SGER: Computational Topology for Surface Reconstruction

康涅狄格州的Thomas PetersU通过计算代数学家和经典拓扑学家之间富有成效的互动，繁荣了曲面重建的研究。对无组织点数据的初始重建技术依赖于视觉检查来验证拓扑正确性。最近的进展首先通过同态下的等价，然后通过更强的环境同伦，得到了保证正确拓扑形式的充分性充分条件。这些理论上的进步已经导致了用于逆向工程的曲面重建的改进和对这些应用中所考虑的曲面的重要子集的医学建模。这引发了一个思考性的统一智力问题：经典微分几何、拓扑学和纽结理论的当代修改是否会导致计算拓扑学的更广泛范围，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古典数学和计算机科学之间已经成功的协同。需要研究生的支持，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。

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.