International audienc

We consider a phase retrieval problem, where we want to reconstruct a n-dimensional vector from its phaseless scalar products with m sensing vectors, independently sampled from complex normal distributions. We show that, with a suitable initalization procedure, the classical algorithm of alternating projections succeeds with high probability when m ≥ Cn, for some C > 0. We conjecture that this result is still true when no special initialization procedure is used, and present numerical experiments that support this conjecture.

ABSTRACT. We study convex relaxation algorithms for phase retrieval on imaging problems. We show that exploiting structural assumptions on the signal and the observations, such as sparsity, smoothness or positivity, can significantly speed-up convergence and improve recovery performance. We detail numerical results in molecular imaging experiments simulated using data from the Protein Data Bank (PDB).

We consider the phase retrieval problem in which one tries to reconstruct a function from the modulus of its wavelet transform. We study the unicity and stability of the reconstruction.In the case where the wavelets are Cauchy wavelets, we prove that the modulus of the wavelet transform uniquely determines the function up to a global phase. We show that the reconstruction operator is continuous but not uniformly continuous. We describe how to construct pairs of functions which are far away in L 2 -norm but whose wavelet transforms are very close, in modulus. The principle is to modulate the wavelet transform of a fixed initial function by a phase which varies slowly in both time and frequency. This construction seems to cover all the instabilities that we observe in practice; we give a partial formal justification to this fact.Finally, we describe an exact reconstruction algorithm and use it to numerically confirm our analysis of the stability question.