Internet

A Nonparametric Riemannian Framework on Tensor Field with Application to Foreground Segmentation
Rui Caseiro, Jo˜ o F. Henriques, Pedro Martins and Jorge Batista a Institute for Systems and Robotics - University of Coimbra, Portugal
{ruicaseiro, henriques, pedromartins, batista}@isr.uc.pt

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 nonparametricnature 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.

1. Introduction
Foreground detection is a crucial aspect in the understanding and analysis of video sequences. It is often described as the process that subdivides an image intoregions of interest-object and background. This task usually relies on the extraction of suitable features that are highly discriminative. Statistical modeling in color/intensity space is a widely used approach for background modeling to foreground detection. However, there are situations where statistical modelling directly on image values isn’t enough to achieve a good discrimination (e.g. dynamicscenes, illumination variation). Thus the image may be converted into a more information rich form, such as a tensor field, to yield latent discriminating features (e.g. color, gradients, filters responses). Texture is one of the most important features, therefore its consideration can greatly improve image analysis. The structure tensor [3, 10] has been introduced for 1

such texture analysis asa fast local computation providing a measure of the presence of edges and their orientation. For the sake of brevity, the related work description will be neither rigorous nor complete, but we want outline some of the key ideas (refer to [5] for a survey). Over the years, several background models have been proposed. These models can be broadly divided into pixel-wise and blockwise models. Thepixel-wise models relie on the separation of statistical model for each pixel and the model is learned entirely from each pixel history. In the block-wise models, the pixel model depends not only on that pixel but also on the nearby pixels. They consider spatial information an essential element to understand the scene structure. Spatial variation information, such as gradient feature, helps improvethe realiability of structure change detection. The pixel model also depends on its neighbors, taking advantage of the correlation existing between neighbouring pixels. Stauffer [21] proposed a parametrically approach in which each color pixel is represented as a GMM. The parameters are updated using an online Kmeans. However, in some scenarios, parametric models don’t accurately explain thephysical processes, i.e. can’t model the nonparametric nature of complex, time varying, non-static backgrounds. One needs to employ nonparametric estimation techniques that don’t make any assumptions about the pdf, except the mild assumption that pdf are smooth functions, and can represent arbitrary pdfs given sufficient data. Elgammal [8] proposed the KDE for modeling the pixel density, from its past...