CAREER: Conformal Geometry Applied to Shape Analysis and Geometric Modeling

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Conformal structure is a natural structure associated with Riemann surfaces (General surfaces are Riemann surfaces) and plays fundamental roles in geometry and physics. Conformal structure has proven useful in graphics and vision, for example, conformal parameterization provides high-quality texture mapping without local distortion, and it is used in surface matching and morphing with applications such as brain mapping.The investigator explores the potential of conformal geometry in computer graphics and vision and ultimately makes the interdisciplinary field of computational conformal geometry accessible and useful to the society. With respect to its direct impacts, the investigator applies conformal geometry to geometric modeling and shape analysis. Specifically, 1.Construct shape spaces based on Teichmuller space theory, where the space of all surfaces is modeled as a finite dimensional manifold, each point represents a conformal equivalence class of surfaces (a Riemann surface), and the metric of the shape space measures the deviation of conformal structures of the two shapes. An additional related goal is to build a geometric search engine. 2. Find a systematic way to generalize geometric constructions defined on planar domains to manifolds, such as manifold triangular BSplines and manifold Powell- Sabin surfaces. Design new surface subdivision schemes by inserting knots into manifold BSpline surfaces. With aspect to education, the investigator finds an effective way to teach conformal geometry (Riemann surface theory) to non-math majors by implementing practical algorithms to visualize the abstract concepts and establish the understanding of the profound theories.Conformal geometry theory is fully developed but very abstract. The investigator builds and disseminates a concrete set of software tools to compute and visualize the conformal structure of arbitrary real surfaces, which makes the theory accessible to students and its practical applications useful to the broader community. Manifold BSpline tools based on conformal geometry bridge the gap between traditional polygonal meshes in graphics and spline surfaces in CAGD. The generic geometric search engine is applied to a geometric database and an Internet search engine. The investigator complements the software development with a systematic development of the classic material in a context that permits integration into the curriculum of non-math majors. The fields of computer graphics, vision, scientific computing, medical imaging, mathematics and physics all benefit from the research and education of computational conformal geometry directly. Computational conformal geometry has already made impacts on the graphics industry and will be more broadly applied in the future.