges no matter how fine the grid is [6], and the performance of RCI would degrade significantly.

The effect of general dictionary mismatch, which is the direct consequence of off-grid effect, is analyzed in [7–10]. This mismatch causes the performance of conventional sparse reconstruction methods to degrade considerably [3, 6, 7, 11–14]. An intuitive way to sidestep off-grid effect is to work directly on the continuous parameter space, that is, atomic norm minimization approach [15], continuous basis pursuit (CBP) [16]. Considering the off-grid problem, several algorithms have been proposed to alleviate the effect. One simple approach is to use refinement strategy and decrease the grid size [17]. Nevertheless, a finer grid may enhance the coherence between the columns of dictionary and increase the computational complexity and instability of reconstruction [14]. Modeling the off-grid problem as a multiplicative perturbation, the sparse total least squares (S-TLS) [18] and joint correlation-parameterization (JCP) [3] algorithms are proposed. To explore the structure of dictionary mismatch, the support-constrained orthogonal matching pursuit (SCOMP) [13] and joint sparse signal recovery methods [19] are proposed based on the first-order Taylor expansion to utilize the support constraint. Lately, from the sparse Bayesian learning (SBL) perspective, several approaches are proposed, such as off-grid sparse Bayesian inference (OGSBI) [20], sparse adaptive calibration recovery via iterative maximum a posteriori (SACR-iMAP) [6], and block SBL (BSBL) [21]. 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 [20].

Although the performance of OGSBI and SACR-iMAP is outstanding, the prior distribution of* off-grid error* (OGE) is not fully utilized; the point estimate rather than distribution of OGE is obtained. Adopting the ideas of variational expectation-maximization (EM) and variational Bayesian inference (VBI) described in [7, 22, 23], the off-grid RCI is investigated in the framework of variational sparse Bayesian learning (VSBL). As for SBL, inference in SBL model is not tractable in closed form; thus approximations are needed, such as maximizing the marginal likelihood (MML), EM-SBL, and VBI [24]. Compared to other approximations, VBI has several advantages when applied to SBL. First, the distributions rather than point estimates of the unobserved variables can be obtained. Second, VSBL allows obtaining analytical approximations to the posterior distributions of interest even when their exact expressions are intractable. Finally, the VSBL methodology allows using different prior distributions and is a deterministic approximate inference framework that can be applied to many models.

In this paper, based on the first-order Taylor expansion, the off-grid RCI model in range-azimuth space is reformed to be sparsely approximated using atoms from three different dictionaries, and meanwhile the three groups of approximation coefficients share the same support. By assigning appropriate priors to the approximation coefficients, such a model can be conveniently manipulated to recover sparse coefficients under the VSBL framework. Compared with the aforementioned algorithm discussed above, the proposed algorithm imposes a group structure on the coefficient vector and explores the group-sparse structure using VSBL. Numerical experiments show that the algorithm realizes the target reconstruction robustly and achieves both high-resolution and outstanding imaging quality in the presence of off-grid scatterers and is also simple to implement.

The rest of the paper is organized as follows. In Section 2, the off-grid RCI model in the range-azimuth space is presented. Section 3 presents the off-grid variational sparse Bayesian learning (OG-VSBL) algorithm in detail. In Section 4, the performance of the presented algorithm is verified by numerical examples and compared with some existing sparse reconstruction methods. Finally, some conclusions and future work are discussed in Section 5.

Notations used in this paper are as follows. We use boldface lowercase letters for vectors and boldface uppercase letters for matrices. , , , and denote the transpose, conjugate transpose, inverse, and pseudoinverse of a vector or matrix, respectively. and are the Hadamard product and vectorization operation, separately. and denote the norm and norm of a vector. is the determinant of a matrix; is a matrix with the elements of a vector on the main diagonal. Finally, denotes the expectation of a variable.

#### 2. Problem Formulation

##### 2.1. Signal Model

As a novel imaging technique, RCI has shown its potentials in high-resolution, staring, and instantaneous imaging [1]. In RCI, the target is illuminated by time-space independent signals, and then the echo components reflected by scatterers at different positions are mutually independent, which could result in the super-resolution within a beam. The RCI can be realized by a multiple-input multiple-output (MIMO) radar system to transmit time-independent and group-orthogonal waveforms (e.g., stochastic waveforms) [1]. While compared with conventional MIMO radar which focuses on multiple paths or multiple observation angles, RCI needs the interference of transmitted waveforms to make the wavefront show spatial fluctuations, thus the spatial variety of detecting signals increases. In addition, the components of each path are separated utilizing the waveform orthogonality in conventional MIMO radar, while the components are not separated in the whole RCI procedure.

In this paper, a RCI system with transmitters and receivers is considered; each transmitter emits an independent stochastic waveform. Without loss of generality, the target is assumed to be composed of several ideal point scattering centers without the spatial property for an enough high carrier frequency, which is widely used in the imaging radar system. Furthermore, the scatterers are widely separated to provide some performance guarantee.

The target scene is considered to be a 2D space using polar coordinate as illustrated in Figure 1. The sparse-based RCI methods discretize the continuous target scene and generate a number of grid-cells. Thus the scene is discretized in range-azimuth space with azimuth cells, range cells, and associated cell size , . Hence, the number of grid-cells is . Assume that the scatterers are initially located at the grid-cell centers; thus the th scatterer is located at the range-azimuth pair . As the scatterers possess nonzero complex scattering coefficient which is proportional to the radar cross section (RCS), the associated means that there is no scatterer at the th grid-cell center. Denote by the index set of scatterer locations.