Toward a mathematical theory of the crystallographic phase retrieval problem

TL;DR

This paper investigates the fundamental uniqueness conditions in crystallographic phase retrieval, proposing conjectures that relate sparsity levels to unique signal recovery, and introduces computational methods to support these claims.

Contribution

It formulates the first mathematical theory for the crystallographic phase retrieval problem, establishing conjectures on uniqueness based on sparsity and symmetry considerations.

Findings

Proposes a computational technique to verify uniqueness for specific (K,N) pairs.

Conjectures that signals are uniquely determined by Fourier magnitude when K <= N/2.

Suggests that solutions are measure zero sets for K < N/2, indicating uniqueness in a generic sense.

Abstract

Motivated by the X-ray crystallography technology to determine the atomic structure of biological molecules, we study the crystallographic phase retrieval problem, arguably the leading and hardest phase retrieval setup. This problem entails recovering a K-sparse signal of length N from its Fourier magnitude or, equivalently, from its periodic auto-correlation. Specifically, this work focuses on the fundamental question of uniqueness: what is the maximal sparsity level K/N that allows unique mapping between a signal and its Fourier magnitude, up to intrinsic symmetries. We design a systemic computational technique to affirm uniqueness for any specific pair (K,N), and establish the following conjecture: the Fourier magnitude determines a generic signal uniquely, up to intrinsic symmetries, as long as K<=N/2. Based on group-theoretic considerations and an additional computational technique, we formulate a second conjecture: if K<N/2, then for any signal the set of solutions to the crystallographic phase retrieval problem has measure zero in the set of all signals with a given Fourier magnitude. Together, these conjectures constitute the first attempt to establish a mathematical theory for the crystallographic phase retrieval problem.