Matching 3D Shapes Using 2D Conformal Representations
Xianfeng Gu and Baba C. Vemuri
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.