Principal Geodesic Analysis for the Study of Nonlinear Statistics of Shape (2004)
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1 Abstract
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. While this is certainly sufficient for geometric models that are parameterized by a set of landmarks or a dense collection of boundary points, it does not handle more complex representations of shape. We have been developing representations of geometry based on the medial axis description or m-rep. While the medial representation provides a rich language for variability in terms of bending, twisting, and widening, the medial parameters are not elements of a Euclidean vector space. They are in fact elements of a nonlinear Riemannian symmetric space. In this paper, we develop the method of principal geodesic analysis, a generalization of principal component analysis to the manifold setting. We demonstrate its use in describing the variability of medially-defined anatomical objects. Results of applying this framework on a population of hippocampi in a schizophrenia study are presented.
2 NOTES
This is the paper that proposed the Principal Geodesic Analysis (PGA). Their application is quite specific, and I’m not aware of it still being used nowadays nor have I any contact with it. That said, the main idea refer to how PGA works, which has its specific note in here: Principal Geodesic Analysis (PGA).