卷积曲面在计算机动画中的应用研究

A convolution surface is an iso-surface in a scalar field defined by convolving a skeleton, comprising points, curves, surfaces, or volumes, with a potential function. Convolution surface has the advantage of being crease-free and bulge-free over other kinds of implicit surfaces. While convolution surfaces are attractive for modeling natural phenomena and objects of complex evolving topology, the analytical evaluation of integrals of convolution models still poses some open problems. Line segments can be considered the most fundamental since they can approximate curve skeletons. Existing analytical models for line-segment skeletons assumes uniform weight distributions and thus can only produce constant-radius convolution surfaces. For Cauchy kernel, we propse an analytical solution for convolving line-segment skeletons with a variable kernel modulated by a polynomial function, thus allowing generalized cylindrical convolution surfaces to be conveniently modeled. Its computation requirement is competitive compared with the case of uniform weight distribution. We also derive the closed-form formulae for most classical kernel functions, namely Gaussian, inverse linear, inverse squared, and quartic functions, and compare their computational complexity. A new competitive kernel function that has a smoothness control parameter is proposed. We propse some novel analytical convolution solutions for arcs and quadratic spline curves with a varying kernel. In addition, we approximate planar higher-degree polynomial spline curves with optimal arc splines within a prescribed tolerance and sum the potential function of each arc to approximate the field for the entire spline curve. We propose a new fast distance surface computation approach based on optimized arc splines approximation for 2D curves skeletons. By taking star-shape objects as special implicit surfaces, we propose a new three-dimensional deformation method using directional polar coordinates. Experiments show the new methods presented in this project are of import value in geometric modeling and computer animation.

元球造型和动画技术是近十年来才兴起的一种与传统的参数曲面造型和动画完全不同的新方法，它在计算机动画中得到了越来越广泛的应用。卷积曲面是元球从零维骨架向高维骨架的推广。本项课题旨在研究基于多项式卷积核的解析卷积曲面模型，卷积曲面之间的ｍｏｒｐｈｉｎｇ模型，动态卷积曲面的快速光线跟踪模型。