Phase retrieval of complex and vector-valued functions

摘要

The phase retrieval problem in the classical setting is to reconstruct real/complex functions from magnitudes of their Fourier/frame measurements. In this paper, we consider a new phase retrieval paradigm in the vector-valued setting, which is motivated by complex conjugate phase retrieval of vectors in the complex range space of a real matrix and functions in the complex Paley-Wiener space, and also by determination of a vector field defined on a graph from their relative magnitudes between neighboring vertices. In this paper, we provide several characterizations to determine complex/vector-valued functions f in a linear space S of (in)finite dimensions, up to a trivial ambiguity, from the magnitudes vertical bar vertical bar phi(f)vertical bar vertical bar of their linear measurements phi(f), phi is an element of Phi, and we apply the characterizations for the recovery of complex functions in a shift-invariant space from their phaseless evaluations and vector fields on a graph from their absolute magnitudes at vertices and relative magnitudes between neighboring vertices. In this paper, we also discuss the affine phase retrieval of vector-valued functions in a linear space and phase retrieval in the quaternion setting. (c) 2022 Elsevier Inc. All rights reserved.