### Matching 3D Shapes Using 2D Conformal Representations

Xianfeng Gu and Baba C. Vemuri
CVPR 2006.
Matching 3D shapes is a fundamental problem in Medical Imaging with many applications including, but not limited to, shape deformation analysis, tracking etc. Matching 3D shapes poses a computationally challenging task. The problem is especially hard when the transformation sought is diffeomorphic and non-rigid between the shapes being matched. In this paper, we propose a novel and computationally efficient matching technique which guarantees that the estimated non-rigid transformation between the two shapes being matched is a diffeomorphism. Specifically, we propose to conformally map each of the two 3D shapes onto the canonical domain and then match these 2D representations over the class of diffeomorphisms. The representation consists of a two tuple ($\lambda$, $H$), where, $\lambda$ is the conformal factor required to map the given 3D surface to the canonical domain (a sphere for genus zero surfaces) and $H$ is the mean curvature of the 3D surface. Given this two tuple, it is possible to uniquely determine the corresponding 3D surface. This representation is one of the most salient features of the work presented here. The second salient feature is the fact that 3D non-rigid registration is achieved by matching the aforementioned 2D representations. We present convincing results on real data with synthesized deformations and real data with real deformations.