Simultaneous Blind Deconvolution and Phase Retrieval
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- December 27, 2021
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We consider the task of recovering two real or complex L-vectors from phaseless Fourier measurements of their circular convolution. Our method is a novel convex relaxation that is based on a lifted matrix recovery formulation that allows a non trivial convex relaxation of the bilinear measurements from convolution. We prove that if the two signals belong to known random subspaces of dimensions K and N, then they can be recovered up to the inherent scaling ambiguity with L e (K + N)log2 L phaseless measurements. Our method provides the rst theoretical recovery guarantee for this problem by a computationally ecient algorithm and does not require a solution estimate to be computed for initialization. Our proof is based on Rademacher complexity estimates. Additionally, we provide an alternating direction method of multipliers (ADMM) implementation and provide numerical experiments that verify the theory. Figure (left) shows restriction of the hyperbolic constraint to the rst quadrant and figure (right) is an abstract illustration of the geometry of the convex relaxation.
Applications
An important application domain where blind deconvolution from phaseless Fourier measurements arises is the Visible Light Communication (VLC). In visible light communication, there is always a diculty associated with measuring phase information in the received light. The proposed convex program solves this dicult inverse problem and recovers the true solution.
Related Papers
[2] A. Ahmed, A. Aghasi, and P. Hand Simultaneous Phase Retrieval and Blind Deconvolution via Convex Programming, arXiv preprint arXiv:1904.12680, 2019.