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

合作研究：拓扑复杂 3D 形状的建模和处理

• 批准号: 0138445
• 负责人: Denis Zorin
• 金额: $ 22.5万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-05-01 至 2006-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0138445&HistoricalAwards=false
• 关键词: Collaborative; Research; Modeling; Processing; Topologically

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.

DMS-0138445Denis Zorindms-0221666MATHIEU DESBRUNDMS-0220905PETER SCHRODERTHIS是由货物计划UnderDMS-0138445资助的合作项目从工程和制造到医学和生物学。 凭借强大的计算资源的广泛可用性，并获得了更好的获取技术，例如3D激光扫描和体积MRI或CAT成像，应用中使用的几何形状正在变得越来越复杂。这种复杂性的一个方面是拓扑，即孔和隧道的存在以及一个和零维表面特征（例如折痕和尖峰）的网络。 拓扑复杂的形状的典型例子是穿孔板或人体血管系统。几何形状的传​​统表现最糟糕，充其量是笨拙且效率低下的复杂拓扑。 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。这种结构通常具有极其复杂的形状和拓扑（例如，血管或神经系统，面部乳胶，折叠的蛋白质分子）。我们开发的表示和算法将导致新的有效方法来操纵和处理此类结构的计算机表示。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方面具有专业知识的团队正在研究携带基本数学工具和高效算法的方法，以挑战为COMPLEXTOPOLOGY提供有效，准确的计算表示和算法。 特别是，我们正在研究（i）在现有模型以及用于表面重建的原始数据中去除拓扑噪声的理论和实践算法； （ii）体积数据集中的拓扑发现； （iii）构建几何形状的多分辨率表示形式，这些表示可以有意义地在更粗的分辨率下赋予良好的层拓扑，以实现强大的多尺度技术进行渲染，修改和仿真。特别关注探索拓扑量表的度量，这对于开发的大多数算法至关重要。 团队将要开发的算法将在两个领域中立即应用：计算机辅助设计和医学可视化。 在CAD中，潜在应用的示例包括拓扑清理，简化与手动构造模型集成的扫描3D数据以及用于交互式概念设计表示的多尺度表示。