3D shape deformations : a lie group based approach

Abstract

3D shapes are ubiquitous in many fundamental tasks of computer graphics and geometry processing. For many applications, new shapes have to be generated from the existing ones, for which it it imperative to understand and model shape of an object and its deformation. This thesis focuses on shape deformations and its applications. Real world 3D objects undergo complex, often non-rigid deformations. One way to model such deformations is using local affine transformations. It is thus important for applications like 3D animation, to understand the structure of affine transformations and come up with robust and efficient computational tools on the set of affine transformations. With such tools, applications like interactive shape deformation and mesh interpolation can be effectively dealt with. In this thesis, an interpolation framework for affine transformations, based on a Lie group representation of a tetrahedron is proposed. The proposed framework provides a intuitive closed form interpolation in all cases in contrast to existing approaches and preserves properties like isometry, reversibility, and monotonic change of volume. The proposed Lie group representation of the tetrahedron is extended to represent triangular and tetrahedral meshes. A detailed analysis of the invariance of the representation and interpolation to some choices made, is provided in the thesis. We demonstrate the applicability of the framework for several applications like interactive shape deformation, shape interpolation, morphing, and deformation transfer. The proposed interactive shape deformation algorithm is close to being real-time, while the mesh interpolation algorithm is able to handle nonregistered meshes and large deformation cases. The interactive shape deformavi tion algorithm is amenable to data-driven methods, and we hope to explore datadriven methods using our mesh representation in near future.