EAGER: Exploring Spline Theory for Shapes of Arbitrary Topology

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Title: EAGER: Exploring Spline Theory for Shapes of Arbitrary TopologyPI: Hong Qin, SUNY at Stony Brook (Stony Brook University)Project Abstract: The goal of this research is to clearly articulate a novel spline-driven computational method (splines are piecewise polynomials that satisfy certain continuity requirements) for the geometric modeling of shapes that are of arbitrary topological type. In order to achieve this objective, the research is exploring a new mathematical theory of splines over shapes of arbitrarily complicated topology and geometry. The flexible and effective construction of splines defined over an arbitrary manifold has immediate impact for computer-aided geometric design, visual information processing, and computer graphics. Moreover, this theory-centered research will enable a more accurate, more efficient, and easier-to-use software system for processing geometric and scientific datasets. The research is demonstrating that splines are a powerful data modeling, analysis, and simulation tool that not only continue to be applicable to geometric and shape modeling, but also have important extensions to a general data modeling and analysis framework.Traditionally, splines have their mathematical root in approximation theory. Continuous representations such as splines can enable compact representation, analysis (especially quantitative analysis), simulation, and digital prototyping. In order to bridge the large gap between conventional spline formulations and the strong demand to accurately and efficiently model acquired datasets towards quantitative analysis and finite element simulation, our research effort centers on the unexplored mathematical theory of manifold splines that will extend popular spline schemes to effectively represent objects over arbitrary topology. In particular, we are conducting a comprehensive study of new theoretical foundations that can transform the spline-centric representations to the accurate and effective modeling of surfaces of arbitrary topology.