Shape Spaces, Metrics and Their Applications to Brain Anatomy

We construct a framework for the analysis of shapes in Euclidean space of any dimension. In this framework, a shape is represented as a continuous map from a reference Riemannian manifold M. To quantify global shape differences, the framework employs a Sobolev-type metric considering information of both the position and the first-order derivative at each point of the shape. Since first-order derivatives are very sensitive to small variations, the derivative term in the metric is smoothed out to reduce noise by a heat operator, which is constructed using the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on M. A pre-shape space is constructed as an unit sphere of an inner product space, where each shape can be viewed as a point on it. A geodesic shape distance on the pre-shape space and an extrinsic distance are given based on the proposed metric. Besides global measures, we develop an energy function to quantify local shape divergence. Specifically, it is modified to reflect the magnitude of local shrinkage or expansion. In practical computations, the framework based on continuous representations is discretized using simplicial complex. To address issues in statistical shape analysis for a population of shapes, we present algorithms to calculate the mean shape and to perform principal component analysis on the tangent plane at the mean on the pre-shape space. The shape models and statistical tools are applied to three data sets of magnetic resonance (MR) scans of the hippocampus to study blindness and Alzheimer's disease. The shape of a hippocampus is represented using either a triangular mesh to represent its contour surface or a cubical mesh to represent its whole volume.