Finite alphabet phase retrieval

TL;DR

This paper investigates the finite alphabet phase retrieval problem, showing that for generic alphabets, signal recovery from Fourier magnitudes reduces to a combinatorial problem involving difference sets, with implications for X-ray crystallography.

Contribution

It establishes a novel connection between finite alphabet phase retrieval and combinatorial difference set problems, especially for sparse signals including zero, relevant to biological imaging.

Findings

Recovery from Fourier magnitudes is equivalent to difference set analysis.

The results apply to sparse signals with zero in the alphabet.

The approach is relevant for X-ray crystallography applications.

Abstract

We consider the finite alphabet phase retrieval problem: recovering a signal whose entries lie in a small alphabet of possible values from its Fourier magnitudes. This problem arises in the celebrated technology of X-ray crystallography to determine the atomic structure of biological molecules. Our main result states that for generic values of the alphabet, two signals have the same Fourier magnitudes if and only if several partitions have the same difference sets. Thus, the finite alphabet phase retrieval problem reduces to the combinatorial problem of determining a signal from those difference sets. Notably, this result holds true when one of the letters of the alphabet is zero, namely, for sparse signals with finite alphabet, which is the situation in X-ray crystallography.