Ambiguities in one-dimensional phase retrieval from Fourier magnitudes

Abstract

In many scientific areas, such as astronomy, electron microscopy, and crystallography, one is faced with the problem to recover an unknown signal from the magnitudes of its Fourier transform. Unfortunately, this phase retrieval problem is complicated by the well-known ambiguousness. In order to find the original signal within the solution set, one hence requires further information about the unknown signal. The dissertation on hand aims to characterize the complete solution set of the one-dimensional phase retrieval problem and to investigate how far additional data or a priori conditions can ensure uniqueness. For this purpose, we firstly restrict ourselves to the recovery of complex-valued discrete-time signals with finite support. Using a novel approach, we here give a complete characterization of all occurring ambiguities. Moreover, we show that each further solution of the discrete-time phase retrieval problem can be described by an appropriate convolution representation of the original signal and by suitable rotations, shifts, and conjugations and reflections of the appearing factors. Using our characterization of the solution set, we investigate different a priori condition in order to reduce the number of ambiguities or even to receive a unique solution. Firstly, we consider the assumption that the unknown signal only possesses real and non-negative components. Although this can avoid the appearance of ambiguities in certain cases, the non-negativity cannot ensure uniqueness in general. Further, if we have access to additional magnitudes or phases of the unknown signal in the time domain, we can show that almost all signals with finite support can be uniquely recovered. An analogous result can be obtained by exploiting additional interference measurements. Here we study the interference of the unknown signal with a known or an unknown reference and with modulations of the signal itself. Furthermore, we analyse the continuous-time phase retrieval problem. If the unknown signal possesses a specific structure, we can here transfer most of our previous findings. Based on this observation, we study the relation between the continuous-time and discrete-time problem. For arbitrary continuous-time signals, we can again avoid undesirable ambiguities by employing appropriate interference measurements. Finally, we consider the recovery of an unknown signal from its Fresnel magnitudes. Here the complete solution set can be characterized similarly to the Fourier setting. Moreover, we transfer most of the ideas to enforce uniqueness by using the close relation between the Fourier and Fresnel transform.