SGER: Computational Topology for Surface Reconstruction

SGER：表面重建的计算拓扑

• 批准号: 0226504
• 负责人: Thomas Peters
• 金额: $ 10万
• 依托单位: University of Connecticut
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-10-01 至 2005-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0226504&HistoricalAwards=false
• 关键词: SGER; Computational; Topology; Surface; Reconstruction

ABSTRACT0226504Thomas PetersU of ConnecticutResearch on surface reconstruction has prospered by productive interactions between computationalgeometers and classical topologists. The initial reconstruction techniques on unorganized pointdata relied upon visual inspection to verify topological correctness. Recent advances have led torigorous sufficient conditions that guarantee correct topological form first, by equivalence underhomeomorphisms and subsequently by the stronger ambient isotopy. These theoretical advanceshave led to improved surface reconstructions for reverse engineering and medical modeling for asignificant subset of the surfaces considered in those applications. This prompts a speculativeunifying intellectual question: Can contemporary modifications of classical difierential geometry,topology and knot theory lead to a broader scope for computational topology that will result infurther algorithmic improvements? More specifically, can these classical mathematical tools beapplied to extend the domain of surface reconstruction algorithms? This research will explorehow this already successful synergy between classical mathematics and computer science can beaccelerated. Graduate student support is requested and this successful project outcome will thenserve as an exemplar for further student involvement in interdisciplinary work between computerscientists and mathematicians.

计算机几何学家和经典拓扑学家之间富有成效的相互作用使表面重构的研究繁荣起来。初始的散乱点数据重建技术依赖于视觉检查来验证拓扑正确性。最近的进展导致torigorous充分条件，保证正确的拓扑形式，首先，等价同胚，随后由更强的环境合痕。这些理论的进步导致了改进的表面重建的逆向工程和医学建模的一个重要子集的表面考虑在这些应用程序。这引发了一个推测性的统一的智力问题：当代经典微分几何，拓扑学和纽结理论的修改是否会导致更广泛的计算拓扑学范围，从而导致进一步的算法改进？更具体地说，这些经典的数学工具是否可以应用于扩展曲面重建算法的领域？这项研究将探索如何加速经典数学和计算机科学之间已经成功的协同作用。要求研究生的支持，这一成功的项目成果将成为学生进一步参与计算机科学家和数学家之间跨学科工作的典范。