Manifold Kernel Sparse Representation of Symmetric Positive-Definite Matrices and Its Applications

The symmetric positive-definite (SPD) matrix, as a connected Riemannian manifold, has become increasingly popular for encoding image information. Most existing sparse models are still primarily developed in the Euclidean space. They do not consider the non-linear geometrical structure of the data sp...

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Published in:	IEEE transactions on image processing Vol. 24; no. 11; pp. 3729 - 3741
Main Authors:	Wu, Yuwei, Jia, Yunde, Li, Peihua, Zhang, Jian, Yuan, Junsong
Format:	Journal Article
Language:	English
Published:	United States IEEE 01.11.2015
Subjects:	Algorithms Biometric Identification Covariance matrices Databases, Factual Dictionaries Face recognition Geometry Humans Image classification Image Processing, Computer-Assisted - methods Kernel Kernel sparse coding Machine Learning Manifolds Measurement Region covariance descriptor Riemannian manifold Sparse matrices Symmetric Positive Definite Matrices Visual tracking Kernel sparse coding image classification visual tracking face recognition region covariance descriptor symmetric positive definite matrices Riemannian manifold
ISSN:	1057-7149, 1941-0042
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Abstract	The symmetric positive-definite (SPD) matrix, as a connected Riemannian manifold, has become increasingly popular for encoding image information. Most existing sparse models are still primarily developed in the Euclidean space. They do not consider the non-linear geometrical structure of the data space, and thus are not directly applicable to the Riemannian manifold. In this paper, we propose a novel sparse representation method of SPD matrices in the data-dependent manifold kernel space. The graph Laplacian is incorporated into the kernel space to better reflect the underlying geometry of SPD matrices. Under the proposed framework, we design two different positive definite kernel functions that can be readily transformed to the corresponding manifold kernels. The sparse representation obtained has more discriminating power. Extensive experimental results demonstrate good performance of manifold kernel sparse codes in image classification, face recognition, and visual tracking.
AbstractList	The symmetric positive-definite (SPD) matrix, as a connected Riemannian manifold, has become increasingly popular for encoding image information. Most existing sparse models are still primarily developed in the Euclidean space. They do not consider the non-linear geometrical structure of the data space, and thus are not directly applicable to the Riemannian manifold. In this paper, we propose a novel sparse representation method of SPD matrices in the data-dependent manifold kernel space. The graph Laplacian is incorporated into the kernel space to better reflect the underlying geometry of SPD matrices. Under the proposed framework, we design two different positive definite kernel functions that can be readily transformed to the corresponding manifold kernels. The sparse representation obtained has more discriminating power. Extensive experimental results demonstrate good performance of manifold kernel sparse codes in image classification, face recognition, and visual tracking.
Author	Yunde Jia Peihua Li Jian Zhang Junsong Yuan Yuwei Wu
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SubjectTerms	Algorithms Biometric Identification Covariance matrices Databases, Factual Dictionaries Face recognition Geometry Humans Image classification Image Processing, Computer-Assisted - methods Kernel Kernel sparse coding Machine Learning Manifolds Measurement Region covariance descriptor Riemannian manifold Sparse matrices Symmetric Positive Definite Matrices Visual tracking
Title	Manifold Kernel Sparse Representation of Symmetric Positive-Definite Matrices and Its Applications
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