Shape Variation of Medial Axis Representations via Principal Geodesic Analysis on Symmetric Spaces

Abstract

Statistical shape analysis of anatomical structures plays an important role in many medical image analysis applications. For instance, shape statistics are useful in understanding the structural changes in anatomy that are caused by growth and disease. Classical statistical techniques can be applied to study shape representations that are parameterized by a linear space, such as landmark data or boundary meshes, but they cannot 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 elements of a nonlinear Riemannian symmetric space. Therefore, linear statistical methods are not applicable in the m-rep setting, and statistical methods for analyzing manifold data are needed. This chapter presents a general method called principal geodesic analysis (PGA) for computing the variability of manifold-valued data. PGA is a direct generalization of principal component analysis (PCA) and is based solely on intrinsic properties of the underlying manifold, such as the notion of geodesic curves and distance. We demonstrate the use of PGA to describe the shape variability of medial representations, and results are shown on a hippocampus data set. The applicability of PGA is also shown within a 3D image segmentation framework based on a Bayesian posterior optimization of deformable medial models.