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

这是由dms -0138445、DMS-0221666和DMS-0220905下的CARGO项目资助的合作项目。复杂几何的精确计算表示在许多学科中都非常重要，从工程、制造到医学和生物学。随着强大的计算资源的广泛可用性和更好的采集技术，如3D激光扫描和体积MRI或CAT成像，应用中使用的几何形状变得越来越复杂。这种复杂性的一个方面是拓扑结构，即洞和隧道的存在，以及一维和零维表面特征（如折痕和尖峰）的网络。拓扑复杂形状的典型例子是穿孔板或人体血管系统。传统的几何表示在最坏的情况下是弱的，在最好的情况下是笨重和低效的，以表示这种复杂的拓扑结构。宏观和微观生物结构的建模对于医学研究、培训和治疗支持变得越来越重要。这些结构通常具有极其复杂的形状和拓扑结构（例如，血管或神经系统、面部肌肉、折叠的蛋白质分子）。我们开发的表示和算法将产生新的有效方法来操纵和处理这些结构的计算机表示。在这个项目中，一个拥有数值分析、几何建模、离散算法和计算机图形学专业知识的团队正在研究如何将基本的数学工具和高效的算法引入复杂拓扑表面，以应对创建高效、准确的计算表示和算法的挑战。特别是，我们正在研究以下方面的理论和实用算法：(1)去除现有模型中的拓扑噪声以及用于表面重建的原始数据；（II）体积数据集的拓扑发现；（III）构建几何的多分辨率表示，可以在较粗的分辨率下有意义地抽象精细层次的拓扑结构，从而实现强大的多尺度渲染、修改和模拟技术。特别注意探索拓扑尺度的度量，这对大多数正在开发的算法是至关重要的。该团队开发的算法将立即应用于两个领域：计算机辅助设计和医学可视化。在CAD中，潜在应用的例子包括拓扑清理，将扫描的3D数据与手工构建的模型集成的简化，以及使用多尺度表示进行交互式概念设计表示。

MS-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.