The extended Kalman filter (EKF) is probably the most widely used estimation algorithm for nonlinear systems. However, more than 35 years of experience in the estimation community has shown that is difficult to implement, difficult to tune, and only reliable for systems that are almost linear on the time scale of the updates. Many of these difficulties arise from its use of linearization. To overcome this limitation, the unscented transformation (UT) was developed as a method to propagate mean and covariance information through nonlinear transformations. It is more accurate, easier to implement, and uses the same order of calculations as linearization. This paper reviews the motivation, development, use, and implications of the UT.

/pdf/unscented-filtering-and-nonlinear-estimation-44xbx5prtw.pdf

Unscented filtering and nonlinear estimation

The problem of tracking curves in dense visual clutter is challenging. Kalman filtering is inadequate because it is based on Gaussian densities which, being unimo dal, 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.

/pdf/c-ondensation-conditional-density-propagation-forvisual-2zlkn4lnac.pdf

C ONDENSATION —Conditional Density Propagation forVisual Tracking

A new approach toward target representation and localization, the central component in visual tracking of nonrigid 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 a few of the potential applications: exploitation of background information, Kalman tracking using motion models, and face tracking.

/pdf/kernel-based-object-tracking-2pj4gwhvr7.pdf

Kernel-based object tracking

The simultaneous localization and map building (SLAM) problem asks if it is possible for an autonomous vehicle to start in an unknown location in an unknown environment and then to incrementally build a map of this environment while simultaneously using this map to compute absolute vehicle location. Starting from estimation-theoretic foundations of this problem, the paper proves that a solution to the SLAM problem is indeed possible. The underlying structure of the SLAM problem is first elucidated. A proof that the estimated map converges monotonically to a relative map with zero uncertainty is then developed. It is then shown that the absolute accuracy of the map and the vehicle location reach a lower bound defined only by the initial vehicle uncertainty. Together, these results show that it is possible for an autonomous vehicle to start in an unknown location in an unknown environment and, using relative observations only, incrementally build a perfect map of the world and to compute simultaneously a bounded estimate of vehicle location. The paper also describes a substantial implementation of the SLAM algorithm on a vehicle operating in an outdoor environment using millimeter-wave radar to provide relative map observations. This implementation is used to demonstrate how some key issues such as map management and data association can be handled in a practical environment. The results obtained are cross-compared with absolute locations of the map landmarks obtained by surveying. In conclusion, the paper discusses a number of key issues raised by the solution to the SLAM problem including suboptimal map-building algorithms and map management.

/pdf/a-solution-to-the-simultaneous-localization-and-map-building-2pp3i7uhdj.pdf

A solution to the simultaneous localization and map building (SLAM) problem

Dynamic Bayesian Networks: Representation, Inference and Learning by Kevin Patrick Murphy Doctor of Philosophy in Computer Science University of California, Berkeley Professor Stuart Russell, Chair 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 ), 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.

/pdf/dynamic-bayesian-networks-representation-inference-and-yy4fhfxdcj.pdf

Dynamic bayesian networks: representation, inference and learning

Tracking and data association

The problem of associating data with targets in a cluttered multi-target environment is discussed and applied to passive sonar tracking. The probabilistic data association (PDA) method, which is based on computing the posterior probability of each candidate measurement found in a validation gate, assumes that only one real target is present and all other measurements are Poisson-distributed clutter. In this paper, a new theoretical result is presented: the joint probabilistic data association (JPDA) algorithm, in which joint posterior association probabilities are computed for multiple targets (or multiple discrete interfering sources) in Poisson clutter. The algorithm is applied to a passive sonar tracking problem with multiple sensors and targets, in which a target is not fully observable from a single sensor. Targets are modeled with four geographic states, two or more acoustic states, and realistic (i.e., low) probabilities of detection at each sample time. A simulation result is presented for two heavily interfering targets illustrating the dramatic tracking improvements obtained by estimating the targets' states using joint association probabilities.

Sonar tracking of multiple targets using joint probabilistic data association

The Probabilistic Data Association (PDA) method, which is based on computing the posterior probability of each candidate measurement found in a validation gate, assumes that only one real target is present and all other measurements are Poisson-distributed clutter. In this paper, some new theoretical results are presented on the Joint Probabilistic Data Association (JPDA) algorithm, in which joint posterior probabilities are computed for multiple targets in Poisson clutter. The algorithm is applied to a passive sonar tracking problem wlth multiple sensors and targets, in which a target is not fully observable from a single sensor. Targets are modeled with four geographic states, two or more acoustic states, and realistic (i.e. low) probabilities of detection at each sample time. Simulation results are presented for two heavily interfering targets; these illustrate the dramatic improvements obtained by computing joint probabilities.

Multi-target tracking using joint probabilistic data association

The adaptive optimization of detection thresholds for tracking in clutter is investigated for the probabilistic data association (PDA) filter. Earlier work on this problem by T.E. Fortmann et al. (1985) involved an approximate steady-state analysis of the state error covariance and is only suitable for time-invariant systems. Furthermore, the method requires numerous assumptions and approximations about the error covariance update equation, and uses a cumbersome graphical optimization algorithm. In this work we propose two adaptive schemes for threshold optimization, namely prior and posterior optimization algorithms which minimize the mean-square state estimation error over detection thresholds which depend on data up to the previous and current time-step, respectively. These algorithm are suitable for real-time implementation in time-varying systems. Some simulation results are presented.

Adaptive detection threshold optimization for tracking in clutter

Adaptive detection threshold optimization for improved downstream tracker performance with a Probabilistic Data Association Filter (PDAF) is investigated. Prior and posterior optimization algorithms which minimize the mean-square estimation error over the signal processor's detection thresholds and which depend on observations up to the previous and current iteration, respectively, are given. These algorithms are suitable for real-time implementation. Simulation results are presented.

Thomas Fortmann

Papers

Tracking and data association

Sonar tracking of multiple targets using joint probabilistic data association

Multi-target tracking using joint probabilistic data association

Adaptive detection threshold optimization for tracking in clutter

Adaptive Detection Threshold Optimization for Tracking in Clutter