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 S and a closed curve y 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 [Beta] as [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. Index Terms - Gaussian curvature, differential geometry, photometric invariant, photometric data, shape recovery, curve orientation.