Photometric Computation of the Sign of Gaussian Curvature Using a Curve-Orientation Invariant

Abstract

The authors compute the sign of Gaussian curvature using a purely geometric definition. Consider a point p on a smooth surface S and a closed curve /spl gamma/ on S which encloses p. The image of /spl gamma/ on the unit normal Gaussian sphere is a new curve /spl beta/. The Gaussian curvature at p is defined as the ratio of the area enclosed by /spl gamma/ over the area enclosed by /spl beta/ as /spl gamma/ contracts to p. The sign of Gaussian curvature at p is determined by the relative orientations of the closed curves /spl gamma/ and /spl beta/. They directly compare the relative orientation of two such curves from intensity data. They 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. Their method is albedo invariant. They assume diffuse reflectance, but the nature of the diffuse reflectance can be general and unknown. Simulations, as well as empirical results, demonstrate the accuracy of the technique.