An Elastic Riemannian Framework for Shape Analysis of Curves and Tree-Like Structures

Abstract

Shape analysis of complex structures formed by Euclidean curves and trees are of interest in many scientific domains. The difficulty in this analysis comes from: (1) Manifold representations of curves and trees, and (2) need for the analysis to be invariant to certain shape-preserving transformations. Additionally, one is faced with the difficult task of registering points and parts across objects during shape comparisons. We present a Riemannian framework to solve this problem as follows: we select an elastic Riemannian metric that is invariant to the action of re-parameterization group and use a square-root velocity function to transform this metric into the $\mathbb {L}^2$ norm. Re-parameterization of objects is considered to control registrations across objects, and an inherited distance on the quotient space of shape representations modulo shape-preserving transformations forms the shape metric. The resulting framework is used to compute geodesic paths and sample means, and to discover modes of variability in sample shapes. We demonstrate these ideas using some simple examples involving planar shapes and neuron morphology.