The generic crystallographic phase retrieval problem
Provides theoretical guarantees for phase retrieval in crystallography and imaging when the signal is sparse in a generic basis.
The paper proves that a sparse signal can be uniquely recovered from its power spectrum up to sign, with sparsity thresholds of ~N/2 for generic vectors and ~N/4 for all vectors, extending prior results to generic bases.
In this paper we consider the problem of recovering a signal $x \in \mathbb{R}^N$ from its power spectrum assuming that the signal is sparse with respect to a generic basis for $\mathbb{R}^N$. Our main result is that if the sparsity level is at most $\sim\! N/2$ in this basis then the generic sparse vector is uniquely determined up to sign from its power spectrum. We also prove that if the sparsity level is $\sim\! N/4$ then every sparse vector is determined up to sign from its power spectrum. Analogous results are also obtained for the power spectrum of a vector in $\mathbb{C}^N$ which extend earlier results of Wang and Xu \cite{arXiv:1310.0873}.