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Don't take it lightly: Phasing optical random projections with unknown operators
Sidharth Gupta · Remi Gribonval · Laurent Daudet · Ivan Dokmanić

Wed Dec 11 05:00 PM -- 07:00 PM (PST) @ East Exhibition Hall B + C #112
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. Such optics-based matrix multiplications can be much faster and energy-efficient than their CPU or GPU counterparts, yet two difficulties must be resolved: 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, conjugating or 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 in both numerical simulations and 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 improving the measured magnitudes.

Author Information

Sidharth Gupta (University of Illinois at Urbana-Champaign)
Remi Gribonval (INRIA)
Laurent Daudet (LightOn)

Laurent is CTO at LightOn, a startup developing optical coprocessors for Machine Learning. Laurent is currently on leave from his position of full Professor of Physics at the Université Paris Diderot, Paris. He is a graduate from Ecole Normale Supérieure and holds a PhD in Applied Mathematics from Marseille University. He is a former fellow of the Institut Universitaire de France, which recognizes the top 2% of French university professors for their research excellence. In parallel, he held various academic positions : Visiting Scholar at Stanford University, USA, Visiting Senior Lecturer at Queen Mary University of London, UK, Visiting Professor at the National Institute for Informatics in Tokyo, Japan. Laurent has been a consultant to various small and large companies, and is a co-inventor in several patents. Laurent is also a Senior Member of the IEEE and an elected member of the IEEE AASP Technical Committee as well as a former Associate Editor of IEEE TSALP. He has co-authored more than 50 peer-reviewed journal articles, and over 120 conference proceedings.

Ivan Dokmanić (University of Basel)

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