Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matrices

2014/12/12 by Sadeep Jayasumana, Jayasumana, Sadeep, Richard Hartley +7 · 3 citations
Computer Science · Engineering · #Advanced Image and Video Retrieval Techniques #Automated Road and Building Extraction #Computer Vision and Pattern Recognition (cs.CV) #FOS: Computer and information sciences #Medical Image Segmentation Techniques #Video Surveillance and Tracking Methods #cs.CV

paper · pdf · doi:10.48550/arxiv.1412.4172

Published in CVPR 2013. arXiv admin note: substantial text overlap with arXiv:1412.0265

arxiv created 2014/12/13 · openalex publication_date 2014/12/13 · arxiv updated 2014/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Symmetric Positive Definite (SPD) matrices have become popular to encode image information. Accounting for the geometry of the Riemannian manifold of SPD matrices has proven key to the success of many algorithms. However, most existing methods only approximate the true shape of the manifold locally by its tangent plane. In this paper, inspired by kernel methods, we propose to map SPD matrices to a high dimensional Hilbert space where Euclidean geometry applies. To encode the geometry of the manifold in the mapping, we introduce a family of provably positive definite kernels on the Riemannian manifold of SPD matrices. These kernels are derived from the Gaussian ker- nel, but exploit different metrics on the manifold. This lets us extend kernel-based algorithms developed for Euclidean spaces, such as SVM and kernel PCA, to the Riemannian manifold of SPD matrices. We demonstrate the benefits of our approach on the problems of pedestrian detection, ob- ject categorization, texture analysis, 2D motion segmentation and Diffusion Tensor Imaging (DTI) segmentation.

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