  In this paper, we propose a novel space-time geometric representation of
human landmark configurations and derive tools for comparison and
classification. We model the temporal evolution of landmarks as parametrized
trajectories on the Riemannian manifold of positive semidefinite matrices of
fixed-rank. Our representation has the benefit to bring naturally a second
desirable quantity when comparing shapes, the spatial covariance, in addition
to the conventional affine-shape representation. We derived then geometric and
computational tools for rate-invariant analysis and adaptive re-sampling of
trajectories, grounding on the Riemannian geometry of the underlying manifold.
Specifically, our approach involves three steps: (1) landmarks are first mapped
into the Riemannian manifold of positive semidefinite matrices of fixed-rank to
build time-parameterized trajectories; (2) a temporal warping is performed on
the trajectories, providing a geometry-aware (dis-)similarity measure between
them; (3) finally, a pairwise proximity function SVM is used to classify them,
incorporating the (dis-)similarity measure into the kernel function. We show
that such representation and metric achieve competitive results in applications
as action recognition and emotion recognition from 3D skeletal data, and facial
expression recognition from videos. Experiments have been conducted on several
publicly available up-to-date benchmarks.
