EAGER: Exploring Spline Theory for Shapes of Arbitrary Topology

标题：EIGER：探索任意拓扑形状的样条线理论PI：洪钦，SUNY在石溪(石溪大学)项目摘要：这项研究的目标是清楚地阐明一种新的样条驱动的计算方法(样条线是满足一定连续性要求的分段多项式)，用于任意拓扑类型的形状的几何建模。为了实现这一目标，该研究正在探索一种新的关于任意复杂拓扑和几何形状上的样条线的数学理论。定义在任意流形上的样条曲线的灵活而有效的构造对计算机辅助几何设计、视觉信息处理和计算机图形学有着直接的影响。此外，这种以理论为中心的研究将使处理几何和科学数据集的软件系统更准确、更高效、更易于使用。研究表明，Spline是一种强大的数据建模、分析和仿真工具，不仅继续适用于几何和形状建模，而且对一般数据建模和分析框架有重要的扩展。连续表示(如样条线)可以实现紧凑表示、分析(尤其是定量分析)、仿真和数字原型制作。为了弥补传统的样条公式之间的巨大差距，以及对获取的数据集进行准确而有效的建模的强烈需求，我们的研究工作集中在尚未探索的流形样条数学理论上，该理论将扩展流行的样条方案，以有效地表示任意拓扑上的对象。特别是，我们正在对新的理论基础进行全面的研究，这些新的理论基础可以将以样条为中心的表示转换为任意拓扑的曲面的准确和有效的建模。

Title: EAGER: Exploring Spline Theory for Shapes of Arbitrary TopologyPI: Hong Qin, SUNY at Stony Brook (Stony Brook University)Project Abstract: The goal of this research is to clearly articulate a novel spline-driven computational method (splines are piecewise polynomials that satisfy certain continuity requirements) for the geometric modeling of shapes that are of arbitrary topological type. In order to achieve this objective, the research is exploring a new mathematical theory of splines over shapes of arbitrarily complicated topology and geometry. The flexible and effective construction of splines defined over an arbitrary manifold has immediate impact for computer-aided geometric design, visual information processing, and computer graphics. Moreover, this theory-centered research will enable a more accurate, more efficient, and easier-to-use software system for processing geometric and scientific datasets. The research is demonstrating that splines are a powerful data modeling, analysis, and simulation tool that not only continue to be applicable to geometric and shape modeling, but also have important extensions to a general data modeling and analysis framework.Traditionally, splines have their mathematical root in approximation theory. Continuous representations such as splines can enable compact representation, analysis (especially quantitative analysis), simulation, and digital prototyping. In order to bridge the large gap between conventional spline formulations and the strong demand to accurately and efficiently model acquired datasets towards quantitative analysis and finite element simulation, our research effort centers on the unexplored mathematical theory of manifold splines that will extend popular spline schemes to effectively represent objects over arbitrary topology. In particular, we are conducting a comprehensive study of new theoretical foundations that can transform the spline-centric representations to the accurate and effective modeling of surfaces of arbitrary topology.