Tuesday, May 31, 2016

Phase Retrieval: Robust phase retrieval with the swept approximate message passing (prSAMP) algorithm / On The 2D Phase Retrieval Problem

Today, we have two preprints on phase retrieval. The first one investigates the use of an AMP solver with a binary measurmeent matrix while the second looks at how a 2D phase retrieval problem can be solved with a analytical approach when it is unique (with no sparsity assumption).

Robust phase retrieval with the swept approximate message passing (prSAMP) algorithm by Boshra Rajaei, Sylvain Gigan, Florent Krzakala, Laurent Daudet

In phase retrieval, the goal is to recover a complex signal from the magnitude of its linear measurements. While many well-known algorithms guarantee deterministic recovery of the unknown signal using i.i.d. random measurement matrices, they suffer serious convergence issues some ill-conditioned matrices. As an example, this happens in optical imagers using binary intensity-only spatial light modulators to shape the input wavefront. The problem of ill-conditioned measurement matrices has also been a topic of interest for compressed sensing researchers during the past decade. In this paper, using recent advances in generic compressed sensing, we propose a new phase retrieval algorithm that well-adopts for both Gaussian i.i.d. and binary matrices using both sparse and dense input signals. This algorithm is also robust to the strong noise levels found in some imaging applications.

On The 2D Phase Retrieval Problem by Dani Kogan, Yonina C. Eldar, Dan Oron

The recovery of a signal from the magnitude of its Fourier transform, also known as phase retrieval, is of fundamental importance in many scientific fields. It is well known that due to the loss of Fourier phase the problem in 1D is ill-posed. Without further constraints, there is no unique solution to the problem. In contrast, uniqueness up to trivial ambiguities very often exists in higher dimensions, with mild constraints on the input. In this paper we focus on the 2D phase retrieval problem and provide insight into this uniqueness property by exploring the connection between the 2D and 1D formulations. In particular, we show that 2D phase retrieval can be cast as a 1D problem with additional constraints, which limit the solution space. We then prove that only one additional constraint is sufficient to reduce the many feasible solutions in the 1D setting to a unique solution for almost all signals. These results allow to obtain an analytical approach (with combinatorial complexity) to solve the 2D phase retrieval problem when it is unique.

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