Abstract
We propose a flag manifold representation as a framework for exposing geometric structure in a large data set. We illustrate the approach by building pose flags for pose identification in digital images of faces and action flags for action recognition in video sequences. These examples illustrate that the flag manifold has the potential to identify common features in noisy and complex datasets.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
T. Arias, A. Edelman, S.T. Smith, The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl 20, 303–353 (1998)
J.R. Beveridge, B. Draper, J-M. Chang, M. Kirby, H. Kley, C. Peterson, Principal angles separate subject illumination spaces in YDB and CMU-PIE. IEEE TPAMI 29(2), 351–363 (2008)
A. Björck, G.H. Golub, Numerical methods for computing angles between linear subspaces. Math. Comput. 27, 579–594 (1973)
J.-M. Chang, M. Kirby, H. Kley, C. Peterson, J.R. Beveridge, B. Draper, Recognition of digital images of the human face at ultra low resolution via illumination spaces, in Computer Vision – ACCV 2007, Tokyo, ed. by Y. Yagi, S.B. Kang, I.S. Kweon, H. Zha. Lecture Notes in Computer Science, vol. 4844 (Springer, 2007), pp. 733–743
B. Draper, M. Kirby, J. Marks, T. Marrinan, C. Peterson, A flag representation for finite collections of subspaces of mixed dimensions. Linear Algebra Appl. 451, 15–32 (2014)
T. Geller, Seeing is not enough. Commun. ACM 54(10), 15–16 (2011)
I.M. James, The Topology of Stiefel Manifolds. London Math Society Lecture Note Series, vol. 24 (Cambridge University Press, Cambridge/New York, 1977)
F. Nater, H. Grabner, L. Van Gool, Exploiting simple hierarchies for unsupervised human behavior analysis, in IEEE CVPR, San Francisco, 2010, pp. 2014–2021
X. Pennec, Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. J. Math. Imaging Vis. 25, 127–154 (2006)
T. Sim, S. Baker, M. Bsat The CMU pose, illumination, and expression (PIE) database, in Proceedings of 5th International Conference on Automatic Face and Gesture Recognition, Washington, D.C., 2002
P. Turaga, A. Veeraraghavan, A. Srivastava, R. Chellappa, Statistical computations on Grassmann and Stiefel manifolds for image and video-based recognition. IEEE TPAMI 33(11), 2273–2286 (2011)
V.S. Varadarajan, Lie Groups, Lie Algebras, and their Representations. Graduate Texts in Mathematics, vol. 102 (Springer, New York, 1984)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Marrinan, T., Beveridge, J.R., Draper, B., Kirby, M., Peterson, C. (2015). Flag Manifolds for the Characterization of Geometric Structure in Large Data Sets. In: Abdulle, A., Deparis, S., Kressner, D., Nobile, F., Picasso, M. (eds) Numerical Mathematics and Advanced Applications - ENUMATH 2013. Lecture Notes in Computational Science and Engineering, vol 103. Springer, Cham. https://doi.org/10.1007/978-3-319-10705-9_45
Download citation
DOI: https://doi.org/10.1007/978-3-319-10705-9_45
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-10704-2
Online ISBN: 978-3-319-10705-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)