Riemannian Shape Analysis Based on Meridian Curves

Comparing different shapes is a fundamental problem in Computational Anatomy (CA), where a rigorous and intrinsic distance metric is key for a shape analysis system to work effectively and consistently. In this paper, we propose a shape comparison and classification framework that consists of two major components. A meridian-based shape representation, stemmed from spectral graph theory, possesses the merit of being able to capture the salient structure property along the direction of maximal shape variations. The meridian extraction algorithm, relying on a discrete approximation of the gradient of the induced Fiedler function, can also be utilized for other purposes, e.g., mesh generation for three-dimensional objects. After projecting the 3D meridians onto a multi-dimensional sphere, similarity/dissimilarity between shapes can be computed based on a Riemannian spherical distance metric. Group statistics, as well as object classification/clustering, can be readily carried out. We demonstrate the effectiveness of our framework with subcortical structures extracted from human brain MR images.