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

DMS-0138445 Denis Zorin DMS-0221666 Mathieu Desbrun DMS-0220905Peter Schroder这是一个由DMS-0138445、DMS-0221666和DMS-0220905下的货运计划资助的合作项目，复杂几何的精确计算表示在从工程和制造到医学和生物的许多学科中都非常重要。随着强大的计算资源的广泛使用和更好的采集技术，如3D激光扫描和体积磁共振成像或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，即存在孔洞和隧道，以及由一维和零维表面特征组成的网络，如折痕和尖峰。拓扑复杂形状的典型例子是穿孔板或身体的血管系统。传统的几何表示在表示这种复杂的拓扑时，往坏了说是很弱的，往好了说是笨重和低效的。宏观和微观生物结构的建模对于医学研究、培训和治疗支持正变得越来越重要。这种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导致操作和处理此类结构的计算机表示的新的有效方法。在这个项目中，一个在数值分析、几何建模、离散算法和计算机图形学方面具有专业知识的团队正在研究如何利用基本的数学工具和高效的算法来应对为复杂拓扑学的表面创建高效和准确的计算表示和算法的挑战。特别是，我们正在研究用于(I)去除现有模型中的拓扑噪声以及用于曲面重建的原始数据中的拓扑噪声的理论和实用算法；(Ii)体数据集中的拓扑发现；(Iii)几何的多分辨率表示的构建，其能够以更粗的分辨率有意义地抽象精细层次的拓扑，从而使得强大的多尺度技术能够用于渲染、修改和模拟。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来说是至关重要的。该团队将开发的算法将立即在两个领域得到应用：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、用于将扫描的3D数据与手动构建的模型集成的简化以及用于交互式概念设计表示的多尺度表示。

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.