CCF: AF: Small: Volumetric Mesh Mapping

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With the rapid development of volumetric acquisition and computational technologies in numerous applications, such as industrial inspection and medical imaging, there is a growing need for tools for processing such datasets to analyze the topology and geometry, including volumetric mapping to canonical structures, volumetric registration, volumetric feature extraction, geometric database indexing, volumetric parameterization, etc. In this project the PI and his team will develop rigorous algorithms for computing the topology and geometry for general mesh volumes, a nontrivial and fundamentally more challenging problem than for surfaces. To capture the tunnels, handles, and voids of a volume, homology and cohomology groups need to be computed; to describe the tangling, twisting, and linking patterns among the handles, tunnels, and voids, fundamental groups need to be computed as well. Because it is NP-hard to verify whether two fundamental groups are isomorphic, conventional algebraic topological methods are inadequate and geometric structures such as Ricci flows need to be incorporated. Thus, a major focus of this project is development of computational algorithms for Ricci flows. On the other hand, it is highly desirable to map one or more volumes to a canonical domain, in order to support database indexing and volume registration, yet it is an open problem to obtain canonical geometric structures for volumes using computational methodologies. The PIs are confident that Ricci flows are the key to solving this problem, too. Project outcomes will include rigorous computational algorithms for processing volumetric data based on 3-manifold topology and geometric structures, which will be developed in a sequence of interrelated steps as follows: design and implementation of algorithms to compute topological invariants, including homology, cohomology groups, and fundamental groups; design and implementation of algorithms to compute canonical geometric structures, Riemannian metrics with constant section curvatures for discrete 3-manifolds, based on curvature flow and differential forms; design and implementation of algorithms to incorporate canonical geometric structures to the topological invariants, such as finding the closed geodesics and minimal surfaces as homotopy class representatives; design and implementation of algorithms for volumetric mapping to canonical structures, volumetric registration, volumetric feature extraction, and volumetric parameterization; design and implementation of parallel version of the above algorithms, and use of a GPU cluster for speed up; and investigation of the complexity, the stability, and the convergence rate of the above algorithms.Broader Impacts: The PIs will build and disseminate a concrete set of software tools for computing and visualizing the topology and geometric structures for mesh volumes, including volumetric parameterization, volumetric registration, volumetric mapping to canonical structures, fundamental groups computation, and topological and geometric feature extraction. Diverse disciplines such as engineering, science, medicine, computer graphics, vision, scientific computing, and mathematics, as well as a host of industrial applications, will directly benefit from these tools, which can be used for volumetric texture mapping, spline volume construction, volumetric deformation, etc.