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Mathematics Colloquium

Martin Rumpf
University of Bonn

Title: From Riemannian geometry on shell space to realtime surface editing and character animation
Date: Friday, September 25, 2020
Place and Time: Zoom, 3:35-4:25 pm

Abstract. Geometric optimization problems are at the core of many applications in geometry processing. The choice of a data representation fitting such optimization problems allows to considerably simplify the problem. The talk will discuss a natural embedding of the space of 2D triangulated surfaces in high dimensions. A Riemannian metric on this space which reflects membrane and bending distortion. Thereby, the vector of edge lengths and dihedral angles of a mesh represents nonlinear rotation-invariant coordinates and a set of integrability conditions implicitly defines the manifold of triangular surfaces. These coordinates are then used to construct low-dimensional nonlinear deformation space to capture the variability of non-rigid shapes from a data set of example poses. In the core of this approach is a Sparse Principal Geodesic Analysis (SPGA). The resulting modes represent characteristic articulations of the shape and usually come with a decomposing into low-dimensional widely decoupled subspaces. For example, for human models, one expects distinct, localized modes for the bending of elbow or knee whereas some more modes are required to represent shoulder articulation. This decoupling property can be used to construct a superposition of shape submanifolds resulting from the decoupling. In a preprocessing stage, samples of the individual subspaces are computed, and, in an online phase, these are interpolated multilinearly. This accelerates the construction of nonlinear deformations and makes the method applicable for interactive, realtime applications.