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Jensen-Bregman LogDet Divergence with Application to Efficient Similarity Search for Covariance Matrices
Sept. 2013 (vol. 35 no. 9)
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
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:
video surveillance,computer vision,covariance matrices,image retrieval,pattern clustering,video surveillance,Jensen-Bregman LogDet divergence,similarity search,covariance matrices,computer vision applications,Riemannian metric,Riemannian manifold,distance computations,nearest neighbor retrieval,dissimilarity measure,JBLD,large covariance datasets,k-means clustering algorithm,big data analytics,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 and Machine Intelligence, vol. 35, no. 9, pp. 2161-2174, Sept. 2013, doi:10.1109/TPAMI.2012.259
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