Geodesic Paths in Shape Space

形状空间中的测地线路径

基本信息

批准号：
212212052
负责人：
Professor Dr. Martin Rumpf
金额：
--
依托单位：
Institut für Numerische Mathematik
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2012
资助国家：
德国
起止时间：
2011-12-31 至 2020-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/212212052?language=en
关键词：
Geodesic Paths Shape Space

项目摘要

This project provides robust and flexible tools for the quantitative analysis of shapes in the interplay between applied geometry and numerical simulation. Here, shapes are curved surfaces that physically represent shell-type geometries. In isogeometric analysis, one faces a wide range of low- and moderate-dimensional descriptions of complicated and realistic geometries. Thus, the geometric description of shapes is flexible, ranging from simple piecewise linear to subdivision-generated spline type surface representations. The fundamental tool for a quantitative shape analysis is the computation of a distance between two shapes as objects in a high- or even infinite-dimensional Riemannian shape space. Beyond the Riemannian distance, we develop a fully fletched Riemannian calculus including the geometric exponential map, the geometric logarithm, parallel transport, covariant derivative as well as Riemannian splines. Furthermore, we investigate the statistical analysis of large data sets of shapes. We aim at applying these methods in the context of surface animation in computer graphics and geometry processing. In the final year of the NFN Geometry+Simulation we will continue to pursue the following approaches. This is an updated description of the project's objectives following the original application of the corresponding subproject in the NFN Geometry+Simulation submitted to FWF: (A) a principal geodesic analysis (PGA) of discrete shells based on nonlinear rigid body motion invariant coordinates,(B) a reduced bases approach derived from the PGA for real-time manipulation and animation of detailed triangular models,(C) an isogeometric approach for the discretization of elastic shell energies based on a G^1 multi-patch B-spline parametrization,(D) the implementation of these nonlinear energies in the software framework G+Smo developed project--overarching within the NFN.

该项目为应用几何和数值模拟之间相互作用的形状定量分析提供了强大而灵活的工具。在这里，形状是物理上代表壳型几何形状的曲面。在等几何分析中，人们面临着一系列复杂和现实几何的低维和中维描述。因此，形状的几何描述是灵活的，范围从简单的分段线性到细分生成的样条类型表面表示。定量形状分析的基本工具是计算高维甚至无限维黎曼形状空间中的两个形状作为对象之间的距离。除了黎曼距离之外，我们还开发了完整的黎曼微积分，包括几何指数图、几何对数、并行传输、协变导数以及黎曼样条。此外，我们研究了大型形状数据集的统计分析。我们的目标是在计算机图形和几何处理中的表面动画背景下应用这些方法。在 NFN 几何+仿真的最后一年，我们将继续采用以下方法。这是在提交给 FWF 的 NFN 几何+仿真中相应子项目的原始应用之后对该项目目标的更新描述：(A) 基于非线性刚体运动不变坐标的离散壳的主测地线分析 (PGA)，( B) 源自 PGA 的简化基础方法，用于详细三角形模型的实时操作和动画，(C) 基于 G^1 的弹性壳能量离散化的等几何方法多面片 B 样条参数化，(D) 在 G+Smo 开发项目的软件框架中实现这些非线性能量 - NFN 内的总体。