Structured sparse signal recovery algorithms and their applications
Date of Issue2014
School of Electrical and Electronic Engineering
Centre for Signal Processing
Getting more sophisticated, the theory of sparse representation (SR) and the successful SR based applications in various aspects of signal processing have been extensively investigated and discussed over the past several decades. Besides sparsity, underlying structures of the signal have been considered and exploited recently to enhance the performance of the standard SR recovery methods. While a large number of recovery algorithms have been proposed to incorporate different signal structures in the literature and have achieved reasonably good performance, there is still a need for further improvement and extending them for more real world applications. This thesis focuses on various issues of structured sparse representations, including theoretical performance analysis for certain existing structured sparse recovery methods, sparse recovery algorithm developments for signal with specific structure and their applications. One of the most commonly exploited structures in the literature is the block sparsity. For signals with block sparsity, a greedy recovery method, known as block orthogonal greedy algorithm (BOGA), is attractive due to its computational efficiency. BOGA has been proven to successfully recover block sparse signals in noiseless environments and outperforms its standard counterpart, i.e., orthogonal greedy algorithm (OGA) in terms of achieving smaller mean square error (MSE) by using less measurements. The associated stability problem dealing with noisy signals is re-studied in this thesis. It is demonstrated that the recovery conditions of the BOGA previously reported can be relaxed by using a different definition of the block-coherence. The theoretical results obtained provide a generalization of those reported by Tseng for block-sparse signal and serve as a complement of the BOGA reported by Eldar for noisy signals. Motivated by the success of the BOGA, new detection architectures for harmonic tonals are proposed for tonal detection in low signal to noise ratio (SNR) environments. The distributions of the test statistics of detection architectures under the orthogonal dictionary are analyzed theoretically and comprehensively based on the theory of order statistics for the first time. Detection performances are also analyzed and compared theoretically and experimentally. Significant improvements on detection performance in low SNR environments are shown over the conventional detectors that do not consider the block structure. As a natural generation of block sparse signal, hierarchical sparse signal, exhibiting two levels of sparsity, i.e., block sparsity among different blocks and internal sparsity within individual block, is of interest. It is mainly recovered by solving a convex optimization problem with two user-selected regularization parameters. To avoid the subjective parameter selection, a recovery method is developed under a Bayesian framework. To enforce the two-level hierarchical sparsity, the variance of the sparse coefficient is modeled by the product of two kinds of random variables controlling the block sparsity and the internal sparsity, respectively. Under the proposed probabilistic model, variational Bayesian inference is used to recover the coefficient vector from the noise corrupted data. Numerical simulation and experimental results show that the proposed method outperforms those state-of-the-art recovery methods for hierarchical sparse signals. Both the block sparse and the hierarchical sparse models require a known block partition which is often unrealistic. For real world applications, nonzero elements of the sparse coefficient vector may cluster randomly with unknown locations and sizes. We propose to impose this spatial cluster structure statistically in a Bayesian framework and successfully apply it to Inverse Synthetic Aperture Radar (ISAR) imaging by using the concept of spatial continuity. Following the Bayesian compressive sensing theory, a hierarchical Bayesian prior is imposed on the scatterers on the range-Doppler plane to encourage the sparsity. Then a correlated prior is used to statistically encourage a two dimensional cluster structure of the scatterers in the target region. However, the posterior distribution is intractable under the formulated probabilistic model. To overcome this difficulty, the Gibbs sampling strategy is used for Bayesian inference and the parameters of the signal model are inferred efficiently from samples obtained by the Gibbs sampler. Both the synthetic and the experimental results demonstrate that the proposed algorithm exploits efficiently the continuity of the target scene and can achieve enhanced ISAR images in the scenarios of limited measurements and low SNR compared with other reported algorithms for ISAR imaging problems.