Abstract

Phase retrieval refers to the problem of recovering a signal from phaseless measurements. Gabor phase retrieval, in particular, is concerned with reconstruction from the absolute value of the Gabor transform and has applications in the time-frequency analysis of audio signals. From a mathematical point of view, phase retrieval (from frame coefficients) is a challenging problem as it has been shown to be unstable in infinite dimensional Hilbert spaces and severely ill-conditioned in finite dimensional spaces. However, it has also been shown that one can relax the classical stability regime to so-called semi-global phase reconstruction and obtain a stability result for phase retrieval from the continuous Gabor transform in this setting. Recently, we were able to adapt semi-global phase reconstruction to the discrete case and to prove a promising stability result for the discrete Gabor transform. In this contribution, we survey selected highlights from recent research on phase retrieval from frame coefficients with emphasis on phase retrieval from Gabor measurements. In particular, we review results on semi-global stability of phase retrieval in the infinite dimensional case and present our result in the finite dimensional setting.