A Framework for Shape Analysis via Hilbert Space Embedding

TL;DR

This paper introduces a novel kernel-based framework for 2D shape analysis on Kendall's shape manifold, enabling the use of Euclidean algorithms for shape classification, clustering, and retrieval.

Contribution

It develops a positive definite kernel on Kendall's shape manifold, allowing kernel methods to be directly applied to shape analysis tasks.

Findings

Kernel enables Euclidean algorithms on shape data

Improved shape classification accuracy

Effective shape clustering and retrieval

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.