Phase retrieval for wavelet transforms

TL;DR

This paper introduces a multiscale iterative algorithm for phase retrieval from wavelet transform modulus, enabling precise, noise-stable reconstruction of signals with linear complexity, applicable to large audio and non-audio signals.

Contribution

It presents a novel phase retrieval algorithm leveraging holomorphic extension of wavelet transforms, improving efficiency and stability in signal reconstruction.

Findings

Reconstruction is precise and stable to noise.

Algorithm has linear complexity, suitable for large signals.

Effective on both audio and non-audio signals.

Abstract

We describe a new algorithm to solve a particular phase retrieval problem, that has wide applications in audio processing: the reconstruction of a function from its scalogram, that is from the modulus of its wavelet transform. It is a multiscale iterative algorithm. To efficiently propagate phase information from low to high frequencies, it uses an equivalent formulation of the phase retrieval problem that involves the holomorphic extension of the wavelet transform. Our numerical results, on audio and non-audio signals, show that reconstruction is precise and stable to noise. The algorithm has a linear complexity in the size of the signal, up to logarithmic factors, and can thus be used on large signals.