2019-07-03: Don't take it lightly: Phasing optical random projections with unknown operators https://arxiv.org/abs/1907.01703v1We showed that measurement phase retrieval can be cast as a problem in distance geometry, and that the unknown phase of random projections can be recovered even without knowing the transmission matrix of the medium

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In this paper we tackle the problem of recovering the phase of complex linear
measurements when only magnitude information is available and we control the
input. We are motivated by the recent development of dedicated optics-based
hardware for rapid random projections which leverages the propagation of light
in random media. A signal of interest $\mathbf{\xi} \in \mathbb{R}^N$ is mixed
by a random scattering medium to compute the projection $\mathbf{y} =
\mathbf{A} \mathbf{\xi}$, with $\mathbf{A} \in \mathbb{C}^{M \times N}$ being a
realization of a standard complex Gaussian iid random matrix. Two difficulties
arise in this scheme: only the intensity ${|\mathbf{y}|}^2$ can be recorded by
the camera, and the transmission matrix $\mathbf{A}$ is unknown. We show that
even without knowing $\mathbf{A}$, we can recover the unknown phase of
$\mathbf{y}$ for some equivalent transmission matrix with the same distribution
as $\mathbf{A}$. Our method is based on two observations: first, changing the
phase of any row of $\mathbf{A}$ does not change its distribution; and second,
since we control the input we can interfere $\mathbf{\xi}$ with arbitrary
reference signals. We show how to leverage these observations to cast the
measurement phase retrieval problem as a Euclidean distance geometry problem.
We demonstrate appealing properties of the proposed algorithm on both numerical
simulations and in real hardware experiments. Not only does our algorithm
accurately recover the missing phase, but it mitigates the effects of
quantization and the sensitivity threshold, thus also improving the measured
magnitudes.