Phase Retrieval: An Overview of Recent Developments

The problem of phase retrieval is a classic one in optics and arises when one is interested in recovering an unknown signal from the magnitude (intensity) of its Fourier transform. While there have existed quite a few approaches to phase retrieval, recent developments in compressed sensing and convex optimization-based signal recovery have inspired a host of new ones. This work presents an overview of these approaches. Since phase retrieval, by its very nature, is ill-posed, to make the problem meaningful one needs to either assume prior structure on the signal (e.g., sparsity) or obtain additional measurements (e.g., masks, structured illuminations). For both the cases, we review conditions for the identifiability of the signal, as well as practical algorithms for signal recovery. In particular, we demonstrate that it is possible to robustly and efficiently identify an unknown signal solely from phaseless Fourier measurements, a fact with potentially far-reaching implications.