Discrete Differential Operators for Computer Graphics

This thesis presents a family of discrete differential operators. Since these operators are derived taking into account the continuous notions of differential geometry, they possess many similar properties. This family consists of firstand second-order properties, both geometric and parametric. These operators are then analyzed and their practical use is tested in several example applications. First, the operators are used in a smoothing application. Due to the properties of the operators, the resulting smoothing algorithm is general, efficient and robust to sampling problems. The smoothing can be applied to many different inputs ranging from images to surfaces to volume data. Second, a surface remeshing technique using the operators is presented. Given the operators, we present an algorithm that resamples a surface mesh according to several geometric criteria (integrated curvature, directional curvature, geometric distortion). The resulting algorithm is efficient, general and user-tunable. Next, a surface mesh parameterization technique is presented. Using geometric invariants associated with the discrete operators, we present an efficient, tunable parameterization algorithm that is robust to sampling irregularities in the input model. Using the properties of the differential operators allows us to make a parameterization algorithm that relies only on geometric information and not the original parameterization of the input model. Finally, we conclude and present future work including physical simulation and sampling theory.