A Framework for Shape Analysis via Hilbert Space Embedding

Abstract

We propose a framework for 2D shape analysis using positive definite kernels defined on Kendall's shape manifold. Different representations of 2D shapes are known to generate different nonlinear spaces. Due to the nonlinearity of these spaces, most existing shape classification algorithms resort to nearest neighbor methods and to learning distances on shape spaces. Here, we propose to map shapes on Kendall's shape manifold to a high dimensional Hilbert space where Euclidean geometry applies. To this end, we introduce a kernel on this manifold that permits such a mapping, and prove its positive definiteness. This kernel lets us extend kernel-based algorithms developed for Euclidean spaces, such as SVM, MKL and kernel PCA, to the shape manifold. We demonstrate the benefits of our approach over the state-of-the-art methods on shape classification, clustering and retrieval.