Lazarus and Verroust  give an excellent survey of previous work on the 3D morphing problem. There are two major classes of 3D morphing technique: volume based approach and surface based approach. In this paper, we focus on surface based approach. For more related work of volume-based approach, please refer to . The surface based approach works on boundary representations such as polyhedral meshes or patch complexes. Methods for this approach usually create an interpolation mesh, which is a common embedding of both source and target meshes. This interpolation mesh is then geometrically deformed to create morphed shapes. This common embedding solves the correspondence problem, associating the vertices or triangles between the source mesh and target mesh. This is a key issue of surface based approach and will be the focus of our paper.
A lot of work has been published in the correspondence issue. Kent et al  introduce parameterizations for solving the correspondence problem. Their approach projected star-shaped objects onto spheres to accomplish parameterization. Similarly, Alexa  employs a relaxation method to embed polyhedral shapes on the spheres. Lazarus and Verroust  introduce skeletons for cylinder-like objects. This approach is an extension of  for objects that are star-shaped around an axis. Parent  presents a recursive algorithm that automatically finds a correspondence between the surfaces of two objects of equivalent topologies. Decarlo et al.  present a method to transform objects with different topologies. The user must identify a sparse control mesh on each surface of objects and this control mesh specifies how to transform one surface to another. Therefore, it will become very complicated and difficult to transform complex shapes. Kanai et al.  and Zockler et al.  utilize mesh parameterization technique such as harmonic mapping to embed a region of a mesh to a 2D convex polygon. Gregory et al.  apply a user-specified control mesh to decompose the surface into a large number of disk-like patches. However, this task is very slow and tedious. Lee et al.  employ the MAPS algorithm  to parameterize input meshes over simple base domains and an additional harmonic map bringing the latter into the correspondence. Their approach can have fold-over problem and user interaction is required to manually fix this problem.