This article presents novel results concerning the recovery of signals from
undersampled data in the common situation where such signals are not sparse in
an orthonormal basis or incoherent dictionary, but in a truly redundant
dictionary. This work thus bridges a gap in the literature and shows not only
that compressed sensing is viable in this context, but also that accurate
recovery is possible via an L1-analysis optimization problem. We introduce a
condition on the measurement/sensing matrix, which is a natural generalization
of the now well-known restricted isometry property, and which guarantees
accurate recovery of signals that are nearly sparse in (possibly) highly
overcomplete and coherent dictionaries. This condition imposes no incoherence
restriction on the dictionary and our results may be the first of this kind. We
discuss practical examples and the implications of our results on those
applications, and complement our study by demonstrating the potential of
L1-analysis for such problems