Human Activity as a Manifold-Valued Random Process

Most of the previous shape-based human activity models are built with either a linear assumption or an extrinsic interpretation of the nonlinear geometry of the shape space, both of which proved to be problematic on account of the nonlinear intrinsic geometry of the associated shape spaces. In this paper, we propose an intrinsic stochastic modeling of human activity on a shape manifold. More importantly, within an elegant and theoretically sound framework, our work effectively bridges the nonlinear modeling of human activity on a nonlinear space, with the classic stochastic modeling in a Euclidean space, and thereby provides a foundation for a more effective and accurate analysis of the nonlinear feature space of activity models. From a video sequence, human activity is extracted as a sequence of shapes. Such a sequence is considered as one realization of a random process on a shape manifold. Different activities are then modeled as manifold valued random processes with different distributions. To address the problem of stochastic modeling on a manifold, we first construct a nonlinear invertible map of a manifold valued process to a Euclidean process. The resulting process is then modeled as a global or piecewise Brownian motion. The mapping from a manifold to a Euclidean space is known as a stochastic development. The advantage of such a technique is that it yields a one-one correspondence, and the resulting Euclidean process intrinsically captures the curvature on the original manifold. The proposed algorithm is validated on two activity databases and compared with the related works on each of these. The substantiating results demonstrate the viability and high-accuracy of our modeling technique in characterizing and classifying different activities.