Two-step PR-scheme for recovering signals in detectable union of cones by magnitude measurements
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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 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 Γ-number of measurements and with very low complexity, where Γ (≤ 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.
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