SIGN OF GAUSSIAN CURVATURE FROM CURVE ORIENTATION IN PHOTOMETRIC SPACE

We compute the sign of Gaussian curvature using a purely geometric def inition. Consider a point: p on a smooth surface Sand a closed curve g amma on S which encloses p. The image of gamma on the unit normal Gaus sian sphere is a new curve beta. The Gaussian curvature at p is define d as the ratio of the area enclosed by gamma over the area enclosed by pas gamma contracts to p. The sign of Gaussian curvature at p is dete rmined by the relative orientations of the closed curves gamma and bet a. We directly compute the relative orientation of two such curves fro m intensity data. We employ three unknown illumination conditions to c reate a photometric scatter plot. This plot is in one-to-one correspon dence with the subset of the unit Gaussian sphere containing the mutua lly illuminated surface normals. This permits direct computation of th e sign of Gaussian curvature without the recovery of surface normals. Our method is albedo invariant. We assume diffuse reflectance, but the nature of the diffuse reflectance can be general and unknown. Error a nalysis on simulated images shows the accuracy of our technique. We al so demonstrate the performance of this methodology on empirical data.