MSPA-MCS: Collaborative Research: Computer Graphics and Visualization Using Conformal Geometry

The proposed research is to apply conformal surface theory to various geometry representations to compute conformal structures. Using computed conformal structures, new geometry representation and analysis tools can be developed, which will pave the road for advances in multiple fronts of science and engineering. The work proposed herein will especially explore these potentials in computer graphics and visualization. Building up conformal structures recasts many three dimensional (3D) geometric problems into two dimensions (2D) and leads to efficient approaches for a number of fundamental geometric problems. These approaches can then immediately benefit a wide range of applications, such as surface classification, surface matching and shape analysis, geometric modeling, simulation, graphics rendering and visualization. Performing conformal parameterization requires solving large least squares problems. To further push the application of the conformal structure to interactive or time critical operations, the research team will investigate novel iterative methods for least-squares problems based on sparse QR and incomplete sparse QR algorithms. Various scientific and engineering applications concern about key operations such as modeling, design, analysis, simulation, and graphics rendering. All these operations are built on top a foundation of geometry representation. In this project, scientists aim to revolutionize this foundation by introducing a conformal structure uniquely characterizing geometry surfaces. Many laws of physics are governed by conformal structures. For example, heat diffusion and electromagnetic field distribution on surfaces, tension in soap bubbles and parts of string theory in theoretical physics are determined by conformal surface structures. Encouraged by the existing success, the scientists strive to explore and unveil the potentials of conformal structures for computer graphics, geometric modeling and much of scientific computing. To illustrate this potential, consider one aspect of conformal structures, namely the canonical flattening of a surface into a plane, resulting in an image like representation of seemingly complicated three dimensional geometry. Overall, the tools developed in this project can help boost the development of effective techniques to deal with the emerging problems related to scientific simulation, data exploration, and identity matching or shape analysis for surveillance and biological discovery.