Improved Coherence Index-Based Bound in Compressive Sensing

Abstract

Within the compressive sensing paradigm, sparse signals can be reconstructed based on a reduced set of measurements. The reliability of the solution is determined by its uniqueness. With its mathematically tractable and feasible calculation, the coherence index is one of very few CS metrics with considerable practical importance. In this paper, we propose an improvement of the coherence-based uniqueness relation for the matching pursuit algorithms. Starting from a simple and intuitive derivation of the standard uniqueness condition, based on the coherence index, we derive a less conservative coherence indexbased lower bound for signal sparsity. The results are generalized to the uniqueness condition of the l0-norm minimization for a signal represented in two orthonormal bases.

On the sparsity bound for the existence of a unique solution in compressive sensing by the Gershgorin theorem

“…In this paper, a relaxed coherence index condition will be derived for the existence of the unique solution of the compressive sensing problem, using the Gershgorin disk theorem. This result guarantees the unique solution existence, but not its reconstruction, meaning that the obtained bound can be relaxed as compared to the one introduced in [27]. The new result for the sparsity bound will be related to the classical one and those proposed in [27], as well as illustrated on numerical examples.…”

“…This result guarantees the unique solution existence, but not its reconstruction, meaning that the obtained bound can be relaxed as compared to the one introduced in [27]. The new result for the sparsity bound will be related to the classical one and those proposed in [27], as well as illustrated on numerical examples.…”

“…Improved Bound: The coherence index bound is, by definition, pessimistic since it takes the worst value µ for all µ mk in (12). Like in [27], when the coherence index was analyzed, we may improve the coherence index-based bound in the Gershgorin disc theorem derivation using the sum of the (2K − 1) largest absolute values instead of using (2K − 1) times the largest absolute value µ, that is…”

“…An approach to improve the coherence index-based bound has been proposed in [27] by analyzing the initial estimate and the support uncertainty principle as in [28,29]. The approach presented in [27] guarantees unique reconstruction of a sparse signal using the orthogonal matching pursuit approach.…”

“…An approach to improve the coherence index-based bound has been proposed in [27] by analyzing the initial estimate and the support uncertainty principle as in [28,29]. The approach presented in [27] guarantees unique reconstruction of a sparse signal using the orthogonal matching pursuit approach. In this paper, a relaxed coherence index condition will be derived for the existence of the unique solution of the compressive sensing problem, using the Gershgorin disk theorem.…”