We investigate the recovery of vectors from magnitudes of frame coefficients
when the frames have a low redundancy, meaning a small number of frame vectors
compared to the dimension of the Hilbert space. We first show that for vectors
in d dimensions, 4d-4 suitably chosen frame vectors are sufficient to uniquely
determine each signal, up to an overall unimodular constant, from the
magnitudes of its frame coefficients. Then we discuss the effect of noise and
show that 8d-4 frame vectors provide a stable recovery if part of the frame
coefficients is bounded away from zero. In this regime, perturbing the
magnitudes of the frame coefficients by noise that is sufficiently small
results in a recovery error that is at most proportional to the noise level.Comment: 12 pages AMSLaTeX, 1 figur