Fermat paths are the paths that satisfy Fermat’s principle, meaning they have locally stationary pathlengths. Fermat paths are either specular paths or at object boundary. They are purely geometric, and are invariant to BRDF. In this work, we use Fermat paths for shape reconstruction.

After detecting Fermat paths and their corresponding pathlengths, what can we tell about the points corresponding to these paths?

First, the point must be on a sphere with the collocated virtual source and detector as its center, and half the Fermat pathlength as its radius. Next, we derive the ray constraint which we call the “Fermat flow” constraint. Then, we conduct sphere-ray intersection to pin down the point.

We use a time-domain, full-frame optical coherent tomography system. We use this system to perform confocal scans under both the looking-around-the-corner and looking-through-diffuser settings. We use spatially and temporally incoherent LED illumination, which allows us to combine transient imaging with diagonal probing. In the context of confocal scanning, this means that we can simultaneously collect transients at all points on the visible surface without scanning, as transient measurements taken at one point will not be contaminated with light emanating from a different point. Our implementation has depth resolution of 10um.