  In this paper, we study the problem of facial expression recognition using a
novel space-time geometric representation. We describe the temporal evolution
of facial landmarks as parametrized trajectories on the Riemannian manifold of
positive semidefinite matrices of fixed-rank. Our representation has the
advantage to bring naturally a second desirable quantity when comparing shapes
-- the spatial covariance -- in addition to the conventional affine-shape
representation. We derive then geometric and computational tools for
rate-invariant analysis and adaptive re-sampling of trajectories, grounding on
the Riemannian geometry of the manifold. Specifically, our approach involves
three steps: 1) facial landmarks are first mapped into the Riemannian manifold
of positive semidefinite matrices of rank 2, to build time-parameterized
trajectories; 2) a temporal alignment is performed on the trajectories,
providing a geometry-aware (dis-)similarity measure between them; 3) finally,
pairwise proximity function SVM (ppfSVM) is used to classify them,
incorporating the latter (dis-)similarity measure into the kernel function. We
show the effectiveness of the proposed approach on four publicly available
benchmarks (CK+, MMI, Oulu-CASIA, and AFEW). The results of the proposed
approach are comparable to or better than the state-of-the-art methods when
involving only facial landmarks.
