Statistics of Shape: Simple Statistics on Interesting Spaces
Sarang Joshi
Florida State University
Dept. of Electrical Engineering
A primary goal of statistical shape analysis is to describe the variability within a population of geometric objects. A standard technique for computing such descriptions is principal component analysis (PCA). However, PCA is limited in that it applies only to data lying in a Euclidean vector space. The statistical framework is well understood when object parameters are elements of a Euclidean vector space, as is the case when objects are described via landmarks or as a dense collection of boundary points.
We have been developing geometric representations based on the medial axis description, or m-rep. Although this representation has proven effective, the medial parameters are not naturally elements of a Euclidean space. In this talk, the ideas of principal component analysis are extended to nonlinear curved spaces, in particular Lie groups and symmetric spaces.
We develop the method of principal geodesic analysis (PGA), a generalization of principal component methodology to Riemannian manifolds. Examples comparing linear PCA analysis and nonlinear PGA analysis are presented.