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

Chapter 4: Algorithm Overview 46These equations for continuous functions of x can be extended to contours of imageswithout much error by using the difference operator. We identified two key methodsof computing curvature for 2D contours from [Oddo, 1992] and [Abidi, 1995]. Oddofollows the strict definition of curvature in the continuous case and extends it to thedigitized curves. He argues that the turning angle at every pixel on the boundary withtwo other points at a fixed pixel distance is proportional to the curvature at that pixel.Abidi [Abidi, 1995] uses a method based on polar coordinates to estimate curvature.We have used a second order differential operator on the boundary contour toapproximate curvature which is also proportional to the second derivative of afunction for our implementation.Curvature estimation on surfaces is a more challenging research area. After a detailedsurvey of the literature we would like to emphasize the fact that most of the researchon curvature estimation is in the context of range images with very little of it suited forthe general problem on surface meshes. Flynn et al. [Flynn and Jain, 1989] and Suk etal. [Suk and Bhandarkar, 1992] offer us with surveys on curvature from rangemethods. These methods give an insight of the fundamental problems that we mightencounter with surface meshes. One of the major assumptions that we make withrange images that prevents us from extending them to triangle meshes is that of aregular grid structure and consistent topology that might not be always the case withpolygonal meshes. In the next few paragraphs, we present a brief survey on differentcurvature measures on triangle meshes. Triangle meshes are the most common outputof 3D scanners and is assumed as a piecewise approximation to a surface.Surface fitting methods try to apply concepts of differential geometry on surfaceapproximations. An analytic surface is fit to the region of interest and curvature iscomputed from that functional approximation. Surface fitting methods do not differmuch from the curvature-from-range methods because of the planar topology of fittedsurfaces and range images. Surface fitting aside, researchers have tried to estimatecurvature using curve fitting methods as well. A family of curves is fit around a pointon the surface and the ensemble is used to compute principal curvatures. Besl and Jain[Besl and Jain, 1986] construct a local parameterization of the surface and estimatecurvature by fitting orthogonal polynomials followed by a series of convolutionoperations. Stokely and Wu [Stokely and Wu, 1992] present five practical solutions,the characterized Sander-Zucker approach, two novel methods based on direct surfacemapping, a piecewise linear manifold technique, and a turtle geometry method. One ofthe new methods, called the cross patch (CP) method, is shown to be very fast, robustin the presence of noise, and is based on a proper surface parameterization, providedthe perturbations of the surface over the patch neighborhood are isotropicallydistributed. Kresk et al. [Kresk et al., 1998] summarize their experience with circlefitting, paraboloid fitting and the Dupin cyclide method. These three methods do notassume that the sample points be on a regular grid. They accept the speed andaccuracy of the circle fitting method but doubt the robustness on dense polygonal