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Geodätische Pfade im Raum der Shapes

Fachliche Zuordnung Mathematik
Förderung Förderung von 2012 bis 2021
Projektkennung Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 212212052
 
Erstellungsjahr 2021

Zusammenfassung der Projektergebnisse

In this project, which was part of the National Research Network (NFN) Geometry + Simulation of the Austrian FWF, we have studied efficient tools for modeling and simulation with (discrete) surfaces. However, our objectives and applications were widely spread. First, we dealt with a variety of surface representations: subdivision surfaces as a popular tool in geometric modeling and animation, triangular surfaces combined with discrete differential geometric calculus frequently used in computer graphics, and finally implicit shell models naturally appearing when directly processing surfaces encoded in images. Second, we considered various distinct directions in surface processing, i.e. analysis and simulation of problems involving a single surface and problems formulated in the shape space of surfaces modeled as a Riemannian manifold. Concretely, we made advances in the following five fields of interest: a) Geodesic Calculus and Shape Statistics in the Space of Discrete Shells. We considered the shape space of discrete surfaces, i.e. triangle meshes, as a Riemannian manifold. Physically, we treated single (discrete) surfaces as thin elastic shells that can undergo distortions caused by external forces. A Riemannian metric is defined as the rate of viscous dissipation caused by tangential stretching or normal bending. Based on the notion of a time-discrete geodesic path, we built an entire time-discrete geodesic calculus consisting of discrete exponential and logarithm maps and a tool for performing discrete parallel transport. Furthermore, we proposed a time-discrete approximation of Riemannian (cubic) splines and develop a Principal Geodesic Analysis (PGA). Lastly, we introduce a novel Sparse Principal Geodesic Analysis (SPGA) that yields spatially localized deformation components and use them to synthesize new nonlinear deformations with interactive frame rates. To this end, we used the edge lengths and dihedral angles of a triangle mesh a primary degrees of freedom. b) Isogeometric Methods in Shape Spaces. We investigated the robust and efficient implementation of an isogeometric discretization approach to partial differential equations on surfaces using subdivision methodology. Thereby, efficiency relies on the proper choice of a numerical quadrature scheme which preserves the expected higher order consistency. Moreover, we studied spline constructions in the Riemannian manifold of subdivision surfaces. c) Higher Order Space and Time Discretization of Geometric PDEs. Subdivision surfaces are ideally suited for the simulation of geometric evolution problems often phrased in terms of a gradient flow of an appropriate geometric energy. We considered general energies which involve first and second derivatives of the surface parametrization leading to fourth-order PDEs for the corresponding Euler-Lagrange equations. Since subdivision elements yield a higher order spatial approximation, an implicit Euler time discretization would result in a time-error dominated scheme. Therefore, we also applied implicit Runge–Kutta methods as higher order time discretization to balance the error terms. d) A Variational Model for Thin Viscous Films on Surfaces. The motion of a thin viscous film of fluid on a curved surface exhibits many intricate visual phenomena, which are challenging to simulate using existing techniques. Based on an existing variational formulation for a lubrication model on smooth surfaces, we have developed a corresponding model for triangle meshes. e) Shape-aware Matching of Implicit Surfaces. In joint work with Otmar Scherzer’s group (Subproject 04 of the NFN), we developed and mathematically analyzed a method to match implicit surfaces, where we made use of a shell deformation approach that is phrased in a level set formulation. In particular, we ensured weak lower semicontinuity of the objective functional and present a comprehensive existence theory.

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