Phase Retrieval: Uniqueness and Stability

Phase-Retrieval in Shift-Invariant Spaces with Gaussian Generator

“…we have either g = f or g = − f , thus the only ambiguity is the (global) sign of f . Our proof yields a reconstruction procedure (see Section 2), but it is well-known that the phase-retrieval problem in infinite dimensions is necessarily ill-posed [4,6,13].…”

“…Particular problems concern the recovery of a function f from its phaseless Fourier measurements |f (ξ )| or the recovery of a function f from its phaseless (or unsigned) samples | f (λ)|. For general information about the phase-retrieval problem we refer to the surveys [13,18].…”

A note on the phase retrieval of holomorphic functions

“…Furthermore, if F and G are rational functions, then they have same zeros and poles in C \ {0}, thus there exists c ∈ T and m ∈ Z such that G = cz m F . But then (7) implies that -on one hand |F (ρe it )| = |c|ρ m |F (ρe it )| for all t thus |c|ρ m = 1.…”

“…Then F and G have the same zeros and poles in C \ {0}, with the same multiplicities. In particular, if F and G are rational functions that satisfy (7), then G = cF outside the poles with c ∈ T.…”

“…[4] then used this result with conformal mappings to show that a polynomial of degree at most n − 1 can be determined by its magnitude at 4n − 4 well-chosen points in the complex plane. More information on phase retrieval problems from holomorphic measurements can be found in [7].…”