Phase Retrieval under Sparsity Constraints
Phase loss problems occur in various fields, particularly those involving imaging and optics. In this talk, we present our work on phase retrieval for sparse signals in one and multiple dimensions, under both discrete and continuous settings. We first establish the conditions under which the phase of a sparse signal can be uniquely recovered, extending and improving previous results related to the "turnpike problem" in combinatorics. We then present a reconstruction algorithm, called the "peeling algorithm", that solves the phase retrieval problem for discrete and continuous sparse signals. On a discrete domain and at high sparsity levels, the proposed algorithm outperforms Charge Flipping, a state of the art method widely used in X-ray crystallography.
(Joint work with Juri Ranieri, Amina Chebira, and Martin Vetterli)