Benchmark Problems for Phase Retrieval

Abstract

In recent years, the mathematical and algorithmic aspects of the phase retrieval problem have received considerable attention. Many papers in this area mention crystallography as a principal application. In crystallography, the signal to be recovered is periodic and comprised of atomic distributions arranged homogeneously in the unit cell of the crystal. The crystallographic problem is both the leading application and one of the hardest forms of phase retrieval. We have constructed a graded set of benchmark problems for evaluating algorithms that perform this type of phase retrieval. The data, publicly available online 1 , is provided in an easily interpretable format. We also propose a simple and unambiguous success/failure criterion based on the actual needs in crystallography. Baseline runtimes were obtained with an iterative algorithm that is similar but more transparent than those used in crystallography. Empirically,the runtimes grow exponentially with respect to a new hardness parameter: the sparsity of the signal autocorrelation. We also review the algorithms used by the leading software packages. This set of benchmark problems, we hope, will encourage the development of new algorithms for the phase retrieval problem in general, and crystallography in particular.

Toward a Mathematical Theory of the Crystallographic Phase Retrieval Problem

“…• Additional information: In many setups, the scientists possess some additional information about the underlying signal; this information may significantly alleviate the reconstruction process. For example, some X-ray crystallography algorithms incorporate knowledge of the minimum atom-atom distance, the presence of a known number of heavy atoms, or even the expected histogram of the signal values [26]. Such information may allow recovering a nonsparse signal even in the regime K > N/2.…”

“…This research thread led to new theoretical, statistical, and computational results in a variety of fields, such as algebraic geometry, statistics, and convex and non-convex optimization. Nevertheless, its contribution to the crystallographic phase retrieval problem is disputable: none of the algorithms that were developed for randomized sensing matrices have been successfully implemented to X-ray crystallography [26]. In contrast, the algorithms that are used routinely by practitioners are based on variations of the Douglas-Rachford splitting scheme.…”

“…Phase retrieval algorithms and computational complexity. While this work focuses on the question of uniqueness, we would like to briefly discuss phase retrieval algorithms and the computational complexity of the crystallographic phase retrieval problem; we refer the reader to [27,24,26,42] for further insights.…”

“…The mathematical model of X-ray crystallography is introduced and discussed at length in [26]. For completeness, we provide a concise summary.…”

On signal reconstruction from FROG measurements

“…Phase retrieval is the problem of estimating a signal from its Fourier magnitude. This problem plays a key role in many scientific and engineering applications, among them X-ray crystallography, speech recognition, blind channel estimation, alignment tasks and astronomy [1,2,3,4,5,6]. Optical applications are of particular interest since optical devices, such as a charge-coupled device (CCD) and the human eye, cannot detect phase information of the light wave [7].…”