Probablility and Geometry: Applications of Stochastic Models to Geometric Computation

离散概率分布的经典理论与当前计算机图形学和计算机辅助设计(CAGD)中流行的参数曲线和曲面之间有着深刻而基本的联系。Bernstein多项式很久以前就被Bernoulli用来表示二项分布，但现在它们作为Bezier曲线和曲面的混合函数在科学和工程应用中普遍存在。同样，在逼近理论和CAGD中采用B-Spline表示非均匀有理B-Spline(NURB)曲线和曲面之前，Laplace在概率统计中早就知道并应用了B-Spline。泊松分布是二项分布的极限，而高斯密度函数是均匀B-Spline的极限。这些经典的密度和分布接近于二项分布和B-样条分布，这表明它们在科学可视化和几何建模中也有广泛的应用。这项建议的目的是应用这些和其他经典的概率分布来建模自由曲线和曲面，这些曲线和曲面代表了自然科学和工程中的问题的解决方案，这些问题无法通过标准的Bezier和B-Spline技术进行满意的建模。应用包括设计高次多项式曲线和曲面的数值稳定近似，以及研究经典偏微分方程如热方程。还将开发用于快速数据压缩和复杂形状重建的新的高斯小波。将建立一个基于泊松、高斯和其他经典密度和分布的曲线和曲面的原型设计/建模系统，以研究这些问题以及在计算科学和工程中的更多应用。这项研究可能会通过引入概率和统计的新方法和表示法来重振几何建模和计算几何，增加可以解析表示的形状的范围和可以计算解决的问题的范围。通过借鉴近似论和CAGD中的技术，这项研究也有望为一些经典的概率分布提供新的线索。

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.