  We consider a family of structural descriptors for visual data, namely
covariance descriptors (CovDs) that lie on a non-linear symmetric positive
definite (SPD) manifold, a special type of Riemannian manifolds. We propose an
improved version of CovDs for image set coding by extending the traditional
CovDs from Euclidean space to the SPD manifold. Specifically, the manifold of
SPD matrices is a complete inner product space with the operations of
logarithmic multiplication and scalar logarithmic multiplication defined in the
Log-Euclidean framework. In this framework, we characterise covariance
structure in terms of the arc-cosine kernel which satisfies Mercer's condition
and propose the operation of mean centralization on SPD matrices. Furthermore,
we combine arc-cosine kernels of different orders using mixing parameters
learnt by kernel alignment in a supervised manner. Our proposed framework
provides a lower-dimensional and more discriminative data representation for
the task of image set classification. The experimental results demonstrate its
superior performance, measured in terms of recognition accuracy, as compared
with the state-of-the-art methods.
