Docteur de l'université Automatic Segmentation and Shape Analysis ...

36 Chapter 2 Literature Review
In parameterization based methods, the shape landmarks are mapped to the pa-
rameter space bijectively, which establishes a correspondence between each land-
mark point and a parameter. Depending on the topology of the object, the under-
lying domain of parameters is usually chosen to be a simpler space homeomorphic
to the representation of the shape. The correspondence between different shapes is
then defined by associating the landmark points with the same parameter. Most
common genus 0 surfaces (i.e. homeomorphic to the sphere S 2 ) are usually pa-
rameterized by spherical coordinates mapping the landmarks to the points on the
unit sphere. Commonly used parameterization algorithms are angle-preserving
conformal mapping (Haker et al., 2000; Gu and Yau, 2003; Gu et al., 2004), and
area-preserving mapping (Brechbühler et al., 1995). Based on the spherical pa-
rameterization, the correspondence can be determined by aligning the first order
ellipsoid of the SPHARM shape decomposition (Kelemen et al., 1999). Since the
parameterization of each shape is independent, there is no guarantee of ‘optimal’
correspondence between two parameterizations. Reparameterization, i.e. transfor-
mation in the parameterization space, is needed to improve the correspondence.
For a shape model with given correspondence, the determinant of the covariance
matrix is used by Kotcheff and Taylor (1998) as an objective function for the opti-
mization the model correspondence and compactness. The minimum description
length (MDL, Davies et al., 2002) based on information theoretic principles is used
as the cost function in search for optimal correspondence, which is to be solved
by genetic algorithm (Davies et al., 2002) or Nelder-Mead simplex (Davies et al.,
2003). A simplified MDL by Thodberg (2003) gives comparable performance as
the original, with analytic form that can be minimized more efficiently using the
steepest gradient (Ericsson and Åström, 2003; Hladůvka and Bühler, 2005). To
reduce the computational complexity of reparameterization on the sphere, the
‘shape image’ technique is developed by embedding the sphere to a rectangular
region, such that the interpolation can be carried out on a 2D grid (Davies et al.,
2008b). Full details of the SSM optimization has been published as a monograph
(Davies et al., 2008a).