Deriving RIP sensing matrices for sparsifying dictionaries
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Compressive sensing involves the inversion of a mapping SD ∈ℝ^m × n, where m < n, S is a sensing matrix, and D is a sparisfying dictionary. The restricted isometry property is a powerful sufficient condition for the inversion that guarantees the recovery of high-dimensional sparse vectors from their low-dimensional embedding into a Euclidean space via convex optimization. However, determining whether SD has the restricted isometry property for a given sparisfying dictionary is an NP-hard problem, hampering the application of compressive sensing. This paper provides a novel approach to resolving this problem. We demonstrate that it is possible to derive a sensing matrix for any sparsifying dictionary with a high probability of retaining the restricted isometry property. In numerical experiments with sensing matrices for K-SVD, Parseval K-SVD, and wavelets, our recovery performance was comparable to that of benchmarks obtained using Gaussian and Bernoulli random sensing matrices for sparse vectors.
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