Phase retrieval of bandlimited functions for the wavelet transform

TL;DR

This paper proves new uniqueness results for wavelet phase retrieval, showing that real-valued bandlimited signals can be uniquely recovered from wavelet magnitude data, including sampled and complex-valued wavelets.

Contribution

It introduces the first uniqueness results for sampled wavelet phase retrieval with complex-valued wavelets and Cauchy wavelet measurements.

Findings

Unique recovery of real-valued bandlimited signals from wavelet magnitudes.

First uniqueness results for sampled wavelet phase retrieval with complex wavelets.

Uniqueness established for phase retrieval from sampled Cauchy wavelet transforms.

Abstract

We study the recovery of square-integrable signals from the absolute values of their wavelet transforms, also called wavelet phase retrieval. We present a new uniqueness result for wavelet phase retrieval. To be precise, we show that any wavelet with finitely many vanishing moments allows for the unique recovery of real-valued bandlimited signals up to global sign. Additionally, we present the first uniqueness result for sampled wavelet phase retrieval in which the underlying wavelets are allowed to be complex-valued and we present a uniqueness result for phase retrieval from sampled Cauchy wavelet transform measurements.