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.

Compressive sensing is a promising technique in data acquisition field. A central problem in compressive sensing is that for a given sparse signal, we wish to recover it accurately, efficiently and stably from very few measurements. Inspired by mathematical analysis, we introduce a combining scheme between stability and robustness in reconstruction problems using compressive sensing. By choosing appropriate parameters, we are able to construct a condition for reconstruction map to perform properly.