Intrinsic Mean Shift for Clustering on Stiefel and Grassmann Manifolds

Abstract

The mean shift algorithm, which is a nonparametric density estimator for detecting the modes of a distribution on a Euclidean space, was recently extended to operate on analytic manifolds. The extension is extrinsic in the sense that the inherent optimization is performed on the tangent spaces of these manifolds. This approach specifically requires the use of the exponential map at each iteration. This paper presents an alternative mean shift formulation, which performs the iterative optimization "on" the manifold of interest and intrinsically locates the modes via consecutive evaluations of a mapping. In particular, these evaluations constitute a modified gradient ascent scheme that avoids the computation of the exponential maps for Stiefel and Grassmann manifolds. The performance of our algorithm is evaluated by conducting extensive comparative studies on synthetic data as well as experiments on object categorization and segmentation of multiple motions.