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Particle representations of information and uncertainty

The particle filter (PF) has revolutionized the theory and applications in nonlinear filtering. It introduced a new paradigm for representing information using samples in the state space, quite different from the classical Gaussian distributions provided by Kalman filters. We have previously contributed to both the theory and practice of PF, and one of the first reported real-time applications of the PF was for car navigation without GPS.

We are now facing a second wave of development, where particle representations are instrumental, but where classical methods such as KF and system identification methods are invoked. CADICS aims at being in the forefront in this development in several core areas.

1. Learning dynamical models using particle methods

The particle methods are allowing us to learn dynamical models we could previously not learn. This holds true both for maximum likelihood and Bayesian approaches. The recent combination of sequential Monte Carlo (SMC) methods and Markov chain Monte Carlo (MCMC) methods results in a very powerful family of inference algorithms, the so-called particle MCMC methods. Within CADICS we contributing to the further development of these methods enabling us to compute the full posterior distribution of all unknown states and parameters given the measurements.

2. Marginalized particle methods

The basic particle methods struggles when the state dimension increases. Marginalization is a key tool to mitigate the complexity and get real-time algorithms. The key idea in the Rao-Blackwellized (marginalized) particle methods is to solve the easy problems with simple algorithms, and leave the hard problems to the particle methods. This leads to a mixed representation of information with samples for a few dimensions, and continuous parametric distributions in the other dimensions. We have derived solutions to both the filtering and the smoothing problem.

One case occurs when the underlying state space model can be split into one conditional linear Gaussian part where the KF/KS applies, and one nonlinear part solved by the PF/PS.

We are investigating this and more intrinsic cases, where also the noise distributions are estimated in a Bayesian framework. For instance, the mean and covariance of the process and measurement noises, respectively, can be estimated analytically, when a PF is used to estimate the state.

2.1 Tracking migrating birds

Low weight light-loggers detecting the time for sunset and sunrise have been used to track migrating birds. We have used the particle filter to estimate the position and a Kalman filter to estimate the velocity. The method and field results are available here

3. Probability Hypothesis Density (PHD) filtering

Multiple target tracking is a research area that has been driven by military needs ever since the radar was invented in 1905. The application required tracks of each target, the operating distance made all targets look like point objects, and the hardware required detection of the received radio waves corresponding to one range measurement. The theory is still concerned with the implications of this underlying assumptions, and main focus is on Kalman filter single target algorithms, data association and handling clutter from false detections. However, the situation is today drastically changed, and tracking multiple objects now appears in many other disciplines, including computer vision and robotics.  The PHD filter appeared recently as a complete framework for multi-sensor multi-target tracking, where the target intensity is the the main output. The PHD recursion resemble the Bayesian optimal filter, and similar numerical implementations are used today, such as PF and filter banks.

CADICS contributes to both the theory and practice of PHD filtering, with both automotive and robotics applications, investigating vision, radar and laser scanners as sensors.
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