Elastic functional coding of human actions: From vector-fields to latent variables

Human activities observed from visual sensors often give rise to a sequence of smoothly varying features. In many cases, the space of features can be formally defined as a manifold, where the action becomes a trajectory on the manifold. Such trajectories are high dimensional in addition to being non-linear, which can severely limit computations on them. We also argue that by their nature, human actions themselves lie on a much lower dimensional manifold compared to the high dimensional feature space. Learning an accurate low dimensional embedding for actions could have a huge impact in the areas of efficient search and retrieval, visualization, learning, and recognition. Traditional manifold learning addresses this problem for static points in ℝⁿ, but its extension to trajectories on Riemannian manifolds is non-trivial and has remained unexplored. The challenge arises due to the inherent non-linearity, and temporal variability that can significantly distort the distance metric between trajectories. To address these issues we use the transport square-root velocity function (TSRVF) space, a recently proposed representation that provides a metric which has favorable theoretical properties such as invariance to group action. We propose to learn the low dimensional embedding with a manifold functional variant of principal component analysis (mfPCA). We show that mf-PCA effectively models the manifold trajectories in several applications such as action recognition, clustering and diverse sequence sampling while reducing the dimensionality by a factor of ∼ 250×. The mfPCA features can also be reconstructed back to the original manifold to allow for easy visualization of the latent variable space.