Toward a mathematical theory of the crystallographic phase retrieval problem

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.