Sparse Signal Separation in Redundant Dictionaries

Abstract—We formulate a unified framework for the separa-tion of signals that are sparse in “morphologically ” different redundant dictionaries. This formulation incorporates the so-called “analysis ” and “synthesis ” approaches as special cases and contains novel hybrid setups. We find corresponding coherence-based 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 x̃ ‖Ψx̃‖1 subject to ‖y −Ax̃‖2 ≤ ε by delivering corresponding coherence-based recovery results. I