PhaseSense — Signal Reconstruction from Phase-Only Measurements via Quadratic Programming

We consider the problem of reconstructing a complex-valued signal from its phase-only measurements. This framework can be considered as a generalization of the well-known one-bit compressed sensing paradigm where the underlying signal is known to be sparse. In contrast, the proposed formalism does not rely on the assumption of sparsity and hence applies to a broader class of signals. The optimization problem for signal reconstruction is formulated by first splitting the linear measurement vector into its phase and magnitude components and subsequently using the non-negativity property of the magnitude component as a constraint. The resulting optimization problem turns out to be a quadratic program (QP) and is solved using two algorithms: (i) alternating directions method of multipliers; and (ii) projected gradient-descent with Nesterov's momentum. Due to the inherent scale ambiguity of the phase-only measurement, the underlying signal can be reconstructed only up to a global scale-factor. We obtain high accuracy for reconstructing 1-D synthetic signals in the absence of noise. We also show an application of the proposed approach in reconstructing images from the phase of their measurement coefficients. The underlying image is recovered up to a peak signal-to-noise ratio exceeding 30 dB in several examples, indicating an accurate reconstruction.