A Nonparametric Riemannian Framework on Tensor Field with Application to Foreground Segmentation

Abstract

Background modelling on tensor field has recently been proposed for foreground detection tasks. Taking into account the Riemannian structure of the tensor manifold, recent research has focused on developing parametric methods on the tensor domain e.g. gaussians mixtures (GMM) [7]. However, in some scenarios, simple parametric models do not accurately explain the physical processes. Kernel density estimators (KDE) have been successful to model, on Euclidean sample spaces the nonparametric nature of complex, time varying, and non-static backgrounds [8]. Founded on the mathematically rigorous KDE paradigm on general Riemannian manifolds [15], we define a KDE specifically to operate on the tensor manifold. We present a mathematically-sound framework for nonparametric modeling on tensor field to foreground segmentation. We endow the tensor manifold with two well-founded Riemannian metrics, i.e. Affine-Invariant and Log-Euclidean. Theoretical aspects are defined and the metrics are compared experimentally. Theoretic analysis and experimental results demonstrate the promise/effectiveness of the framework.