Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matrices

TL;DR

This paper introduces positive definite kernels on the Riemannian manifold of SPD matrices, enabling kernel methods like SVM and PCA to better encode the geometry of SPD data in various computer vision and imaging tasks.

Contribution

It proposes a family of positive definite kernels derived from the Gaussian kernel that incorporate manifold geometry, extending kernel methods to SPD matrices.

Findings

Improved pedestrian detection accuracy

Enhanced object categorization performance

Better segmentation in DTI imaging

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.