SGER: Computational Topology for Surface Reconstruction

[0226504] thomas PetersU（康涅狄格州）计算几何学者和经典拓扑学家之间富有成效的相互作用使表面重建的研究蓬勃发展。初始的无组织点数据重建技术依赖于视觉检查来验证拓扑的正确性。最近的进展首先通过等价的欠同胚性，然后通过更强的环境同位素，提出了保证正确拓扑形式的严格的充分条件。这些理论的进步已经导致了改进的表面重建，用于逆向工程和医学建模，为这些应用中考虑的表面的一个重要子集。这引发了一个推测性的统一的智力问题：经典微分几何，拓扑和结理论的当代修改是否会导致计算拓扑的更广泛范围，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可以应用于扩展表面重建算法的领域吗？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。研究生的支持是需要的，这个成功的项目结果将作为一个典范，为学生进一步参与计算机科学家和数学家之间的跨学科工作。

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.