Optimal D-RIP bounds in compressed sensing

Abstract

This paper establishes new bounds on the restricted isometry constants with coherent tight frames in compressed sensing. It is shown that if the sensing matrix A satisfies the D-RIP condition δk < 1/3 or \(\delta _{2k} < \sqrt 2 /2\), then all signals f with D*f are k-sparse can be recovered exactly via the constrained l1 minimization based on y = Af, where D* is the conjugate transpose of a tight frame D. These bounds are sharp when D is an identity matrix, see Cai and Zhang’s work. These bounds are greatly improved comparing to the condition δk < 0.307 or δ2k < 0.4931. Besides, if δk < 1/3 or \(\delta _{2k} < \sqrt 2 /2\), the signals can also be stably reconstructed in the noisy cases.