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
of a population of geometric objects. A standard technique for computing such
descriptions is principal component analysis. However, principal component
analysis is limited in that it only works for data lying in a Euclidean vector
space. 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 talk the ideas of principal component analysis will be extended
to nonlinear curved spaces in particular Lie-Groups and Symmetric spaces. We
develop the method of principal geodesic analysis (PGA), a generalization of the
principal components methodology to Riemannian manifolds.

Linear PCA Analysis

Non-Linear PGA Analysis

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