Projections and phase retrieval

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≥2M−1 subspaces. We also show that this bound is sharp when N=2k+1. The results of this paper answer a number of questions raised in [5].