Optimal D-RIP bounds in compressed sensing

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 δ_k < 1/3 or δ_2k < √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 δ_k < 0.307 or δ_2k < 0.4931. Besides, if δ_k < 1/3 or δ_2k < √2/2, the signals can also be stably reconstructed in the noisy cases.