Vectorial Phase Retrieval
Phase retrieval - namely the recovery of a signal from its absolute Fourier transform coefficients is a problem of fundamental importance in many fields. While in 2-D phase retrieval typically has a unique solution, in 1-D it is often ill-posed, admitting multiple solutions.
In this talk we'll present a novel framework for reconstruction of pairs of signals, from measurements of both their spectral intensities, and of their mutual interferences. We show that for noise-free measurements of compactly supported signals, this new setup, denoted vectorial phase retrieval, admits a unique solution already in 1-D. We then derive a computationally efficient and statistically robust spectral algorithm to solve the vectorial phase retrieval problem, as well as a model selection criteria to estimate the unknown support of the signals. We illustrate the reconstruction performance of our algorithm on several simulated signals. We conclude with some yet unresolved challenges - mathematical, statistical and computational.
Joint work with Oren Raz and Nirit Dudovich (Weizmann Institute of Science).