Conformal Surface Registration with Applications on Face Morphing

Mei-Heng Yueh National Chiao Tung University Xianfeng David Gu Stony Brook University Wen-Wei Lin National Chiao Tung University Chin-Tien Wu National Chiao Tung University Shing-Tung Yau Harvard University

Computational Geometry mathscidoc:1605.09001

Morphing is the process of changing a geometric model or an image into another. The process generally involves rigid body motions and non-rigid deformations. It is well known that there exists a unique conformal mapping from a simply connected surface into a unit disk by the Riemann mapping theorem. On the other hand, a 3D surface deformable model can be built via various approaches such as mutual parameterization from direct interpolation or surface matching using landmarks. In this paper, a numerical methods of 3D surface morphing based on deformable model and conformal mapping is demonstrated. We take the advantage of the unique representation of 3D surfaces by the mean curvatures $H$ and the conformal factors $\lambda$ associated with the Riemann mapping and build up the deformation model by consistently registering the landmarks on the conformal parametric domains. As a result, the correspondence of the $(H, \lambda)$ between two surfaces can be defined and a 3D deformation field can be reconstructed. Furthermore, by composition of the M\"obius transformation and the 3D deformation field, a smooth morphing sequence can be generated over a consistent mesh structure via the cubic spline homotopy. Several numerical experiments on the face morphing are presented to demonstrate the robustness of our approach.
conformal mapping, surface morphing, surface registration
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@inproceedings{mei-hengconformal,
  title={Conformal Surface Registration with Applications on Face Morphing},
  author={Mei-Heng Yueh, Xianfeng David Gu, Wen-Wei Lin, Chin-Tien Wu, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160520142138341876035},
}
Mei-Heng Yueh, Xianfeng David Gu, Wen-Wei Lin, Chin-Tien Wu, and Shing-Tung Yau. Conformal Surface Registration with Applications on Face Morphing. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160520142138341876035.
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