Sparsity pattern recovery using FRI methods

The problem of finding the sparse representation of a signal has attracted a lot of attention over the past years. In particular, uniqueness conditions and reconstruction algorithms have been established by relaxing a non-convex optimisation problem. The finite rate of innovation (FRI) theory is an alternative approach that solves the sparsity problem using algebraic methods based around Prony's algorithm. Recent extensions to this framework have shown that it is possible to recover sparse representations beyond the uniqueness limits, that is, finding all the possible sparse representations that fit the observation for the case of signals which are sparse in the union of Fourier and canonical bases. In this paper, we show the application of such methods to the case of the union of DCT and Haar basis. We present an extension that takes advantage of the even symmetry of the cosine functions to build an algorithm that can operate over the observed vector and in a dual domain. We also analyse the case of the union of frames. Simulation results confirm the validity of this new approach and show that it outperforms state of the art algorithms in a number scenarios.