Iterating the transient light transport matrix for non-line-of-sight imaging

Time-of-flight non-line-of-sight imaging systems measure transient light transport at a visible relay surface to image hidden objects. Such measurements form a transient light transport matrix (TLTM). Because conventional methods capture only lower-dimensional subsets of this matrix, they are largely limited to reconstructing hidden geometry. Here, we show that a full-dimensional first-order TLTM (TLTM-1), measured between visible relay-surface patches, enables recovery of TLTM-2, the time-resolved light transport between hidden-surface patches. Using a dense laser scan and gated 16 × 16 single-photon avalanche diode array, we sample all dimensions of TLTM-1 on a planar relay wall. We then computationally focus virtual illumination and detection independently within the hidden scene, effectively turning the relay surface into a synthetic time-of-flight camera. Experiments and simulations show that TLTM-2 reveals indirect shadows, interreflections, and volumetric scattering. We further demonstrate direct-indirect transport separation, virtual relighting, and dual photography. These results establish an iterative framework for recovering higher-order transport (TLTM-n).