Kernel methods on the riemannian manifold of symmetric positive definite matrices

Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash Harandi

Research output: Chapter in Book/Report/Conference proceedingConference PaperResearchpeer-review

173 Citations (Scopus)

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 kernel, 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, object categorization, texture analysis, 2D motion segmentation and Diffusion Tensor Imaging (DTI) segmentation.

Original languageEnglish
Title of host publicationProceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition
PublisherIEEE Computer Society
Pages73-80
Number of pages8
DOIs
Publication statusPublished - 15 Nov 2013
Externally publishedYes
EventIEEE Conference on Computer Vision and Pattern Recognition 2013 - Portland, United States of America
Duration: 23 Jun 201328 Jun 2013
Conference number: 26th

Conference

ConferenceIEEE Conference on Computer Vision and Pattern Recognition 2013
Abbreviated titleCVPR 2013
CountryUnited States of America
CityPortland
Period23/06/1328/06/13

Keywords

  • Hilbert space embedding
  • kernel methods
  • positive definite kernels
  • Riemannian manifolds
  • RKHS
  • Symmetric positive definite matrices

Cite this