Signal Inpainting from Fourier Magnitudes

TL;DR

This paper introduces novel methods for signal inpainting using Fourier magnitude data, connecting phase retrieval techniques with practical restoration of missing signal segments, demonstrating high recovery success and noise robustness.

Contribution

It formulates the inpainting problem as a phase retrieval challenge and proposes two new algorithms, an alternating minimization and a convex relaxation, for improved signal restoration.

Findings

High probability of successful signal recovery

Robustness to noise in Fourier magnitude measurements

Effective inpainting of speech signal gaps

Abstract

Signal inpainting is the task of restoring degraded or missing samples in a signal. In this paper we address signal inpainting when Fourier magnitudes are observed. We propose a mathematical formulation of the problem that highlights its connection with phase retrieval, and we introduce two methods for solving it. First, we derive an alternating minimization scheme, which shares similarities with the Gerchberg-Saxton algorithm, a classical phase retrieval method. Second, we propose a convex relaxation of the problem, which is inspired by recent approaches that reformulate phase retrieval into a semidefinite program. We assess the potential of these methods for the task of inpainting gaps in speech signals. Our methods exhibit both a high probability of recovering the original signals and robustness to magnitude noise.