Shape analysis of complex symmetric structures: Estimating components of symmetric variation and asymmetry

The method of object symmetry (Mardia et al. 2000) has been elaborated for the studies of shape variation in structures that have an internal plane of bilateral symmetry (e.g. vertebrate skulls and insect bodies). Further work related object symmetry to the structure of the shape tangent space (Kolamunnage & Kent 2003, 2005). For instance, principal component analysis (PCA) can separate a component of symmetric shape variation and components of asymmetric shape variation (Kolamunnage & Kent 2003). The approach of Mardia et al. (2000) for the studies of shape variation in bilaterally symmetric shapes has been extended for the analysis of any type of symmetry (Savriama & Klingenberg 2006). In particular, the method of object symmetry has been generalised to any structure that exhibits more complex internal symmetry (e.g. radial symmetry in corals). Every type of symmetry is associated with a set of symmetry transformations that forms a symmetry group. For example, the identity and reflection are the symmetry transformations that characterise bilateral symmetry. The first step of the analysis is to assemble a dataset from copies of an original configuration of landmarks to which all transformations in the symmetry group of the object have been applied. Thereafter, all configurations in the dataset are superimposed in a single Procrustes fit. The resulting Procrustes mean (consensus) is symmetric. Here we follow the approach of Kolamunnage & Kent (2003, 2005) to explore the patterns of variation in the total shape tangent space for structures with complex internal symmetry by using PCA.