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

Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matrices