Necessary conditions for a unique solution to two-dimensional phase recovery

Abstract

In this paper we show that although in one dimension multiplicity of solutions to the phase reconstruction problem presents a serious problem, in two or more dimensions multiplicity is pathologically rare. We derive from a given solution pair (g,G) necessary conditions for the existence of alternative solution pairs (h,H), and a characterization of their form. The mathematical tools employed are from the theory of functions of two complex variables.

An approach to three-dimensional structures of biomolecules by using single-molecule diffraction images

“…However, this is often not the case. Mathematically, it has been shown that multiple solutions are rare for two-dimensional (2D) and 3D specimens (11). By using this oversampling method, the phase information has recently been retrieved ab initio from the experimental diffraction pattern of a 2D noncrystalline specimen (a series of dots of 1͞10th micron dimensions) (12).…”

A Bayesian Approach to Surface X-ray Diffraction

“…The possibility of inverting a set of oversampled Fourier amplitudes (without phases) depends on the dimensionality: There is no unique solution in 1D [18], but it has been argued that one may be found if oversampling is possible in 2D and 3D [19,20]. In the seminal works [12,13], the object recovered was a 2D projection of 2 non-periodic dimensions.…”

Phase retrieval techniques for radar ambiguity problems

“…In higher dimension an entire function can be irreducible and the above factorisation may be trivial i.e. zero-flipping may be impossible, see [19], [4] for details. Example 1 : Let H be a fixed function with support in −…”