  Statistical classification of actions in videos is mostly performed by
extracting relevant features, particularly covariance features, from image
frames and studying time series associated with temporal evolutions of these
features. A natural mathematical representation of activity videos is in form
of parameterized trajectories on the covariance manifold, i.e. the set of
symmetric, positive-definite matrices (SPDMs). The variable execution-rates of
actions implies variable parameterizations of the resulting trajectories, and
complicates their classification. Since action classes are invariant to
execution rates, one requires rate-invariant metrics for comparing
trajectories. A recent paper represented trajectories using their transported
square-root vector fields (TSRVFs), defined by parallel translating
scaled-velocity vectors of trajectories to a reference tangent space on the
manifold. To avoid arbitrariness of selecting the reference and to reduce
distortion introduced during this mapping, we develop a purely intrinsic
approach where SPDM trajectories are represented by redefining their TSRVFs at
the starting points of the trajectories, and analyzed as elements of a vector
bundle on the manifold. Using a natural Riemannain metric on vector bundles of
SPDMs, we compute geodesic paths and geodesic distances between trajectories in
the quotient space of this vector bundle, with respect to the
re-parameterization group. This makes the resulting comparison of trajectories
invariant to their re-parameterization. We demonstrate this framework on two
applications involving video classification: visual speech recognition or
lip-reading and hand-gesture recognition. In both cases we achieve results
either comparable to or better than the current literature.
