Probablility and Geometry: Applications of Stochastic Models to Geometric Computation

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There is a deep and fundamental connection between the classical theory of discrete probability distributions and the parametric curves and surfaces currently in vogue in Computer Graphics and Computer Aided Design (CAGD). The Bernstein polynomials were introduced long ago by Bernoulli to represent the binomial distribution, but they are now ubiquitous in scientific and engineering applications as the blending functions for Bezier curves and surfaces. Similarly, B-splines were known and applied by Laplace in Probability and Statistics long before they were adopted in Approximation Theory and CAGD for representing non-uniform rational B-spline (NURB) curves and surfaces.The Poisson distribution is the limit of the binomial distribution and the Gaussian density function is the limit of uniform B-splines. The proximity of these classical densities and distributions to the binomial and B-spline distributions suggests that they too have wide ranging applications in Scientific Visualization and Geometric Modeling. The purpose of this proposal is to apply these and other classical probability distributions to model freeform curves and surfaces which represent solutions to problems in the natural sciences and engineering that cannot be satisfactorily modeled by standard Bezier and B-spline techniques. Applications include devising numerically stable approximations to high order polynomial curves and surfaces, and investigating classical partial differential equations such as the heat equation. New Gaussian wavelets for rapid data compression and reconstruction of complex shapes will also be developed.A prototype design/modeling system based on curves and surfaces generated from Poisson, Gaussian, and other classical densities and distributions will be built to investigate these issues as well as to study additional applications in computational science and engineering. This research is likely to reinvigorate Geometric Modeling and Computational Geometry by introducing novel methods and representations from Probability and Statistics, increasing the range of shapes that can be represented analytically and the scope of problems that can be solved computationally. The research is also expected to shed new light on some classical probability distributions by applying to them techniques borrowed from Approximation Theory and CAGD.