Nonlinear Deformation Based on Rigid Cells

Mario Botsch, Mark Pauly, Martin Wicke, Markus Gross, Leif Kobbelt

In this project we developed a new method for 3D surface deformation that achieves intuitive and robust deformations by emulating physically plausible surface behavior inspired by thin shells. The surface mesh is embedded in a layer of volumetric prisms, which are coupled through non-linear, elastic forces. This non-linearity allows for intuitive results even under large scale deformations. To deform the mesh, prisms are rigidly transformed to satisfy user constraints while minimizing the elastic energy.

The rigidity of the prisms prevents degenerations even under extreme deformations, making the method numerically stable. For the underlying geometric optimization we employ both local and global shape matching techniques. Our modeling framework allows for the specification of various geometrically intuitive parameters that provide control over the physical surface behavior. While computationally more involved than previous methods, our approach significantly improves robustness and simplifies user interaction for large, complex deformations.

In a follow-up project we extended the surface-based approach to full nonlinear deformations of solid objects, resulting in a new adaptive space deformation technique for interactive shape modeling. Our volumetric energy formulation based on elastically coupled rigid cells yields intuitive detail preservation even under large deformations. The rigid cells again allow for extremely robust numerical solvers for the resulting nonlinear optimization problem. Scalability is achieved using an adaptive spatial discretization that is decoupled from the resolution of the embedded object. The approach is versatile and easy to implement, supports thin-shell and solid deformations of 2D and 3D objects, and is applicable to arbitrary sample-based representations, such as meshes, triangle soups, or point clouds.

• M. Botsch, M. Pauly, M. Gross, L. Kobbelt, PriMo: Coupled Prisms for Intuitive Surface Modeling, Proceedings of Eurographics Symposium on Geometry Processing (Cagliari, Sardinia, Italy, June 26-28, 2006), pp. 11-20
  [Abstract] [PDF] [Video]
• M. Botsch, M. Pauly, M. Wicke, M. Gross, Adaptive Space Deformations Based on Rigid Cells, Proceedings of Eurographics (Prague, Czech Republic, September 3-7, 2007), Computer Graphics Forum, vol. 26, no. 3, pp. 339-347
  [Abstract] [PDF] [Video]