Project Details
Projekt
Geodesic Paths in Shape Space
Applicant
Professor Dr. Martin Rumpf
Subject Area
Mathematics
Term
from 2012 to 2021
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 212212052
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.
DFG Programme
Research Grants
International Connection
Austria
