Surface curvature from 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 cornerstone of this technique is the use of the integrability constraint which 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 the gradient ratio constants, can be simultaneously solved for from 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 two important applications of this technique: segmentation of the object surface by sign of Gaussian curvature, and further segmentation of non-negative Gaussian curvature into convexity and concavity.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>