To the Graduate Council: I am submitting herewith a thesis written by ...

Chapter 4: Algorithm Overview 47meshes. With paraboloid fitting which is slower than the circle fitting method, theypoint out the systematic error that is introduced by the procedure in estimatingcurvature of smooth and uniformly varying surfaces such as spheres and cylinders.The Dupin Cyclide method turns out slower and inaccurate compared to theparaboloid fitting method.Another approach to curvature estimation is to use the geometry and topology of thesurface approximation to estimate curvature. These methods compute total curvatureas a global feature at each of the vertices of the triangle mesh though theoreticallyeach sample point on the mesh is a singularity. Lin and Perry [Lin and Perry, 1982]use the angle excess around a vertex and extend the Gauss-Bonnet Theorem indifferential geometry to define a total curvature measure. They relate it to theGaussian curvature of the surface. Desbrun et al. in [Desbrun et al., 1999] derive anestimate of mean curvature on a triangle mesh based on the loss of angle approach.Delingette [Delingette, 1999] lays out a framework called simplex meshes as a dual oftriangle meshes for surface representation and formulates curvature measures on thesurface very similar to the angle excess method on the triangle mesh. Gourley in[Gourley,1998] attempts to approximate a curvature metric based on the dispersion offace normals around a vertex while Mangan and Whitaker [Mangan andWhitaker,1999] refine a curvature measure further as the norm of a covariance matrixfor the face normals. Chen and Schmitt [Chen and Schmitt, 1992] formulate aquadratic representation of curvature at each vertex to derive principal curvatures byminimizing least squares. Taubin [Taubin, 1995] enhances Chen’s approach into anelegant algorithm that defines a symmetric matrix that has the same Eigen vectors asthe principal directions and Eigen values that are related by a homogenous lineartransformation to principal curvatures. Talking of Eigen analysis [Page, 2001]proposes the idea of normal vector voting that selects a geodesic neighborhood aroundeach vertex. The triangles in this neighborhood vote to estimate the curvature at thespecified vertex. He collects these votes in a covariance matrix and uses Eigenanalysis of the matrix to estimate curvature. The relative size of the neighborhoodcontrols the trade-off between algorithm robustness and accuracy.We would like to summarize by saying that the surface fitting methods require themost computational effort since they typically employ optimization in the fittingprocess. They are robust to noise but cannot deal with discontinuities. Curve fittingmethods on triangle meshes are extremely sensitive to noise yet very simple. Of themethods discussed in the previous paragraphs, we have decided to perform an analysisas to which of these would help us in characterizing surfaces and their complexity. Wechose to compare Gaussian curvature estimates using the paraboloid fitting method,Taubin’s method, angle deficit as curvature and the Gauss-Bonnet’s extension tocurvature estimation. Our comparison differs from [Surazhsky,2003] and[Meyer,2000] because we not only focus on the absolute error in the estimation ofcurvature, but also the effect of resolution on the same surface and how well each one