  Morphing is the process of changing one figure into another. Some numerical
methods of 3D surface morphing by deformable modeling and conformal mapping are
shown in this study. It is well known that there exists a unique Riemann
conformal mapping from a simply connected surface into a unit disk by the
Riemann mapping theorem. The dilation and relative orientations of the 3D
surfaces can be linked through the M\"obius transformation due to the conformal
characteristic of the Riemann mapping. 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, we take the advantage of the unique representation of 3D
surfaces by the mean curvatures and the conformal factors associated with the
Riemann mapping. By registering the landmarks on the conformal parametric
domains, the correspondence of the mean curvatures and the conformal factors
for each surfaces can be obtained. As a result, we can construct the 3D
deformation field from the surface reconstruction algorithm proposed by Gu and
Yau. Furthermore, by composition of the M\"obius transformation and the 3D
deformation field, the morphing sequence can be generated from the mean
curvatures and the conformal factors on a unified mesh structure by using the
cubic spline homotopy. Several numerical experiments of the face morphing are
presented to demonstrate the robustness of our approach.

Graphics;Computational Geometry