Collaborative Research: Modeling and Processing of Topologically Complex 3D Shapes

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DMS-0138445Denis ZorinDMS-0221666Mathieu DesbrunDMS-0220905Peter SchroderThis is a collaborative project funded by the CARGO program underDMS-0138445, DMS-0221666, and DMS-0220905.Accurate computational representations of complex geometry are of great importance in many disciplines ranging from engineering and manufacturing to medicine and biology. With the wide availability of powerful computational resources and ever better acquisition technologies such as 3D laser scanning and volumetric MRI or CAT imaging the geometries used in applications are becoming increasingly complex. One aspect of this complexity is topology, i.e., the presenceof holes and tunnels and of a network of one and zero-dimensional surface features such as creases and spikes. Typical examples of topologically complex shapes are a perforated plate or the system of blood vessels of the body. Traditional representations of geometry are at worst weak and at best cumbersome and inefficient in representingsuch complex topologies. Modeling of macro- and microscopic biological structures is becoming increasingly important for medical research, training, and treatment support. Such structures often have extremely complex shape and topology (e.g., the blood vessel or the nervous system, facialmuscles, a folded protein molecule). The representations and algorithms we develop will result in new efficient ways of manipulating and processing computer representations of such structures. In this project a team with expertise in numerical analysis, geometric modeling, discrete algorithms and computer graphics is studying ways to bring fundamental mathematical tools and highly efficient algorithms to bear on the challenge of creating efficient and accurate computational representations and algorithms for surfaces of complextopology. In particular we are investigating theory and practical algorithms for (I) removal of topological noise in existing models as well as in raw data used for surface reconstruction; (II) topology discovery in volumetric data sets; (III) construction of multiresolution representations of geometry which can meaningfullyabstract fine level topology at coarser resolutions to enable powerful multiscale techniques for rendering, modification and simulation. Particular attention is paid to exploring measures of topological scale which are crucial to most of the algorithms being developed. The algorithms to be developed by the team will have immediate applications in two areas: Computer-Aided Design and Medical Visualization. In CAD, examples of potential applications include topology cleanup, simplification for integration of scanned 3D data with manually constructed models and use of multiscale representations for interactive conceptual design representations.