Sparse signal recovery via non-convex optimization and overcomplete dictionaries

摘要

In this paper, we address the problem of recovering signals from undersampled data where such signals are not sparse in an orthonormal basis, but in an overcomplete dictionary. We show that if the combined matrix obeys a certain restricted isometry property and if the signal is sufficiently sparse, the reconstruction that relies on ?p minimization with 0 < p < 1 is exact. In addition, under a mild assumption about the dictionary D, we use a similar method [H. Rauhut et al., Compressed sensing and redundant dictionaries, IEEE Trans. Inf. Theory 54(5) (2008) 2210–2219] to derive an estimation of the restricted isometry constant of the composed matrix AD. Finally, the performance of the ?p minimization is testified by some numerical examples.