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 $N\ge 2M-1$ 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 \cite{CCPW:13}.