Phase retrieval of complex and vector-valued functions
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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/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. 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.
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