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Particle Filtering. Sensors and Uncertainty. Real world sensors are noisy and suffer from missing data (e.g., occlusions, GPS blackouts) Use sensor models to estimate ground truth, unobserved variables, make forecasts. X 0. X 1. X 2. X 3. Hidden Markov Model.
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Sensors and Uncertainty • Real world sensors are noisy and suffer from missing data (e.g., occlusions, GPS blackouts) • Use sensor models to estimate ground truth, unobserved variables, make forecasts
X0 X1 X2 X3 Hidden Markov Model • Use observations to get a better idea of where the robot is at time t Hidden state variables Observed variables z1 z2 z3 Predict – observe – predict – observe…
Last Class • Kalman Filtering and its extensions • Exact Bayesian inference for Gaussian state distributions, process noise, observation noise • What about more general distributions? • Key representational issue • How to represent and perform calculations on probability distributions?
Particle Filtering (aka Sequential Monte Carlo) • Represent distributions as a set of particles • Applicable to non-gaussian high-D distributions • Convenient implementations • Widely used in vision, robotics
Simultaneous Localization and Mapping (SLAM) • Mobile robots • Odometry • Locally accurate • Drifts significantly over time • Vision/ladar/sonar • Inaccurate locally • Global reference frame • Combine the two • State: (robot pose, map) • Observations: (sensor input)
General problem • xt ~ Bel(xt) (arbitrary p.d.f.) • xt+1 = f(xt,u,ep) • zt+1 = g(xt+1,eo) • ep ~ arbitrary p.d.f., eo ~ arbitrary p.d.f. Process noise Observation noise
Particle Representation • Bel(xt) = {(wk,xk), k=1,…,n} • wk are weights, xk are state hypotheses • Weights sum to 1 • Approximates the underlying distribution
Monte Carlo Integration • If P(x) ≈ Bel(x) = {(wk,xk), k=1,…,N} • EP[f(x)] = integral[ f(x)P(x)dx ] ≈ Skwkf(xk) • What might you want to compute? • Mean: set f(x) = x • Variance: f(x) = x2 (recover Var(x) = E[x2]-E[x]2) • P(y): set f(x) = P(y|x) • Because P(y) = integral[ P(y|x)P(x)dx ]
Recovering the Distribution • Kernel density estimation • P(x) = Sk wk K(x,xk) • K(x,xk) is the kernel function • Better approximation as # particles, kernel sharpness increases
Filtering Steps • Predict • Compute Bel’(xt+1): distribution of xt+1 using dynamics model alone • Update • Compute a representation of P(xt+1|zt+1) via likelihood weighting for each particle in Bel’(xt+1) • Resample to produce Bel(xt+1) for next step
Predict Step • Given input particles Bel(xt) • Distribution of xt+1=f(xt,ut,e) determined by sampling e from its distribution and then propagating individual particles • Gives Bel’(xt+1)
Update Step • Goal: compute a representation of P(xt+1 | zt+1) given Bel’(xt+1), zt+1 • P(xt+1 | zt+1) = a P(zt+1 | xt+1) P(xt+1) • P(xt+1) = Bel’(xt+1) (given) • Each state hypothesis xk Bel’(xt+1) is reweighted by P(zt+1 | xt+1) • Likelihood weighting: • wkwkP(zt+1|xt+1=xk) • Then renormalize to 1
Update Step • wk wk’ * P(zt+1 | xt+1=xk) • 1D example: • g(x,eo) = h(x) + eo • eo ~ N(m,s) • P(zt+1 | xt+1=xk) = C exp(- (h(x)-zt+1)2 / 2s2) • In general, distribution can be calibrated using experimental data
Resampling • Likelihood weighted particles may no longer represent the distribution efficiently • Importance resampling: sample new particles proportionally to weight
Sampling Importance Resampling (SIR) variant Predict Update Resample
Particle Filtering Issues • Variance • Std. dev. of a quantity (e.g., mean) computed as a function of the particle representation ~ 1/sqrt(N) • Loss of particle diversity • Resampling will likely drop particles with low likelihood • They may turn out to be useful hypotheses in the future
Other Resampling Variants • Selective resampling • Keep weights, only resample when # of “effective particles” < threshold • Stratified resampling • Reduce variance using quasi-random sampling • Optimization • Explicitly choose particles to minimize deviance from posterior • …
Storing more information with same # of particles • Unscented Particle Filter • Each particle represents a local gaussian, maintains a local covariance matrix • Combination of particle filter + Kalman filter • Rao-Blackwellized Particle Filter • State (x1,x2) • Particle contains hypothesis of x1, analytical distribution over x2 • Reduces variance
Recap • Bayesian mechanisms for state estimation are well understood • Representation challenge • Methods: • Kalman filters: highly efficient closed-form solution for Gaussian distributions • Particle filters: approximate filtering for high-D, non-Gaussian distributions • Implementation challenges for different domains (localization, mapping, SLAM, tracking)