Active imaging systems sample the Transient Light Transport Matrix (TLTM) for a scene by sequentially illuminating various positions in this scene using a controllable light source, and then measuring the resulting spatiotemporal light transport with time of flight (ToF) sensors. Time-resolved Non-line-of-sight (NLOS) imaging employs an active imaging system that measures part of the TLTM of an intermediary relay surface, and uses the indirect reflections of light encoded within this TLTM to "see around corners". Such imaging systems have applications in diverse areas such as disaster response, remote surveillance, and autonomous navigation. While existing NLOS imaging systems usually measure a subset of the full TLTM, development of customized gated Single Photon Avalanche Diode (SPAD) arrays \cite{riccardo_fast-gated_2022} has made it feasible to probe the full measurement space. In this work, we demonstrate that the full TLTM on the relay surface can be processed with efficient algorithms to computationally focus and detect our illumination in different parts of the hidden scene, turning the relay surface into a second-order active imaging system. These algorithms allow us to iterate on the measured, first-order TLTM, and extract a \textbf{second order TLTM for surfaces in the hidden scene}. We showcase three applications of TLTMs in NLOS imaging: (1) Scene Relighting with novel illumination, (2) Separation of direct and indirect components of light transport in the hidden scene, and (3) Dual Photography. Additionally, we empirically demonstrate that SPAD arrays enable parallel acquisition of photons, effectively mitigating long acquisition times.

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