Reconstruction of Signals from Magnitudes of Redundant Representations: The Complex Case

TL;DR

This paper addresses the problem of reconstructing complex vectors from magnitude-only measurements in redundant representations, introducing new invertibility results and a robust iterative algorithm with performance analysis.

Contribution

It provides novel invertibility conditions and an iterative reconstruction algorithm that is robust to noise in the complex setting.

Findings

New invertibility results for complex signals

An iterative algorithm for least-square reconstruction

Performance analysis showing robustness to noise

Abstract

This paper is concerned with the question of reconstructing a vector in a finite-dimensional complex Hilbert space when only the magnitudes of the coefficients of the vector under a redundant linear map are known. We present new invertibility results as well an iterative algorithm that finds the least-square solution and is robust in the presence of noise. We analyze its numerical performance by comparing it to the Cramer-Rao lower bound.