CARGO: Multi-Scale Topological Analysis of Time-Evolving Shapes

DMS-0138420Jarek Rossign我们建议开发一套用于自动分析随时间演变的形状的理论基础和一套实用的计算工具。给定一组表示3D形状的边界的曲面，其几何和拓扑随时间变化，我们建议构造一个更高维的多分辨率表示，我们称之为Atlas转换图，简称ATD，它将识别和跟踪3D形状随着时间和分辨率的演变的形态和拓扑特征。我们的ATD还将图表与每个特征相关联，从而提供一个曲面参数化，该曲面参数自然跟随在族中每个形状的分支和句柄之后。图表随着时间和分辨率平滑地演化，并且在它们的公共边界处以拓扑方式粘合在一起以提供连续的映射，S(r，t，u，v；f)，给定特征ff、分辨率Rr、时间Tt和两个参数u和v，将标识表面上的唯一点，并将允许我们跟踪其随Rr和Tt的演变。理论基础和算法设计将导致建立和查询ATD的有效系统的实际实现，远远超出Morse理论、表面分割方法和多分辨率技术的简单扩展，到目前为止，这些技术主要是针对3D中的静态表面进行探索的。演变表面对许多科学和工程学科非常重要，包括医学、发育生物学、细胞生物学、计算流体力学和计算机辅助设计。例如，它们可能代表肿瘤的生长，飞机机翼上方漩涡的位置移动，或者人类胚胎手指头的萌芽。我们建议开发和集成一系列理论和算法工具，用于分析和自动化可视化此类演化。这些工具将使我们能够将不断演变的表面划分为特征，以此为基础对表面的形状及其演变进行高级别的描述。此外，它们将允许我们随着时间的推移跟踪它们的点，从而跟踪它们的表面属性，并通过纹理贴图更好地可视化它们的演变，纹理贴图随着要素的不断演变并突出它们的边界和自然方向。最后，这些工具将帮助我们支持关于不断演变的表面中拓扑变化的时间和性质的查询，这可能对于自动分析和检索科学数据集很重要。为了达到这些结果，我们建议建立一个由时间和分辨率独立或同时控制的表面表示，并将时间/分辨率域分解成单元，其中表面拓扑(组件和通孔的数量)及其特征划分保持不变。为了验证我们的理论贡献并增加它们对社区的影响，我们计划开发一个三角边界动画对象的原型实现。我们希望公开这个实现的源代码和它的编程接口。我们设想探索与科学、工程、医学和生物学领域的应用程序开发人员的合作，以帮助我们改进和验证这一方法。

DMS-0138420Jarek RossignacWe propose to develop the theoretical foundations and a set of practical computing tools for the automatic analysis of time-evolving shapes. Given a family of surfaces that represent the boundary of a 3D shape whose geometry and topology change with time, we propose to construct a higher-dimensional multiresolution representation, which we have named Atlas Transition Diagram, abbreviated ATD, that will identify and track the morphological and topological features of the 3D shape as they evolve with time and with resolution. Our ATD will also associate a chart to each feature, thus providing a surface-parameterization that naturally follows the branches and handles of each shape in the family. The charts evolve smoothly with time and resolution and are topologically glued together at their common boundaries to provide a continuous mapping, S(r,t,u,v;f); f), that, given a feature ff, a resolution rr, a time tt, and two parameters u and v, will identify a unique point on the surface and will allow us to trace its evolution with rr and tt. The theoretical underpinnings and algorithmic designs that will lead to a practical implementation of an efficient system for building and querying ATDs go far beyond simple extensions of Morse theory, of surface segmentation approaches, and of multi-resolution techniques, which have so far been mainly explored for static surfaces in 3D.Evolving surfaces are important to many scientific and engineering disciplines, including medicine, developmental biology, cell biology, computational fluid dynamics and computer aided design. They may for example represent the growth of a tumor, the shifting in position of a vortex over an airplane wing, or the budding of fingers on the hand of a human embryo. We propose to develop and integrate a collection of theoretical and algorithmic tools for the analysis and automated visualization of such evolutions. These tools will allow us to partition the evolving surface into features upon which a high-level description of the shape of the surface and of its evolution will be based. Furthermore, they will allow us to track their points, and thus surface properties, through time and to better visualize their evolutions through texture maps that continuously evolve with the features and highlight their boundaries and natural orientation. Finally, these tools will help us support queries about the time and nature of topological changes in the evolving surface, which may be important for the automatic analysis and retrieval of scientific datasets. To achieve these results, we propose to build a surface representation that is controlled independently or simultaneously by time and resolution and to decompose the time/resolution domain into cells where the surface topology (number of components and through holes) and its partition into features remain constant. To validate our theoretical contributions and to increase their impact on the community, we plan to develop a prototype implementation for animated objects with triangulated boundaries. We expect to make the source code of this implementation and its programming interface publicly available. We envision exploring collaborations with application developers in Science, Engineering, Medicine and Biology to help us refine and validate this approach.