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.
Chapter PDF
References
Comaniciu, D., Meer, P.: Mean shift: A robust approach toward feature space analysis. TPAMI 24, 603–619 (2002)
Subbarao, R., Meer, P.: Nonlinear mean shift over riemannian manifolds. IJCV 84, 1–20 (2009)
Ceting, H., Vidal, R.: Intrinsic mean shift for clustering on stiefel and grassmann manifolds. In: CVPR (2009)
Berard, P., Besson, G., Gallot, S.: Embedding riemannian manifolds by their heat kernel. Geometric And Functional Analysis 4, 373–398 (1994)
Carreira, J., Caseiro, R., Batista, J., Sminchisescu, C.: Semantic Segmentation with Second-Order Pooling. In: Fitzgibbon, A., Lazebnik, S., Sato, Y., Schmid, C. (eds.) ECCV 2012, Part VII. LNCS, vol. 7578, pp. 430–443. Springer, Heidelberg (2012)
Turaga, P., Veeraraghavan, A., Srivastava, A., Chellappa, R.: Statistical computations on grassmann and stiefel manifolds for image and video-based recognition. TPAMI 33, 2273–2286 (2011)
Caseiro, R., Henriques, J.F., Martins, P., Batista, J.: A nonparametric riemannian framework on tensor field application to foreground segmentation. In: ICCV (2011)
Pennec, X.: Statistical Computing on Manifolds for Computational Anatomy. In: L’Habilitation Diriger des Recherche (2006)
Turaga, P., Veeraraghavan, A., Srivastava, A., Chellappa, R.: Statistical Analysis on Manifolds and Its Applications to Video Analysis. In: Schonfeld, D., Shan, C., Tao, D., Wang, L. (eds.) Video Search and Mining. SCI, vol. 287, pp. 115–144. Springer, Heidelberg (2010)
Tuzel, O., Porikli, F., Meer, P.: Simultaneous multiple 3d motion estimation via mode finding on lie groups. In: ICCV (2005)
Sheikh, Y., Khan, E., Kanade, T.: Mode-seeking by medoidshifts. In: ICCV (2007)
Vedaldi, A., Soatto, S.: Quick Shift and Kernel Methods for Mode Seeking. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008, Part IV. LNCS, vol. 5305, pp. 705–718. Springer, Heidelberg (2008)
Subbarao, R., Meer, P.: Nonlinear mean shift for clustering over analytic manifolds. In: CVPR (2006)
Schpolkopf, B., Smola, A.J.: Learning with Kernels Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press (2002)
Lafferty, J., Lebanon, G.: Diffusion kernels on statistical manifolds. Journal of Machine Learning Research 6, 129–163 (2005)
Hamm, J., Lee, D.D.: Grassmann discriminant analysis: a unifying view on subspace-based learning. In: International Conference on Machine Learning (2008)
Harandi, M.T., Sanderson, C., Lovell, B.C.: Graph embedding discriminant analysis on grassmannian manifolds for improved image set matching. In: CVPR (2011)
Avramidi, I.: Heat kernel approach in quantum field theory. Nuclear Physics B 104, 3–32 (2002)
Avramidi, I.: The covariant technique for calculation of the heat kernel asymptotic expansion. Physics Letters B 238, 92–97 (1990)
Avramidi, I.G.: Covariant techniques for computation of the heat kernel. Reviews in Mathematical Physics 11, 947–980 (1999)
Avramidi, I.G.: Analytic and Geometric Methods for Heat Kernel Applications in Finance. New Mexico Institute of Mining and Technology, NM 87801, USA (2007)
O’Neill, B.: Semi-Riemannian Manifolds: With Applications to Relativity. Academic Press (1983)
Tuzel, O., Porikli, F., Meer, P.: Kernel methods for weakly supervised mean shift clustering. In: ICCV (2009)
Kulis, B., Basu, S., Dhillon, I., Mooney, R.: Semi-supervised graph clustering: A kernel approach. In: International Conference on Machine Learning (2005)
Cristianini, N., Shawe-Taylor, J.: Support Vector Machines and Other Kernel-Based Learning Methods. Cambridge University Press (2000)
Narcowich, F.: Generalized hermite interpolation and positive definite kernels on a riemannian manifolds. Journal of Mathematical Analysis 190, 165–193 (1995)
Dyn, N., Narcowich, F., Ward, J.: A framework for interpolation and approximation on riemannian manifolds. In: Approximation Theory and Optimization: Tributes to M. J. D. Powell, pp. 133–144. Cambridge University Press (1997)
Gilkey, P.B.: Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem. Publish or Perish (1984)
Chikuse, Y.: Statistics on Special Manifolds. Springer (2003)
Leibe, B., Schiele, B.: Analyzing appearance and contour based methods for object categorization. In: CVPR (2003)
Krizhevsky, A.: Learning multiple features from tiny images. T. Report (2009)
Tuzel, O., Porikli, F., Meer, P.: Region Covariance: A Fast Descriptor for Detection and Classification. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006, Part II. LNCS, vol. 3952, pp. 589–600. Springer, Heidelberg (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Caseiro, R., Henriques, J.F., Martins, P., Batista, J. (2012). Semi-intrinsic Mean Shift on Riemannian Manifolds. In: Fitzgibbon, A., Lazebnik, S., Perona, P., Sato, Y., Schmid, C. (eds) Computer Vision – ECCV 2012. ECCV 2012. Lecture Notes in Computer Science, vol 7572. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33718-5_25
Download citation
DOI: https://doi.org/10.1007/978-3-642-33718-5_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33717-8
Online ISBN: 978-3-642-33718-5
eBook Packages: Computer ScienceComputer Science (R0)