Two-step PR-scheme for recovering signals in detectable union of cones by magnitude measurements

Abstract

Motivated by the research on sampling problems for a union of subspaces (UoS), we investigate in this paper the phase-retrieval problem for the signals that are residing in a union of (finitely generated) cones (UoC for short) in ${\mathbb{R}}^n$. We propose a two-step PR-scheme: $PR=detection+recovery$. We first establish a sufficient and necessary condition for the detectability of a UoC, and then design a detection algorithm that allows us to determine the cone where the target signal is residing. The phase-retrieval will be then performed within the detected cone, which can be achieved by using at most $\Gamma$-number of measurements and with very low complexity, where $\Gamma (\le n)$ is the maximum of the ranks of the generators for the UoC. Numerical experiments are provided to demonstrate the efficiency of our approach, and to exhibit comparisons with some existing phase-retrieval methods.

References in corpus (3)

• Xampling: Signal Acquisition and Processing in Union of Subspaces
• On signal reconstruction without noisy phase
• Compressed Sensing from Phaseless Gaussian Measurements via Linear Programming in the Natural Parameter Space