The problem of tracking curves in dense visual clutter is challenging. Kalman filtering is inadequate because it is based on Gaussian densities which, being unimodal, cannot represent simultaneous alternative hypotheses. The Condensation algorithm uses "factored sampling", previously applied to the interpretation of static images, in which the probability distribution of possible interpretations is represented by a randomly generated set. Condensation uses learned dynamical models, together with visual observations, to propagate the random set over time. The result is highly robust tracking of agile motion. Notwithstanding the use of stochastic methods, the algorithm runs in near real-time. Contents 1 Tracking curves in clutter 2 2 Discrete-time propagation of state density 3 3 Factored sampling 6 4 The Condensation algorithm 8 5 Stochastic dynamical models for curve motion 10 6 Observation model 13 7 Applying the Condensation algorithm to video-streams 17 8 Conclusions 26 A Non-line...

Increasingly, for many application areas, it is becoming important to include elements of nonlinearity and non-Gaussianity in order to model accurately the underlying dynamics of a physical system. Moreover, it is typically crucial to process data on-line as it arrives, both from the point of view of storage costs as well as for rapid adaptation to changing signal characteristics. In this paper, we review both optimal and suboptimal Bayesian algorithms for nonlinear/non-Gaussian tracking problems, with a focus on particle filters. Particle filters are sequential Monte Carlo methods based on point mass (or “particle”) representations of probability densities, which can be applied to any state-space model and which generalize the traditional Kalman filtering methods. Several variants of the particle filter such as SIR, ASIR, and RPF are introduced within a generic framework of the sequential importance sampling (SIS) algorithm. These are discussed and compared with the standard EKF through an illustrative example.

Mobile robot localization is the problem of determining a robot's pose from sensor data. This article presents a family of probabilistic localization algorithms known as Monte Carlo Localization (MCL). MCL algorithms represent a robot's belief by a set of weighted hypotheses (samples), which approximate the posterior under a common Bayesian formulation of the localization problem. Building on the basic MCL algorithm, this article develops a more robust algorithm called MixtureMCL, which integrates two complimentary ways of generating samples in the estimation. To apply this algorithm to mobile robots equipped with range finders, a kernel density tree is learned that permits fast sampling. Systematic empirical results illustrate the robustness and computational efficiency of the approach.

In this article, we present an overview of methods for sequential simulation from posterior distributions. These methods are of particular interest in Bayesian filtering for discrete time dynamic models that are typically nonlinear and non-Gaussian. A general importance sampling framework is developed that unifies many of the methods which have been proposed over the last few decades in several different scientific disciplines. Novel extensions to the existing methods are also proposed. We show in particular how to incorporate local linearisation methods similar to those which have previously been employed in the determin-istic filtering literature; these lead to very effective importance distributions. Furthermore we describe a method which uses Rao-Blackwellisation in order to take advantage of the analytic structure present in some important classes of state-space models. In a final section we develop algorithms for prediction, smoothing and evaluation of the likelihood in dynamic models.

Modelling sequential data is important in many areas of science and engineering. Hidden Markov models (HMMs) and Kalman filter models (KFMs) are popular for this because they are simple and flexible. For example, HMMs have been used for speech recognition and bio-sequence analysis, and KFMs have been used for problems ranging from tracking planes and missiles to predicting the economy. However, HMMs and KFMs are limited in their “expressive power”. Dynamic Bayesian Networks (DBNs) generalize HMMs by allowing the state space to be represented in factored form, instead of as a single discrete random variable. DBNs generalize KFMs by allowing arbitrary probability distributions, not just (unimodal) linear-Gaussian. In this thesis, I will discuss how to represent many different kinds of models as DBNs, how to perform exact and approximate inference in DBNs, and how to learn DBN models from sequential data. In particular, the main novel technical contributions of this thesis are as follows: a way of representing Hierarchical HMMs as DBNs, which enables inference to be done in O(T) time instead of O(T 3), where T is the length of the sequence; an exact smoothing algorithm that takes O(log T) space instead of O(T); a simple way of using the junction tree algorithm for online inference in DBNs; new complexity bounds on exact online inference in DBNs; a new deterministic approximate inference algorithm called factored frontier; an analysis of the relationship between the BK algorithm and loopy belief propagation; a way of applying Rao-Blackwellised particle filtering to DBNs in general, and the SLAM (simultaneous localization and mapping) problem in particular; a way of extending the structural EM algorithm to DBNs; and a variety of different applications of DBNs. However, perhaps the main value of the thesis is its catholic presentation of the field of sequential data modelling.

This paper analyses the recently suggested particle approach to filtering time series. We suggest that the algorithm is not robust to outliers for two reasons: the design of the simulators and the use of the discrete support to represent the sequentially updating prior distribution. Both problems are tackled in this paper. We believe we have largely solved the first problem and have reduced the order of magnitude of the second. In addition we introduce the idea of stratification into the particle filter which allows us to perform on-line Bayesian calculations about the parameters which index the models and maximum likelihood estimation. The new methods are illustrated by using a stochastic volatility model and a time series model of angles. Some key words: Filtering, Markov chain Monte Carlo, Particle filter, Simulation, SIR, State space. 1 1

A new approach toward target representation and localization, the central component in visual tracking of non-rigid objects, is proposed. The feature histogram based target representations are regularized by spatial masking with an isotropic kernel. The masking induces spatially-smooth similarity functions suitable for gradient-based optimization, hence, the target localization problem can be formulated using the basin of attraction of the local maxima. We employ a metric derived from the Bhattacharyya coefficient as similarity measure, and use the mean shift procedure to perform the optimization. In the presented tracking examples the new method successfully coped with camera motion, partial occlusions, clutter, and target scale variations. Integration with motion filters and data association techniques is also discussed. We describe only few of the potential applications: exploitation of background information, Kalman tracking using motion models, and face tracking. Keywords: non-rigid object tracking; target localization and representation; spatially-smooth similarity function; Bhattacharyya coefficient; face tracking. 1

A general framework for using Monte Carlo methods in dynamic systems is provided and its wide applications indicated. Under this framework, several currently available techniques are studied and generalized to accommodate more complex features. All of these methods are partial combinations of three ingredients: importance sampling and resampling, rejection sampling, and Markov chain iterations. We deliver a guideline on how they should be used and under what circumstance each method is most suitable. Through the analysis of differences and connections, we consolidate these methods into a generic algorithm by combining desirable features. In addition, we propose a general use of Rao-Blackwellization to improve performances. Examples from econometrics and engineering are presented to demonstrate the importance of Rao-Blackwellization and to compare different Monte Carlo procedures. Keywords: Blind deconvolution; Bootstrap filter; Gibbs sampling; Hidden Markov model; Kalman filter; Markov...

The Kalman filter(KF) is one of the most widely used methods for tracking and estimation due to its simplicity, optimality, tractability and robustness. However, the application of the KF to nonlinear systems can be difficult. The most common approach is to use the Extended Kalman Filter (EKF) which simply linearises all nonlinear models so that the traditional linear Kalman filter can be applied. Although the EKF (in its many forms) is a widely used filtering strategy, over thirty years of experience with it has led to a general consensus within the tracking and control community that it is difficult to implement, difficult to tune, and only reliable for systems which are almost linear on the time scale of the update intervals. In this paper a new linear estimator is developed and demonstrated. Using the principle that a set of discretely sampled points can be used to parameterise mean and covariance, the estimator yields performance equivalent to the KF for linear systems yet general...

. A probabilistic method for tracking 3D articulated human gures in monocular image sequences is presented. Within a Bayesian framework, we de ne a generative model of image appearance, a robust likelihood function based on image graylevel dierences, and a prior probability distribution over pose and joint angles that models how humans move. The posterior probability distribution over model parameters is represented using a discrete set of samples and is propagated over time using particle ltering. The approach extends previous work on parameterized optical ow estimation to exploit a complex 3D articulated motion model. It also extends previous work on human motion tracking by including a perspective camera model, by modeling limb self occlusion, and by recovering 3D motion from a monocular sequence. The explicit posterior probability distribution represents ambiguities due to image matching, model singularities, and perspective projection. The method relies only on a...