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Semi-intrinsic Mean Shift on Riemannian Manifolds

  • Rui Caseiro
  • João F. Henriques
  • Pedro Martins
  • Jorge Batista
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7572)

Abstract

The original mean shift algorithm [1] on Euclidean spaces (MS) was extended in [2] to operate on general Riemannian manifolds. This extension is extrinsic (Ext-MS) since the mode seeking is performed on the tangent spaces [3], where the underlying curvature is not fully considered (tangent spaces are only valid in a small neighborhood). In [3] was proposed an intrinsic mean shift designed to operate on two particular Riemannian manifolds (IntGS-MS), i.e. Grassmann and Stiefel manifolds (using manifold-dedicated density kernels). It is then natural to ask whether mean shift could be intrinsically extended to work on a large class of manifolds. We propose a novel paradigm to intrinsically reformulate the mean shift on general Riemannian manifolds. This is accomplished by embedding the Riemannian manifold into a Reproducing Kernel Hilbert Space (RKHS) by using a general and mathematically well-founded Riemannian kernel function, i.e. heat kernel [5]. The key issue is that when the data is implicitly mapped to the Hilbert space, the curvature of the manifold is taken into account (i.e. exploits the underlying information of the data). The inherent optimization is then performed on the embedded space. Theoretic analysis and experimental results demonstrate the promise and effectiveness of this novel paradigm.

Keywords

Riemannian Manifold Vector Bundle Tangent Space Heat Kernel Reproduce Kernel Hilbert Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Rui Caseiro
    • 1
  • João F. Henriques
    • 1
  • Pedro Martins
    • 1
  • Jorge Batista
    • 1
  1. 1.Institute of Systems and RoboticsUniversity of CoimbraPortugal

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