Geometric Flows of Curves in Shape Space for Processing Motion of Deformable Objects

Abstract

We introduce techniques for the processing of motion and animations of non-rigid shapes. The idea is to regard animations of deformable objects as curves in shape space. Then, we use the geometric structure on shape space to transfer concepts from curve processing in R n to the processing of motion of non-rigid shapes. Following this principle, we introduce a discrete geometric flow for curves in shape space. The flow iteratively replaces every shape with a weighted average shape of a local neighborhood and thereby globally decreases an energy whose minimizers are discrete geodesics in shape space. Based on the flow, we devise a novel smoothing filter for motions and animations of deformable shapes. By shortening the length in shape space of an animation, it systematically regularizes the deformations between consecutive frames of the animation. The scheme can be used for smoothing and noise removal, e.g., for reducing jittering artifacts in motion capture data. We introduce a reduced-order method for the computation of the flow. In addition to being efficient for the smoothing of curves, it is a novel scheme for computing geodesics in shape space. We use the scheme to construct non-linear "Bézier curves" by executing de Casteljau's algorithm in shape space.

Latent‐space Dynamics for Reduced Deformable Simulation

“…Linear subspaces have found wide application outside of deformable solid simulation, including fluids [TLP06, SSW * 13], shape deformation [vTSSH13,BvTH16], computational design [XLCB15,MHR * 16,UMK17] and sound simulation [OSG02,JBP06]. However, very small linear subspaces can have difficulty representing even moderately non-linear deformations as seen in Figure 3 jective Dynamics [BEH18] attempts to overcome this limitation by combining model reduction with the very efficient full-space approach of Projective Dynamics [BML * 14].…”

Geometric optimization using nonlinear rotation-invariant coordinates

“…Heeren et al [40,4] propose an alternative physically-based metric on the shape space of triangle meshes that reflects the viscous dissipation required to physically deform a thin shell. Brandt et al [41] derive a discrete curve shortening flow in shape space and use it for processing animations of deformable objects. While the shape spaces study deformations of meshes with fixed connectivity, functional correspondences [42] can be used to blend [43] and analyze [44] pairs of meshes with different connectivity.…”

Splines in the Space of Shells

“…We build on their Riemannian metric. Using this metric, Brandt et al [BvTH16] suggested a scheme that largely accelerates geodesic computation in shell space by resorting to dimensionality reduction in the spatial domain. In a different development, Winkler et al [WDAH10] and Fröhlich and Botsch [FB11] used edge length and dihedral angle coordinates to represent meshes for efficiently computing interpolation paths between two given poses.…”

“…Having the notion of geodesics at hand one can easily transfer this concept to Riemannian manifolds—Beziér curves are simply generated by applying the de Casteljau algorithm with linear interpolation replaced by geodesic interpolation [PN07]. This was done in [ERS*14] for the shape space of images and in [BvTH16] for the space of shells.…”