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Issue No.09 - Sept. (2013 vol.35)
pp: 2161-2174
A. Cherian , Dept. of Comput. Sci. & Eng., Univ. of Minnesota, Minneapolis, MN, USA
S. Sra , Dept. of Empirical Inference, Max Planck Inst. for Intell. Syst., Tubingen, Germany
A. Banerjee , Dept. of Comput. Sci. & Eng., Univ. of Minnesota, Minneapolis, MN, USA
N. Papanikolopoulos , Dept. of Comput. Sci. & Eng., Univ. of Minnesota, Minneapolis, MN, USA
ABSTRACT
Covariance matrices have found success in several computer vision applications, including activity recognition, visual surveillance, and diffusion tensor imaging. This is because they provide an easy platform for fusing multiple features compactly. An important task in all of these applications is to compare two covariance matrices using a (dis)similarity function, for which the common choice is the Riemannian metric on the manifold inhabited by these matrices. As this Riemannian manifold is not flat, the dissimilarities should take into account the curvature of the manifold. As a result, such distance computations tend to slow down, especially when the matrix dimensions are large or gradients are required. Further, suitability of the metric to enable efficient nearest neighbor retrieval is an important requirement in the contemporary times of big data analytics. To alleviate these difficulties, this paper proposes a novel dissimilarity measure for covariances, the Jensen-Bregman LogDet Divergence (JBLD). This divergence enjoys several desirable theoretical properties and at the same time is computationally less demanding (compared to standard measures). Utilizing the fact that the square root of JBLD is a metric, we address the problem of efficient nearest neighbor retrieval on large covariance datasets via a metric tree data structure. To this end, we propose a K-Means clustering algorithm on JBLD. We demonstrate the superior performance of JBLD on covariance datasets from several computer vision applications.
INDEX TERMS
Covariance matrix, Measurement, Symmetric matrices, Manifolds, Eigenvalues and eigenfunctions, Computer vision, Standards,activity recognition, Region covariance descriptors, Bregman divergence, image search, nearest neighbor search, LogDet divergence, video surveillance
CITATION
A. Cherian, S. Sra, A. Banerjee, N. Papanikolopoulos, "Jensen-Bregman LogDet Divergence with Application to Efficient Similarity Search for Covariance Matrices", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.35, no. 9, pp. 2161-2174, Sept. 2013, doi:10.1109/TPAMI.2012.259
REFERENCES
[1] D. Alexander, C. Pierpaoli, P. Basser, and J. Gee, "Spatial Transformations of Diffusion Tensor Magnetic Resonance Images," IEEE Trans. Medical Imaging, vol. 20, no. 11, pp. 1131-1139, Nov. 2001.
[2] H. Zhu, H. Zhang, J. Ibrahim, and B. Peterson, "Statistical Analysis of Diffusion Tensors in Diffusion-Weighted Magnetic Resonance Imaging Data," J. Am. Statistical Assoc., vol. 102, no. 480, pp. 1085-1102, 2007.
[3] M. Chiang, R. Dutton, K. Hayashi, O. Lopez, H. Aizenstein, A. Toga, J. Becker, and P. Thompson, "3D Pattern of Brain Atrophy in HIV/AIDS Visualized Using Tensor-Based Morphometry," Neuroimage, vol. 34, no. 1, pp. 44-60, 2007.
[4] O. Tuzel, F. Porikli, and P. Meer, "Region Covariance: A Fast Descriptor for Detection and Classification," Proc. European Conf. Computer Vision, 2006.
[5] F. Porikli and O. Tuzel, "Covariance Tracker," Proc. IEEE Conf. Computer Vision and Pattern Recognition, 2006.
[6] O. Tuzel, F. Porikli, and P. Meer, "Human Detection via Classification on Riemannian Manifolds," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 1-8, 2007.
[7] J. Malcolm, Y. Rathi, and A. Tannenbaum, "A Graph Cut Approach to Image Segmentation in Tensor Space," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 1-8, 2007.
[8] T. Brox, M. Rousson, R. Deriche, and J. Weickert, "Unsupervised Segmentation Incorporating Colour, Texture, and Motion," Proc. 10th Int'l Conf. Computer Analysis of Images and Patterns, pp. 353-360, 2003.
[9] Y. Pang, Y. Yuan, and X. Li, "Gabor-Based Region Covariance Matrices for Face Recognition," IEEE Trans. Circuits and Systems for Video Technology, vol. 18, no. 7, pp. 989-993, July 2008.
[10] W. Zheng, H. Tang, Z. Lin, and T. Huang, "Emotion Recognition from Arbitrary View Facial Images," Proc. European Conf. Computer Vision, pp. 490-503, 2010.
[11] K. Guo, P. Ishwar, and J. Konrad, "Action Recognition Using Sparse Representation on Covariance Manifolds of Optical Flow," Proc. Seventh IEEE Int'l Conf. Advanced Video and Signal Based Surveillance, pp. 188-195, 2010.
[12] C. Ye, J. Liu, C. Chen, M. Song, and J. Bu, "Speech Emotion Classification on a Riemannian Manifold," Proc. Ninth Pacific Rim Conf. Multimedia: Advances in Multimedia Information Processing, pp. 61-69, 2008.
[13] M. Datar, N. Immorlica, P. Indyk, and V. Mirrokni, "Locality-Sensitive Hashing Scheme Based on P-Stable Distributions," Proc. Ann. Symp. Computational Geometry, pp. 253-262, 2004.
[14] X. Pennec, P. Fillard, and N. Ayache, "A Riemannian Framework for Tensor Computing," Int'l J. Computer Vision, vol. 66, no. 1, pp. 41-66, 2006.
[15] V. Arsigny, P. Fillard, X. Pennec, and N. Ayache, "Log-Euclidean Metrics for Fast and Simple Calculus on Diffusion Tensors," Magnetic Resonance in Medicine, vol. 56, no. 2, pp. 411-421, 2006.
[16] M. Moakher and P. Batchelor, "Symmetric Positive-Definite Matrices: From Geometry to Applications and Visualization," Visualization and Processing of Tensor Fields, Springer, 2006.
[17] R. Bhatia, Positive Definite Matrices. Princeton Univ. Press, 2007.
[18] X. Li, W. Hu, Z. Zhang, X. Zhang, M. Zhu, and J. Cheng, "Visual Tracking via Incremental Log-Euclidean Riemannian Subspace Learning," Proc. IEEE Conf. Computer Vision and Pattern Recognition, 2008.
[19] Q. Gu and J. Zhou, "A Similarity Measure under Log-Euclidean Metric for Stereo Matching," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 1-4, 2009.
[20] Z. Wang, B. Vemuri, Y. Chen, and T. Mareci, "A Constrained Variational Principle for Direct Estimation and Smoothing of the Diffusion Tensor Field from Complex DWI," IEEE Trans. Medical Imaging, vol. 23, no. 8, pp. 930-939, Aug. 2004.
[21] I. Dryden, A. Koloydenko, and D. Zhou, "Non-Euclidean Statistics for Covariance Matrices, with Applications to Diffusion Tensor Imaging," Annals of Applied Statistics, vol. 3, no. 3, pp. 1102-1123, 2009.
[22] F. Nielsen, P. Piro, and M. Barlaud, "Bregman Vantage Point Trees for Efficient Nearest Neighbor Queries," Proc. IEEE Int'l Conf. Multimedia and Expo, pp. 878-881, 2009.
[23] L. Cayton, "Fast Nearest Neighbor Retrieval for Bregman Divergences," Proc. 25th Int'l Conf. Machine Learning, pp. 112-119, 2008.
[24] P. Turaga and R. Chellappa, "Nearest-Neighbor Search Algorithms on Non-Euclidean Manifolds for Computer Vision Applications," Proc. Seventh ACM Indian Conf. Computer Vision, Graphics and Image Processing, pp. 282-289, 2010.
[25] R. Chaudhry and Y. Ivanov, "Fast Approximate Nearest Neighbor Methods for Non-Euclidean Manifolds with Applications to Human Activity Analysis in Videos," Proc. European Conf. Computer Vision, pp. 735-748, 2010.
[26] Z. Chebbi and M. Moakher, "Means of Hermitian Positive-Definite Matrices Based on the Log-Determinant Alpha-Divergence Function," Linear Algebra and Its Applications, vol. 436, pp. 1872-1889, 2012.
[27] B. Kulis, M. Sustik, and I. Dhillon, "Low-Rank Kernel Learning with Bregman Matrix Divergences," J. Machine Learning Research, vol. 10, pp. 341-376, 2009.
[28] A. Banerjee, S. Merugu, I. Dhillon, and J. Ghosh, "Clustering with Bregman Divergences," J. Machine Learning Research, vol. 6, pp. 1705-1749, 2005.
[29] Y. Censor and S.A. Zenios, Parallel Optimization: Theory, Algorithms, and Applications. Oxford Univ. Press, 1997.
[30] F. Nielsen and R. Nock, "On the Centroids of Symmetrized Bregman Divergences," Arxiv preprint arXiv:0711.3242, 2007.
[31] S. Sra, "Positive Definite Matrices and the Symmetric Stein Divergence," http://arxiv.org/abs1110.1773, 2011.
[32] G.H. Golub and C.F. Van Loan, Matrix Computations, third ed. Johns Hopkins Univ. Press, 1996.
[33] P. Fillard, V. Arsigny, N. Ayache, and X. Pennec, "A Riemannian Framework for the Processing of Tensor-Valued Images," Proc. First Int'l Workshop Deep Structure, Singularities, and Computer Vision, pp. 112-123, 2005.
[34] A. Yuille and A. Rangarajan, "The Concave-Convex Procedure," Neural Computation, vol. 15, no. 4, pp. 915-936, 2003.
[35] F. Nielsen and S. Boltz, "The Burbea-Rao and Bhattacharyya Centroids," IEEE Trans. Information Theory, vol. 57, no. 8, pp. 5455-5466, Aug. 2011.
[36] R. Bhatia, Matrix Analysis. Springer Verlag, 1997.
[37] R. Horn and C. Johnson, Matrix Analysis. Cambridge Univ. Press, 1990.
[38] J.D. Lawson and Y. Lim, "The Geometric Mean, Matrices, Metrics, and More," The Am. Math. Monthly, vol. 108, no. 9, pp. 797-812, 2001.
[39] B. Sriperumbudur and G. Lanckriet, "On the Convergence of the Concave-Convex Procedure," Proc. Neural information Processing Systems, vol. 22, pp. 1759-1767, 2009.
[40] P. Ciaccia, M. Patella, and P. Zezula, "M-Tree: An Efficient Access Method for Similarity Search in Metric Spaces," Proc. 23rd Int'l Conf. Very Large Databases, pp. 426-435, 1997.
[41] S. Brin, "Near Neighbor Search in Large Metric Spaces," Proc. 21st Int'l Conf. Very Large Databases, 1995.
[42] D. Bini and B. Iannazzo, "Computing the Karcher Mean of Symmetric Positive Definite Matrices," Linear Algebra and Its Applications, vol. 438, pp. 1700-1710, 2011.
[43] T. Myrvoll and F. Soong, "On Divergence Based Clustering of Normal Distributions and Its Application to HMM Adaptation," Proc. European Conf. Speech Comm. and Technology, pp. 1517-1520, 2003.
[44] E. Maggio, E. Piccardo, C. Regazzoni, and A. Cavallaro, "Particle PHD Filtering for Multi-Target Visual Tracking," Proc. IEEE Int'l Conf. Acoustics, Speech, and Signal Processing, vol. 1, 2007.
[45] R. Caseiro, J. Henriques, and J. Batista, "Foreground Segmentation via Background Modeling on Riemannian Manifolds," Proc. 20th Int'l Conf. Pattern Recognition, pp. 3570-3574, 2010.
[46] K. Dana, B. Van-Ginneken, S. Nayar, and J. Koenderink, "Reflectance and Texture of Real World Surfaces," ACM Trans. Graphics, vol. 18, no. 1, pp. 1-34, 1999.
[47] L. Gorelick, M. Blank, E. Shechtman, M. Irani, and R. Basri, "Actions as Space-Time Shapes," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 29, no. 12, pp. 2247-2253, Dec. 2007.
[48] C. Chen, M. Ryoo, and J. Aggarwal, "UT-Tower Data Set: Aerial View Activity Classification Challenge," http://cvrc.ece.utexas. edu/SDHA2010Aerial_View_Activity.html , 2010.
[49] V. Jain and E. Learned-Miller, "FDDB: A Benchmark for Face Detection in Unconstrained Settings," Technical Report UM-CS-2010-009, Univ. of Massachusetts, Amherst, 2010.
[50] J. Beis and D. Lowe, "Shape Indexing Using Approximate Nearest-Neighbour Search in High-Dimensional Spaces," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 1000-1006, 1997.
[51] F. Nielsen, M. Liu, and B.C. Vemuri, "Jensen Divergence-Based Means of SPD Matrices," Matrix Information Geometry, pp. 111-122, Springer, 2012.
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