Background Modelling on Tensor Field for Foreground Segmentation

The paper proposes a new method to perform foreground detection by means of background modeling using the tensor concept. Sometimes, statistical modelling directly on image values is not enough to achieve a good discrimination. Thus the image may be converted into a more information rich form, such as a tensor field, to yield latent discriminating features. Taking into account the theoretically well-founded differential geometrical properties of the Riemannian manifold where tensors lie, we propose a new approach for foreground detection on tensor field based on data modeling by means of Gaussians mixtures directly on tensor domain. We introduced a online Kmeans approximation of the Expectation Maximization algorithm to estimate the parameters based on an Affine-Invariant metric. This metric has excellent theoretical properties but essentially due to the space curvature the computational burden is high. We propose a novel Kmeans algorithm based on a new family of metrics, called Log-Euclidean, in order to speed up the process, while conserving the same theoretical properties. Contrary to the affine case, we obtain a space with a null curvature. Hence, classical statistical tools usually reserved to vectors are efficiently generalized to tensors in the Log-Euclidean framework. Theoretical aspects are presented and the Affine-Invariant and Log-Euclidean frameworks are compared experimentally. From a practical point of view, results are similar to those of the Affine-Invariant framework but are obtained much faster. Theoretic analysis and experimental results demonstrate the promise and effectiveness of the proposed framework.