Group Representation of Global Intrinsic Symmetries

Abstract

Global intrinsic symmetry detection of 3D shapes has received considerable attentions in recent years. However, unlike extrinsic symmetry that can be represented compactly as a combination of an orthogonal matrix and a translation vector, representing the global intrinsic symmetry itself is still challenging. Most previous works based on point-to-point representations of global intrinsic symmetries can only find reflectional symmetries, and are inadequate for describing the structure of a global intrinsic symmetry group. In this paper, we propose a novel group representation of global intrinsic symmetries, which describes each global intrinsic symmetry as a linear transformation of functional space on shapes. If the eigenfunctions of the Laplace-Beltrami operator on shapes are chosen as the basis of functional space, the group representation has a block diagonal structure. We thus prove that the group representation of each symmetry can be uniquely determined from a small number of symmetric pairs of points under certain conditions, where the number of pairs is equal to the maximum multiplicity of eigenvalues of the Laplace- Beltrami operator. Based on solid theoretical analysis, we propose an efficient global intrinsic symmetry detection method, which is the first one able to detect all reflectional and rotational global intrinsic symmetries with a clear group structure description. Experimental results demonstrate the effectiveness of our approach.