  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.
