Riemannian Structures on Shape Spaces: A Framework for Statistical Inferences

Abstract

Shape features are becoming increasingly important for inferences in image analysis. Shape analysis involves choosing mathematical representations of shapes, deriving tools for quantifying shape differences, and characterizing imaged objects according to the shapes of their boundaries. In this paper, we focus on characterizing shapes of continuous curves, both open and closed, in ℝ². Under appropriate constraints that remove shape-preserving transformations, these curves form infinite-dimensional, non-linear spaces, called shape spaces, on which inferences are posed. We impose two Riemannian metrics on these spaces and study properties of the resulting structures. An important tool in Riemannian analysis of shapes is the construction of geodesic paths in shape spaces. Not only do the geodesics quantify shape differences, but they also provide a framework for computing intrinsic shape statistics. We will present algorithms to compute simple shape statistics — means and covariances — and will derive probability models on shape spaces using local principal component analysis (PCA), called tangent PCA (TPCA). These concepts are demonstrated using a number of applications: (i) unsupervised clustering of imaged objects according to their shapes, (ii) developing statistical shape models of human silhouettes in infrared surveillance images, (iii) interpolation of endo-and epicardial boundaries in echocardiographic image sequences, and (iv) using shape statistics to test phylogenetic hypotheses.