Estimating the state of a system from noisy measurements is a problem which arises in a variety of scientific and industrial areas which include signal processing, communications, statistics and econometrics. Recursive filtering is one way to achieve this by incorporating noisy observations as they become available with prior knowledge of the system model. Bayesian methods provide a general framework for dynamic state estimation problems. The central idea behind this recursive Bayesian estimation is computing the probability density function of the state vector of the system conditioned on the measurements. However, the optimal solution to this problem is often intractable because it requires high-dimensional integration. Although we can use the Kalman lter in the case of a linear state space model with Gaussian noise, this method is not optimum for a non-linear and non-Gaussian system model. There are many new methods of filtering for the general case. The main emphasis of this thesis is on one such recently developed filter, the particle lter [2,3,6]. In this thesis, a detailed introduction to particle filters is provided as well as some guidelines for the efficient implementation of the particle lter. The application of particle lters to various communication channels like detection of symbols over the channels, capacity calculation of the channel are discussed.

This thesis tackles the visual tracking problem as a target contour estimation problem in the face of corrupted measurements. The major aim is to design robust recursive curve filters for accurate contour-based tracking. The state-space representation adopted comprises of a group component and a shape component describing the rigid motion and the non-rigid shape deformation respectively; filtering strategies on each component are then decoupled. The thesis considers two implicit curve descriptors, a classification probability field and the traditional signed distance function, and aims to develop an optimal probabilistic contour observer and locally optimal curve filters. For the former, introducing a novel probabilistic shape description simplifies the filtering problem on the infinite-dimensional space of closed curves to a series of point-wise filtering tasks. The definition and justification of a novel update model suited to the shape space, the derivation of the filtering equations and the relation to Kalman filtering are studied. In addition to the temporal consistency provided by the filtering, extensions involving distributed filtering methods are considered in order to maintain spatial consistency. For the latter, locally optimal closed curve filtering strategies involving curve velocities are explored. The introduction of a local, linear description for planar curve variation and curve uncertainty enables the derivation of a mechanism for estimating the optimal gain associated to the curve filtering process, given quantitative uncertainty levels. Experiments on synthetic and real sequences of images validate the filtering designs.