The numerics of phase retrieval

Abstract

Phase retrieval, i.e. the problem of recovering a function from the squared magnitude of its Fourier transform, arises in many applications, such as X-ray crystallography, diffraction imaging, optics, quantum mechanics and astronomy. This problem has confounded engineers, physicists, and mathematicians for many decades. Recently, phase retrieval has seen a resurgence in research activity, ignited by new imaging modalities and novel mathematical concepts. As our scientific experiments produce larger and larger datasets and we aim for faster and faster throughput, it is becoming increasingly important to study the involved numerical algorithms in a systematic and principled manner. Indeed, the past decade has witnessed a surge in the systematic study of computational algorithms for phase retrieval. In this paper we will review these recent advances from a numerical viewpoint.

Phase Retrieval With Bregman Divergences and Application to Audio Signal Recovery

“…Problem (1) may be tackled with conventional optimization algorithms such as gradient descent [14], [15], alternating projections [16], [17], majorization-minimization [18], alternating direction method of multipliers (ADMM) [19], [20], and leveraging the structure of time-frequency measurements [21], [22]. An extensive review of those algorithms from a numerical perspective can be found in [23]. Convex optimization approaches are also considered in [24]- [27] by lifting the problem to a higher dimensional space (i.e., solving a constrained quadratic problem involving xx H ) and relaxing the rank-one constraint.…”

Total variation-based reconstruction and phase retrieval for diffraction tomography

“…To overcome these problems, more general phase retrieval problems, where the intensity of arbitrary linear measurements are given, have been studied. Under certain assumptions on the linear measurements, the phase retrieval problem becomes unique [11,12,23,24]. Using a tensorial lifting, the problem may be exemplarily solved using semi-definite programming [11,12], tensor-free primal-dual methods [5], or polarization techniques [2,56].…”

Phase retrieval for nilpotent groups

“…Since its introduction phase retrieval has developed into a legitimate branch of applied mathematics, with links to diverse subjects such as harmonic analysis, algebraic geometry, optimization, numerical analysis, etc. We refer to the review papers [10,8] and their extensive lists of references.…”