Stable Fourier Phase Retrieval

A complex-valued signal can be understood as a combination of its in- tensity and its phase information. In principle, to determine the signal, both the intensity and the phase have to be known. In various applications, however, one can only record the intensity information but not the phase. The task of recovering the phase, and therefore the full signal from intensity measurements is known as phase retrieval. To name a concrete and highly relevant example where this problem arises we mention diffraction imaging, where a nano-scale object is illuminated by a x-ray beam and typically only the intensity of the diffracted radiation can be observed. Phase retrieval problems are notoriously difficult to study. In the infinite- dimensional setting such problems are inherently unstable, meaning that very different signals can exhibit very similar intensity measurements, which makes the problem rather ill-posed. Within this project we will conduct a comprehensive quantitative study of various instances of phase retrieval problems, with the aim to pinpoint crucial ingredients which make such problems well-posed. Furthermore, we aim to develop algorithms for the numerical solution of phase retrieval prob- lems. 1

In numerous physical applications, we are confronted with the problem that we can only observe the magnitudes of complex-valued measurements, but not the phases. However, in general the phase contains essential information and cannot simply be neglected. It is therefore necessary to recover the missing phase information from the available phase-less measurements. This type of problem is summarized under the term phase retrieval. In the course of this project, various instances of phase retrieval were investigated in detail. The most important results are listed and described below. A central question deals with phase-less sampling in the so-called Fock space. The Fock space consists of holomorphic functions in the complex plane, which are subject to a corresponding growth condition. Assuming such a function, suppose one can only measure the magnitudes of the evaluation on a certain discrete set of points in the plane. How must this set of points be chosen in order to guarantee that each function is uniquely determined by these measurements? Here we were able to show that point sets with this uniqueness property can be obtained from random perturbations of suitable lattice sets. This construction is the first and so far the only one that provides uniqueness sets of finite density. Another main focus was the question of how phase recovery in Fock space can be practically accomplished. The recovery process is inherently unstable. This means that, under certain circumstances, very different initial functions can lead to very similar measurements. In practice, it must be assumed that measurements are subject to error. If the underlying function is an instability, the reconstruction question is therefore ill-posed, as the solution is de facto ambiguous. We have designed an algorithm that takes (possibly noisy) phase-less samples as input data and returns an estimate of the underlying function (with phase!). This algorithm is accompanied by mathematical results which guarantee that the method delivers accurate results, provided that the disturbances are small and there is no instability. The aspect of stability was also the focus of another sub-project. General results state that instabilities for phase retrieval can be generated by constructing functions that consist of two or more components. It is assumed that the converse of this statement is also true, i.e. that every instability must be of this type. So far, this assumption has only been proven for a single case, namely for the Fock space. In the course of this project, it was possible to prove the assumption for a number of other cases. In particular, this means that the characterization of instabilities is not based on the fact that the underlying functions are holomorphic, but is a much more general fact.