Probablility and Geometry: Applications of Stochastic Models to Geometric Computation

离散概率分布的经典理论与计算机图形学和计算机辅助设计（CAGD）中当前流行的参数曲线和曲面之间有着深刻而基本的联系。伯恩斯坦多项式很久以前就由伯努利引入来表示二项分布，但是它们现在在科学和工程应用中作为Bezier曲线和曲面的混合函数而无处不在。类似地，B样条早在逼近理论和计算机辅助几何设计（CAGD）中用于表示非均匀有理B样条（Non-uniform rational B-spline，RIB）曲线和曲面之前就已经被拉普拉斯（Laplace）应用于概率论和统计学，泊松分布是二项分布的极限，高斯密度函数是均匀B样条的极限。这些经典的密度和分布的接近二项式和B样条分布表明，他们也有广泛的应用在科学可视化和几何建模。这个建议的目的是应用这些和其他经典的概率分布模型的自由曲线和曲面，代表解决问题的自然科学和工程，不能令人满意地模拟标准贝塞尔和B样条技术。应用包括设计数值稳定的近似高阶多项式曲线和曲面，并调查经典的偏微分方程，如热方程。基于泊松、高斯和其他经典密度和分布生成的曲线和曲面的原型设计/建模系统将用于研究这些问题以及研究计算科学和工程中的其他应用。这项研究可能会通过引入新的方法和概率统计表示来重振几何建模和计算几何，增加可以用分析表示的形状范围和可以用计算解决的问题范围。 这项研究也有望通过应用从近似理论和CAGD借来的技术来揭示一些经典的概率分布。

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.