Robust Phase Retrieval Algorithm for Time-Frequency Structured Measurements

TL;DR

This paper introduces a robust phase retrieval algorithm tailored for time-frequency structured measurements, specifically Gabor frames, demonstrating near-optimal measurement efficiency and noise robustness for applications like diffraction imaging and speech recognition.

Contribution

The paper proposes a new phase retrieval algorithm leveraging polarization techniques for Gabor frame measurements, improving robustness and measurement efficiency over existing methods.

Findings

Achieves near-optimal measurement count for phase retrieval.

Demonstrates robustness against measurement noise.

Links geometric properties of measurement frames to reconstruction stability.

Abstract

We address the problem of signal reconstruction from intensity measurements with respect to a measurement frame. This non-convex inverse problem is known as phase retrieval. The case considered in this paper concerns phaseless measurements taken with respect to a Gabor frame. It arises naturally in many practical applications, such as diffraction imaging and speech recognition. We present a reconstruction algorithm that uses a nearly optimal number of phaseless time-frequency structured measurements and discuss its robustness in the case when the measurements are corrupted by noise. We show how geometric properties of the measurement frame are related to the robustness of the phaseless reconstruction. The presented algorithm is based on the idea of polarization as proposed by Alexeev, Bandeira, Fickus, and Mixon.