Stable phase retrieval with low-redundancy frames

2013/02/22 by Bernhard G. Bodmann, Bodmann, Bernhard G., Nathaniel Hammen +1 · 3 citations
Physics and Astronomy · Computer Science · #Advanced X-ray Imaging Techniques #Optical measurement and interference techniques #Laser-Plasma Interactions and Diagnostics

paper · pdf · doi:10.48550/arxiv.1302.5487

Abstract

We investigate the recovery of vectors from magnitudes of frame coefficients when the frames have a low redundancy, meaning a small number of frame vectors compared to the dimension of the Hilbert space. We first show that for vectors in d dimensions, 4d-4 suitably chosen frame vectors are sufficient to uniquely determine each signal, up to an overall unimodular constant, from the magnitudes of its frame coefficients. Then we discuss the effect of noise and show that 8d-4 frame vectors provide a stable recovery if part of the frame coefficients is bounded away from zero. In this regime, perturbing the magnitudes of the frame coefficients by noise that is sufficiently small results in a recovery error that is at most proportional to the noise level.

Citations

• Phase Recovery, MaxCut and Complex Semidefinite Programming

Cited by

• Phase retrieval for the Cauchy wavelet transform
• Phase Retrieval in \mathbb C⁴ Requires Exactly Eleven Measurements
• On Lipschitz Analysis and Lipschitz Synthesis for the Phase Retrieval Problem

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