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Compressive sensing involves the inversion of a mapping $SD \in \mathbb{R}^{m\times n}$, where $m < n$, $S$ is a sensing matrix, and $D$ is a sparisfyingdictionary. The restricted isometry property is a powerful sufficient conditionfor the inversion that guarantees the recovery of high-dimensional sparsevectors from their low-dimensional embedding into a Euclidean space via convexoptimization. However, determining whether $SD$ has the restricted isometryproperty for a given sparisfying dictionary is an NP-hard problem, hamperingthe application of compressive sensing. This paper provides a novel approach toresolving this problem. We demonstrate that it is possible to derive a sensingmatrix for any sparsifying dictionary with a high probability of retaining therestricted isometry property. In numerical experiments with sensing matricesfor K-SVD, Parseval K-SVD, and wavelets, our recovery performance wascomparable to that of benchmarks obtained using Gaussian and Bernoulli randomsensing matrices for sparse vectors.

Deriving RIP sensing matrices for sparsifying dictionaries