Sparse signal separation in redundant dictionaries

Abstract

Abstract-We formulate a unified framework for the separation of signals that are sparse in "morphologically" different redundant dictionaries. This formulation incorporates the socalled "analysis" and "synthesis" approaches as special cases and contains novel hybrid setups. We find corresponding coherencebased recovery guarantees for an 1-norm based separation algorithm. Our results recover those reported in Studer and Baraniuk, ACHA, submitted, for the synthesis setting, provide new recovery guarantees for the analysis setting, and form a basis for comparing performance in the analysis and synthesis settings. As an aside our findings complement the D-RIP recovery results reported in Candès et al., ACHA, 2011, for the "analysis" signal recovery problem minimize x Ψ x 1 subject to y − A x 2 ≤ ε by delivering corresponding coherence-based recovery results.

Stable restoration and separation of approximately sparse signals

“…Deterministic recovery guarantees. Recovery guarantees in the deterministic setting for noiseless measurements and signals being perfectly sparse, i.e., the model z = Ax + Be, have been studied in [12,19,34,35,44,45]. In [34], it has been shown that when A is the discrete Fourier transform (DFT) matrix, B = I M and when the support set of the interference e is known, perfect recovery of x is possible if 2n x n e < M , where n e = e 0 .…”

“…The recovery conditions have been derived using a joint concentration measure and the so-called cluster coherence, which enable the derivation of recovery conditions for the analysis separation problem that explicitly exploit the structure of particular pairs of frames (e.g., wavelets and curvelets). Coherence-based results for hybrid synthesis-analysis problems for pairs of general dictionaries were developed recently in [45].…”

Piecewise Sparse Approximation via Threshold P_OMP with Application to Surface Reconstruction

“…Piecewise sparsity describes a type of structured sparsity which emerges in the application of reconstructing a surface from scattered data in PSI space [13,14,15], data separation/signal decomposition [16,17,18,19], etc. In these applications, data(including image, signal) are represented in different bases and frames, and each frame exhibits its particular geometric tendency or attribute, i.e.…”

Fast Signal Separation of 2-D Sparse Mixture via Approximate Message-Passing

“…Practical solving of (3) has been considered in the works of [4], [5] via the Templates for First-Order Conic Solvers (TFOCS) [7] and the Split Bregman iteration [8], respectively. However, their examples are not in the context of our problem setup since they do not contain the concept of linear mixing by setting A = I.…”