# Parameterization-to-Parameterization Registration

In this strategy, the boundary surfaces are represented by meshes, and they are not registered together but are registered to one base domain. Most of these methods select a sphere as the base domain. A method proposed by Kelemen et al. uses a spherical harmonics mapping (SPHARM) for the registration to the sphere [20]. In a method proposed by Brett and Taylor [21], all shapes are mapped to 2D disks that are then aligned to generate the correspondence. These methods guarantee a diffeomorphism among all datasets, but it is difficult for users to control the resultant correspondences. Other methods have been proposed that control the parameterization through a small set of known or assumed correspondences [2226].

# Population-Based Optimization

As mentioned above, it is not easy to define *good* correspondence. Kotcheff and Taylor [27] proposed a method that generates the correspondences so that the resultant SSM has *good* properties: The authors defined the compactness of the resultant model as an index of *goodness.* The compactness is evaluated through the determinant of the covariance matrix. Although the method improves the performance, the objective function minimized when the corresponding points generated have no theoretical foundation. Using the same strategy, Davies et al. proposed an objective function defined with the minimum description length [28]. This objective function has a theoretical foundation, but the calculation cost is high. Thodberg describes a simplified version of the cost function [29].