Don't take it lightly: Phasing optical random projections with unknown operators

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.