Short Introduction to Particle Filtering by Arthur Pece [ follows my Introduction to Kalman filtering ]

Unscented Kalman filter If the dynamics or observation models are nonlinear, linearization is an option (extended Kalman filter) A better option is to take samples in state space and propagate all samples through the nonlinear equations In the unscented Kalman filter, regular samples are taken d standard deviations apart, in all directions of state space

Unscented sampling

Unscented transformation The n unscented samples are propagated through the dynamical/observation equations to obtain n new samples The mean and covariance of these new samples are taken as the mean and covariance of the new state/observation

Unscented Kalman equations Prediction : x i (t) = Dx i (t-1) + C u(t-1) x(t) =  x i (t) / n A(t) =  x i - x)  x i - x) T /n + N Update: y i = Fx i ; y =  y i / n innovation: v = y - y innov. cov: W =  y i - y)  y i - y) T / n + R Kalman gain, posterior mean, posterior cov: as in standard Kalman filter

Unscented Kalman vs. particle The unscented Kalman filter still uses a Gaussian approximation, which is not always satisfactory In particle filtering, no approximation is used at any stage: densities are always approximated by a set of finite samples

Particle filtering The posterior pdf is approximated by a finite set of particles Each sample (particle) consists of a state, a weight, and possibly other information (a covariance for instance) Each particle is propagated through the dynamical equations with noise (from a random number generator) Each particle has its own predicted observation

ConDensation filter 3 steps: prediction/sampling, observation/weighting, re-sampling Sampling: for each particle, use the dynamical equations to predict the current state from the previous state, then add noise Weighting: set the weight of each particle proportional to the likelihood of its state and normalize the weights to unit sum ---> The weighted density of particles is equal to the product of prior density (from the sampling density) and likelihood (from the weights)

ConDensation sampling and weighting

ConDensation resampling It can be proven that, by iterating the sampling and weighting procedure alone, the weights almost surely diverge until one particle has unit weight and all other particles have zero weight The solution is re-sampling: draw new particles from each of the old particles, with probability equal to the weight of the old particle

ConDensation sampling, weighting, and re-sampling

General particle filter 3 steps: sampling, weighting, re-sampling Sampling: for each particle, sample the new state from a proposal density conditional on the previous state and the observation The main difference between particle filters is the proposal density Weighting: set the particle weight proportional to the ratio of proposal density and posterior pdf, and normalize weights Re-sampling: as before

Kalman particle filter 3 steps: sampling, weighting, re-sampling Main difference from ConDensation: - in ConDensation, the proposal density is the prior pdf - in the KPF, the proposal density is the posterior pdf obtained from a Kalman model

Kalman particle A Kalman particle behaves as a Kalman filter in the sampling stage, except for random noise added at the end The relationship between a Kalman filter and a KPF is the same as that between a Gaussian and a mixture of Gaussians

KPF sampling and weighting

Summary Two approaches to tracking: Kalman filter and particle filter The unscented Kalman filter is an extension of the Kalman filter The Kalman particle filter uses the Kalman filter as a sub-routine of a particle filter