SGER: Computational Topology for Surface Reconstruction

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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.