A Theory of Refractive and Specular 3D Shape by Light-Path Triangulation

We investigate the feasibility of reconstructing an arbitrarily-shaped specular scene (refractive or mirror-like) from one or more viewpoints. By reducing shape recovery to the problem of reconstructing individual 3D light paths that cross the image plane, we obtain three key results. First, we show how to compute the depth map of a specular scene from a single viewpoint, when the scene redirects incoming light just once. Second, for scenes where incoming light undergoes two refractions or reflections, we show that three viewpoints are sufficient to enable reconstruction in the general case. Third, we show that it is impossible to reconstruct individual light paths when light is redirected more than twice. Our analysis assumes that, for every point on the image plane, we know at least one 3D point on its light path. This leads to reconstruction algorithms that rely on an “environment matting” procedure to establish pixel-to-point correspondences along a light path. Preliminary results for a variety of scenes (mirror, glass, etc.) are also presented.