Geodesic Paths in Shape Space

该项目为应用几何和数值模拟之间的相互作用中的形状定量分析提供了强大而灵活的工具。在这里，形状是物理上表示壳型几何形状的曲面。在等几何分析中，人们面临着各种各样的低维和中维描述的复杂和现实的几何形状。因此，形状的几何描述是灵活的，从简单的分段线性细分生成的样条类型的表面表示。定量形状分析的基本工具是在高维甚至无限维黎曼形状空间中计算两个形状之间的距离。在黎曼距离之外，我们发展了一个完整的黎曼演算，包括几何指数映射，几何对数，平行运输，协变导数以及黎曼样条。此外，我们调查的形状的大数据集的统计分析。我们的目标是在计算机图形学和几何处理中的表面动画的背景下应用这些方法。在NFN几何+模拟的最后一年，我们将继续追求以下方法。这是在提交给FWF的NFN几何+模拟中相应子项目的原始应用之后对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）从PGA导出的用于详细三角形模型的实时操纵和动画的缩减基方法，（C）基于G^1多块B样条参数化的弹性壳能量离散化的等几何方法，（D）在G+Smo开发的项目软件框架中实现这些非线性能量-在NFN中占据主导地位。

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.