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In this paper, we study the problem of facial expression recognition using anovel space-time geometric representation. We describe the temporal evolutionof facial landmarks as parametrized trajectories on the Riemannian manifold ofpositive semidefinite matrices of fixed-rank. Our representation has theadvantage to bring naturally a second desirable quantity when comparing shapes-- the spatial covariance -- in addition to the conventional affine-shaperepresentation. We derive then geometric and computational tools forrate-invariant analysis and adaptive re-sampling of trajectories, grounding onthe Riemannian geometry of the manifold. Specifically, our approach involvesthree steps: 1) facial landmarks are first mapped into the Riemannian manifoldof positive semidefinite matrices of rank 2, to build time-parameterizedtrajectories; 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. Weshow the effectiveness of the proposed approach on four publicly availablebenchmarks (CK+, MMI, Oulu-CASIA, and AFEW). The results of the proposedapproach are comparable to or better than the state-of-the-art methods wheninvolving only facial landmarks.

A Novel Space-Time Representation on the Positive Semidefinite Con for Facial Expression Recognition