Conformal Geometry and Its Applications on 3D Shape Matching, Recognition and Stitching
IEEE Transactions on Pattern Analysis and Machine Intelligence
Sen Wang, Yang Wang, Miao Jin, Xianfeng Gu, Dimitris Samaras
3D shape matching is a fundamental issue in computer vision with many applications such as shape
registration, 3D object recognition and classication. However, shape matching with noise, occlusion
and clutter is a challenging problem. In this paper, we analyze a family of quasi-conformal maps
including harmonic maps, conformal maps and least squares conformal maps with regards to 3D shape
matching. As a result, we propose a novel and computationally efcient shape matching framework by
using least squares conformal maps. According to conformal geometry theory, each 3D surface with
disk topology can be mapped to a 2D domain through a global optimization and the resulting map is a
diffeomorphism, i.e., one-to-one and onto. This allows us to simplify the 3D shape-matching problem
to a 2D image-matching problem, by comparing the resulting 2D parametric maps, which are stable,
insensitive to resolution changes and robust to occlusion and noise. Therefore, highly accurate and
efcient 3D shape matching algorithms can be achieved by using the above three parametric maps.
Finally, the robustness of least square conformal maps is evaluated and analyzed comprehensively in
3D shape matching with occlusion, noise and resolution variation. In order to further demonstrate the
performance of our proposed method, we also conduct a series of experiments on two computer vision
applications, i.e., 3D face recognition and 3D non-rigid surface alignment and stitching.