Sign of gaussian curvature from curve orientation in photometric space

We compute the sign of Gaussian curvature using a purely geometric definition. Consider a point: p on a smooth surface Sand a closed curve gamma on S which encloses p. The image of gamma on the unit normal Gaussian sphere is a new curve beta. The Gaussian curvature at p is defined 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 determined by the relative orientations of the closed curves gamma and beta. We directly compute the relative orientation of two such curves from intensity data. We employ three unknown illumination conditions to create a photometric scatter plot. This plot is in one-to-one correspondence with the subset of the unit Gaussian sphere containing the mutually illuminated surface normals. This permits direct computation of the 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 analysis on simulated images shows the accuracy of our technique. We also demonstrate the performance of this methodology on empirical data.