Projections and phase retrieval

Abstract

We characterize collections of orthogonal projections for which it is possible to reconstruct a vector from the magnitudes of the corresponding projections. As a result we are able to show that in an M -dimensional real vector space a vector can be reconstructed from the magnitudes of its projections onto a generic collection of N ≥ 2 M − 1 subspaces. We also show that this bound is sharp when N = 2 k + 1 . The results of this paper answer a number of questions raised in [5] . • We characterize collections of real orthogonal projections which admit phase retrieval. • We prove that 2 M − 1 generic magnitudes suffice for phase retrieval in R M . • We prove that 2 M − 1 magnitudes are necessary when M = 2 k + 1 .