Abstract. This chapter provides a survey of the main methods in non-uniform random variate generation, and highlights recent research on the subject. Classical paradigms such as inversion, rejection, guide tables, and transformations are reviewed. We provide information on the expected time complexity of various algorithms, before addressing modern topics such as indirectly specified distributions, random processes, and Markov chain methods.

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.

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 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...

A framework for positioning, navigation and tracking problems using particle filters (sequential Monte Carlo methods) is developed. It consists of a class of motion models and a general non-linear measurement equation in position. A general algorithm is presented, which is parsimonious with the particle dimension. It is based on marginalization, enabling a Kalman filter to estimate all position derivatives, and the particle filter becomes low-dimensional. This is of utmost importance for highperformance real-time applications. Automotive and airborne applications illustrate numerically the advantage over classical Kalman filter based algorithms. Here the use of non-linear models and non-Gaussian noise is the main explanation for the improvement in accuracy. More specifically, we describe how the technique of map matching is used to match an aircraft's elevation profile to a digital elevation map, and a car's horizontal driven path to a street map. In both cases, real-time implementations are available, and tests have shown that the accuracy in both cases is comparable to satellite navigation (as GPS), but with higher integrity. Based on simulations, we also argue how the particle filter can be used for positioning based on cellular phone measurements, for integrated navigation in aircraft, and for target tracking in aircraft and cars. Finally, the particle filter enables a promising solution to the combined task of navigation and tracking, with possible application to airborne hunting and collision avoidance systems in cars.

Abstract — Recently Rao-Blackwellized particle filters have been introduced as effective means to solve the simultaneous localization and mapping (SLAM) problem. This approach uses a particle filter in which each particle carries an individual map of the environment. Accordingly, a key question is how to reduce the number of particles. In this paper we present adaptive techniques to reduce the number of particles in a Rao-Blackwellized particle filter for learning grid maps. We propose an approach to compute an accurate proposal distribution taking into account not only the movement of the robot but also the most recent observation. This drastically decrease the uncertainty about the robot’s pose in the prediction step of the filter. Furthermore, we present an approach to selectively carry out re-sampling operations which seriously reduces the problem of particle depletion. Experimental results carried out with mobile robots in large-scale indoor as well as in outdoor environments illustrate the advantages of our methods over previous approaches. I.

We present a method for recovering 3D human body motion from monocular video sequences based on a robust image matching metric, incorporation of joint limits and non-self-intersection constraints, and a new sample-and-refine search strategy guided by rescaled cost-function covariances. Monocular 3D body tracking is challenging: besides the difficulty of matching an imperfect, highly flexible, self-occluding model to cluttered image features, realistic body models have at least 30 joint parameters subject to highly nonlinear physical constraints, and at least a third of these degrees of freedom are nearly unobservable in any given monocular image. For image matching we use a carefully designed robust cost metric combining robust optical flow, edge energy, and motion boundaries. The nonlinearities and matching ambiguities make the parameter-space cost surface multi-modal, ill-conditioned and highly nonlinear, so searching it is difficult. We discuss the limitations of CONDENSATION-like samplers, and describe a novel hybrid search algorithm that combines inflated-covariance-scaled sampling and robust continuous optimization subject to physical constraints and model priors. Our experiments on challenging monocular sequences show that robust cost modeling, joint and selfintersection constraints, and informed sampling are all essential for reliable monocular 3D motion estimation.

Abstract—The classical particle filter deals with the estimation of one state process conditioned on a realization of one observation process. We extend it here to the estimation of multiple state processes given realizations of several kinds of observation processes. The new algorithm is used to track with success multiple targets in a bearings-only context, whereas a JPDAF diverges. Making use of the ability of the particle filter to mix different types of observations, we then investigate how to join passive and active measurements for improved tracking. Index Terms—Bayesian estimation, bearings-only tracking, Gibbs sampler, multiple receivers, multiple targets tracking,

We address the problem of multitarget tracking encountered in many situations in signal or image processing. We consider stochastic dynamic systems detected by observation processes. The difficulty lies on the fact that the estimation of the states requires the assignment of the observations to the multiple targets. We propose an extension of the classical particle filter where the stochastic vector of assignment is estimated by a Gibbs sampler. This algorithm is used to estimate the trajectories of multiple targets from their noisy bearings, thus showing its ability to solve the data association problem. Moreover this algorithm is easily extended to multireceiver observations where the receivers can produce measurements of various nature with different frequencies.

Abstract — Recently, Rao-Blackwellized particle filters have been introduced as an effective means to solve the simultaneous localization and mapping problem. This approach uses a particle filter in which each particle carries an individual map of the environment. Accordingly, a key question is how to reduce the number of particles. In this paper, we present adaptive techniques for reducing this number in a Rao-Blackwellized particle filter for learning grid maps. We propose an approach to compute an accurate proposal distribution taking into account not only the movement of the robot but also the most recent observation. This drastically decreases the uncertainty about the robot’s pose in the prediction step of the filter. Furthermore, we present an approach to selectively carry out resampling operations which seriously reduces the problem of particle depletion. Experimental results carried out with real mobile robots in large-scale indoor as well as in outdoor environments illustrate the advantages of our methods over previous approaches. Index Terms — SLAM, Rao-Blackwellized particle filter, adaptive resampling, motion-model, improved proposal