Statistics of Shape via Principal Component Analysis on Lie Groups

Principal component analysis has proven to be useful for understanding geometric variability in populations of parameterized objects. The statistical framework is well understood when the parameters of the objects are elements of a Euclidean vector space. This is certainly the case when the objects are described via landmarks or as a dense collection of boundary points. We have been developing representations of geometry based on the medial axis description or m-rep. Although this description has proven to be effective, the medial parameters are not naturally elements of a Euclidean space. In this paper we show that medial descriptions are in fact elements of a Lie group. We develop methodology based on Lie groups for the statistical analysis of medially-defined anatomical objects.