Coherence Pattern–Guided Compressive Sensing with Unresolved Grids

Abstract

Abstract. Highly coherent sensing matrices arise in discretization of continuum imaging problems such as radar and medical imaging when the grid spacing is below the Rayleigh threshold.Algorithms based on techniques of band exclusion (BE) and local optimization (LO) are proposed to deal with such coherent sensing matrices. These techniques are embedded in the existing compressed sensing algorithms such as Orthogonal Matching Pursuit (OMP), Subspace Pursuit (SP), Iterative Hard Thresholding (IHT), Basis Pursuit (BP) and Lasso, and result in the modified algorithms BLOOMP, BLOSP, BLOIHT, BP-BLOT and Lasso-BLOT, respectively.Under appropriate conditions, it is proved that BLOOMP can reconstruct sparse, widely separated objects up to one Rayleigh length in the Bottleneck distance independent of the grid spacing. One of the most distinguishing attributes of BLOOMP is its capability of dealing with large dynamic ranges.The BLO-based algorithms are systematically tested with respect to four performance metrics: dynamic range, noise stability, sparsity and resolution. With respect to dynamic range and noise stability, BLOOMP is the best performer. With respect to sparsity, BLOOMP is the best performer for high dynamic range while for dynamic range near unity BP-BLOT and Lasso-BLOT with the optimized regularization parameter have the best performance. In the noiseless case, BP-BLOT has the highest resolving power up to certain dynamic range.The algorithms BLOSP and BLOIHT are good alternatives to BLOOMP and BP/Lasso-BLOT: they are faster than both BLOOMP and BP/Lasso-BLOT and shares, to a lesser degree, BLOOMP's amazing attribute with respect to dynamic range.Detailed comparisons with existing algorithms such as Spectral Iterative Hard Thresholding (SIHT) and the frame-adapted BP are given.

Greedy signal space methods for incoherence and beyond

“…It would be also important to mention the relation of the ǫ-OMP and ǫ-thresholding algorithms (Algorithms 2 and 3) to the methods proposed in [24,28]. The notion of excluding coherent atoms in the process of building the representation is used also within these works.…”

“…The notion of excluding coherent atoms in the process of building the representation is used also within these works. In particular, without the extension step, the ǫ-OMP and ǫ-threshodling techniques share a great similarity with the Band-Excluded OMP (BOMP) and Band-Excluded Matched Thresholding (BMT) methods in [24] and the heuristic coherence-inhibiting sparse approximation strategy in [28]. As we have seen in Section 4.4, the use of the extension step deteriorates the performance in the case of separated coefficients as a larger support is processed and therefore the RIP conditions are harder to be satisfied.…”

“…As we have seen in Section 4.4, the use of the extension step deteriorates the performance in the case of separated coefficients as a larger support is processed and therefore the RIP conditions are harder to be satisfied. It is likely that using the techniques in [24,28] would be better suited to deal with separated coefficient vectors.…”

“…This idea is related to the recent literature of super-resolution (see e.g. [23,24,25,26,27, 28] and references therein) and to the ǫ-OMP algorithm [29], which is an extension of OMP. In this work we employ ǫ-OMP (with a post-processing step that adds correlated atoms) as a support selection procedure.…”

Compressed Sensing Of Complex Sinusoids Off The Grid

“…To decrease the computational complexity, the grid can be refined only around the regions where sources are present [3]. For widely separated objects, A. F. Jiang proposed a method based on band exclusion and local optimization [4]. Under some special conditions, the methods proposed in [5,6] can be used.…”