iccv iccv2013 iccv2013-257 iccv2013-257-reference knowledge-graph by maker-knowledge-mining
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Author: Peihua Li, Qilong Wang, Wangmeng Zuo, Lei Zhang
Abstract: The symmetric positive de?nite (SPD) matrices have been widely used in image and vision problems. Recently there are growing interests in studying sparse representation (SR) of SPD matrices, motivated by the great success of SR for vector data. Though the space of SPD matrices is well-known to form a Lie group that is a Riemannian manifold, existing work fails to take full advantage of its geometric structure. This paper attempts to tackle this problem by proposing a kernel based method for SR and dictionary learning (DL) of SPD matrices. We disclose that the space of SPD matrices, with the operations of logarithmic multiplication and scalar logarithmic multiplication de?ned in the Log-Euclidean framework, is a complete inner product space. We can thus develop a broad family of kernels that satis?es Mercer’s condition. These kernels characterize the geodesic distance and can be computed ef?ciently. We also consider the geometric structure in the DL process by updating atom matrices in the Riemannian space instead of in the Euclidean space. The proposed method is evaluated with various vision problems and shows notable per- formance gains over state-of-the-arts.
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