Phase retrieval of complex and vector-valued functions

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 ‖ϕ(f)‖ of their linear measurements ϕ(f),ϕ∈Φ, 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.