Sampling Without Time: Recovering Echoes of Light via Temporal Phase Retrieval

This paper considers the problem of sampling and reconstruction of a continuous-time sparse signal without assuming the knowledge of the sampling instants or the sampling rate. This topic has its roots in the problem of recovering multiple echoes of light from its low-pass filtered and auto-correlated, time-domain measurements. Our work is closely related to the topic of sparse phase retrieval and in this context, we discuss the advantage of phase-free measurements. While this problem is ill-posed, cues based on physical constraints allow for its appropriate regularization. We validate our theory with experiments based on customized, optical time-of-flight imaging sensors. What singles out our approach is that our sensing method allows for temporal phase retrieval as opposed to the usual case of spatial phase retrieval. Preliminary experiments and results demonstrate a compelling capability of our phase-retrieval based imaging device.