Linear Transformation Groups and Shape Space

Abstract

The problem of modelling variabilities of ensembles of objects leads to the study of the ‘shape’ of point sets and of transformations between point sets. Linear models are only able to amply describe variabilities between shapes that are sufficiently close and require the computation of a mean configuration. Olsen and Nielsen (2000) introduced a Lie group model based on linear vector fields and showed that this model could describe a wider range of variabilities than linear models. The purpose of this paper is to investigate the mathematics of this Lie group model further and determine its expressibility. This is a necessary foundation for any future work on inference techniques in this model.

Let Σ^k _m denote Kendall's shape space of sets of k points in m-dimensional Euclidean space (k > m): this consists of point sets up to equivalence under rotation, scaling and translation. Not all linear transformations on point sets give well-defined transformations of shapes. However, we show that a subgroup of transformations determined by invertible real matrices of size k − 1 does act on Σ^k _m. For m > 2, this group is maximal, whereas for m = 2, the maximal group consists of the invertible complex matrices. It is proved that these groups are able to transform any generic shape to any other. Moreover, we establish that for k > m + 1 this may be done via one-parameter subgroups. Each one-parameter subgroup is given by exponentiation of an arbitrary (k − 1) × (k − 1) matrix. Shape variabilities may thus be modelled by elements of a (k − 1)²-dimensional vector space.