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

DMS-0138420 Jarek Rossignac我们建议开发的理论基础和一套实用的计算工具，自动分析的时间演变的形状。给定一系列表示几何形状和拓扑结构随时间变化的3D形状边界的表面，我们建议构建一个更高维度的多分辨率表示，我们将其命名为Atlas Transition Diagram，缩写为ATD，它将识别和跟踪3D形状的形态和拓扑特征随着时间和分辨率的变化而变化。我们的ATD还将图表与每个特征相关联，从而提供自然遵循家族中每个形状的分支和手柄的表面参数化。这些图随着时间和分辨率平滑地演变，并且在它们的公共边界处拓扑地粘合在一起以提供连续映射S（r，t，u，v;f）; f），给定特征ff、分辨率rr、时间tt以及两个参数u和v，该连续映射将标识表面上的唯一点，并且将允许我们用rr和tt来追踪其演变。理论基础和算法设计将导致用于构建和查询ATD的有效系统的实际实现远远超出莫尔斯理论、表面分割方法和多分辨率技术的简单扩展，这些技术迄今为止主要针对3D中的静态表面进行探索。细胞生物学、计算流体动力学和计算机辅助设计。例如，它们可能代表肿瘤的生长，飞机机翼上漩涡位置的变化，或者人类胚胎手上手指的萌芽。 我们建议开发和集成的理论和算法工具的集合，用于分析和自动可视化这样的演变。这些工具将使我们能够将不断演变的表面划分为特征，并基于这些特征对表面形状及其演变进行高级描述。此外，它们将使我们能够跟踪它们的点，从而跟踪表面属性，并通过纹理图更好地可视化它们的演变，这些纹理图随着特征不断演变，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查询的时间和性质的拓扑变化在不断变化的表面，这可能是重要的自动分析和检索的科学数据集。为了实现这些结果，我们建议建立一个表面表示，独立或同时控制的时间和分辨率，并分解的时间/分辨率域到细胞的表面拓扑结构（组件和通孔的数量）和它的分区功能保持不变。为了验证我们的理论贡献，并增加其对社区的影响，我们计划开发一个原型实现的动画对象与三角形边界。我们希望公开此实现的源代码及其编程接口。我们设想与科学、工程、医学和生物学领域的应用开发人员合作，帮助我们完善和验证这种方法。

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.