Phase retrieval in crystallography and optics

Abstract

Phase problems occur in many scientific disciplines, particularly those involving remote sensing using a wave field. Although there has been much interest in phase retrieval in optics and in imaging in general over the past decade, phase retrieval has a much longer history in x-ray crystallography, and a variety of powerful and practical techniques have been developed. The nature of crystallography means that crystallographic phase problems are distinct from those in other imaging contexts, but there are a number of commonalities. Here the principles of phase retrieval in crystallography are outlined and are compared and contrasted with phase retrieval in general imaging. Uniqueness results are discussed, but the emphasis is on phase-retrieval algorithms and areas in which results in one discipline have, and may, contribute to the other.

Vectorial Phase Retrieval of 1-D Signals

“…Property (2) readily follows from (36) for . Similarly, we have that (39) Using (36) yields property (3). The Interference Residual : The following lemma describes some statistical properties of the interference residuals .…”

“…However, other or additional interferences of the form are possible and can easily be incorporated in our approach. 3 The reason is that from these three measurements one may compute , which corresponds to a real-valued signal with support of size , which may then be recovered via the methods of [38].…”

“…Lemma 1: Let be given by (35 (4), when evaluated at the true unknown phases, for . Combining (3) and (4) with (34) and the assumption of a finite support gives, up to second order noise terms (37) That is, at the true phases, the residual values depend both on the noise realization and on the unknown phase vector . The following lemma characterizes some statistical properties of this residual.…”

“…Phase retrieval applications range from X-ray crystallography [3], nuclear magnetic resonance [4], astrophysics [5], lens-less imaging [6]- [11] and radar [12] to characterization of ultrashort pulses [13], [14].…”

Coherent X-ray diffractive imaging on the single-cell-level of microbial samples

“…Good introductions can be found in text book literature such as [6,223] and reviews [216,243,297] and [198], where the reader nds a treatment of the crystallographic phase problem. The reader who is interested in practical introductions with an emphasis on the experimental side is referred to the doctoral theses [95,108,267,293].…”