Phase retrieval of complex and vector-valued functions

Abstract

The phase retrieval problem in the classical setting is to reconstruct real/complex functions from the magnitudes of their Fourier/frame measurements. In this paper, we consider a new phase retrieval paradigm in the complex/quaternion/vector-valued setting, and we provide several characterizations to determine complex/quaternion/vectorvalued 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. Our characterization in the scalar setting implies the well-known equivalence between the complement property for linear measurements Φ and the phase retrieval of linear space S. In this paper, we also discuss the affine phase retrieval of vector-valued functions in a linear space and the reconstruction of vector fields on a graph, up to an orthogonal matrix, from their absolute magnitudes at vertices and relative magnitudes between neighboring vertices.

Inertial Proximal ADMM for Separable Multi-Block Convex Optimizations and Compressive Affine Phase Retrieval

“…The above affine phase retrieval problem arises in holography [41], data separation [24,43], phaseless sampling [20], phase retrieval with background information [25,56], and phase retrieval with reference signal [3,5,6,36,37]. A sufficient and necessary condition on the pair (A, b) of measurement matrix and reference vector is introduced in [19,27] so that any (sparse) real vector x is uniquely determined by its affine quadratic measurements |Ax + b| 2 in (1.7). However the reconstruction of the sparse real vector x ∈ R n from its affine quadratic measurements is highly nonlinear and notoriously difficult to solve numerically and stably.…”