Surface Curvature and Shape Reconstruction from Unknown Multiple Illumination and Integrability

Abstract

We present a multiple illumination technique that directly recovers the viewer-centered curvature matrix, up to a scalar factor, at each mutually illuminated point on a smooth object surface. This technique is completely independent of knowledge of incident illumination orientation, local surface orientation, or diffuse surface albedo. The cornerstore of this technique is the use of theintegrability constraintwhich is a fundamental mathematical property of smooth surfaces. The integrability constraint allows the derivation of an equation at each object point, which is linear in terms of quantities involving the initially unknown parameters of incident illumination orientation. These quantities, which we call thegradient ratio constants, can be simultaneously solved for five or more equations arising from the same number of object points. We show that deriving these gradient ratio constants provides just enough calibration information about incident illumination geometry to compute the viewer-centered curvature matrix at each object point, up to a scalar multiple. We demonstrate three important applications of this technique: (i) segmentation of the object surface by sign of Gaussian curvature, (ii) further segmentation of nonnegative Gaussian curvature into convexity and concavity, and (iii) reconstruction of the surface height, independent of knowledge of incident illumination orientation and diffuse surface albedo.