New Restricted Isometry Condition Using Null Space Constant for Compressed Sensing

Abstract

It has been widely recognized that in compressed sensing, many restricted isometry property (RIP) conditions can be easily obtained by using the null space property (NSP) with its null space constant (NSC) 0<θ≤1 to construct a contradicted method for sparse signal recovery. However, the traditional NSP with θ=1 will lead to conservative RIP conditions. In this paper, we extend the NSP with 0<θ<1 to a scale NSP, which uses a factor τ to scale down all vectors belonged to the Null space of a sensing matrix. Following the popular proof procedure and using the scale NSP, we establish more relaxed RIP conditions with the scale factor τ, which guarantee the bounded approximation recovery of all sparse signals in the bounded noisy through the constrained ℓ₁ minimization. An application verifies the advantages of the scale factor in the number of measurements.