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Particle Filters. Outline. Introduction to particle filters Recursive Bayesian estimation Bayesian Importance sampling Sequential Importance sampling ( SIS ) Sampling Importance resampling ( SIR ) Improvements to SIR On-line Markov chain Monte Carlo Basic Particle Filter algorithm
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Outline • Introduction to particle filters • Recursive Bayesian estimation • Bayesian Importance sampling • Sequential Importance sampling (SIS) • Sampling Importance resampling (SIR) • Improvements to SIR • On-line Markov chain Monte Carlo • Basic Particle Filter algorithm • Example for robot localization • Conclusions
Key Idea of Particle Filters • Idea = we try to have more samples where we expect to have the solution
Motion Model Reminder • Density of samples represents the expected probability of robot location
Global Localization of Robot with Sonarhttp://www.cs.washington.edu/ai/Mobile_Robotics/mcl/animations/global-floor.gif • This is the lost robot problem
Particles are used for probability density function Approximation
Function Approximation • Particle sets can be used to approximate functions • The more particles fall into an interval, the higher the probability of that interval • How to draw samples from a function/distribution?
Importance Sampling Principle weight • w = f / g • f is often calledtarget • g is often calledproposal • Pre-condition:f(x)>0 g(x)>0
Importance sampling: another example of calculating weight samples • How to calculate formally the f/g value?
History of Monte Carlo Idea and especially Particle Filters • First attempts – simulations of growing polymers • M. N. Rosenbluth and A.W. Rosenbluth, “Monte Carlo calculation of the average extension of molecular chains,” Journal of Chemical Physics, vol. 23, no. 2, pp. 356–359, 1956. • First application in signal processing - 1993 • N. J. Gordon, D. J. Salmond, and A. F. M. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” IEE Proceedings-F, vol. 140, no. 2, pp. 107–113, 1993. • Books • A. Doucet, N. de Freitas, and N. Gordon, Eds., Sequential Monte Carlo Methods in Practice, Springer, 2001. • B. Ristic, S. Arulampalam, N. Gordon, Beyond the Kalman Filter: Particle Filters for Tracking Applications, Artech House Publishers, 2004. • Tutorials • M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-gaussian Bayesian tracking,” IEEE Transactions on Signal Processing, vol. 50, no. 2, pp. 174–188, 2002.
What is the problem that we want to solve? • The problem is tracking the state of a system as it evolves over time • Sequentially arriving (noisy or ambiguous) observations • We want to know: Best possible estimate of the hidden variables
Solution: Sequential Update • Storing and processing all incoming measurements is inconvenient and may be impossible • Recursive filtering: • Predict next state pdf from current estimate • Update the prediction using sequentially arriving new measurements • Optimal Bayesian solution: • recursively calculating exact posterior density These lead to various particle filters
Particle Filters • Sequential Monte Carlo methods for on-line learning within a Bayesian framework. • Known as • Particle filters • Sequential sampling-importance resampling (SIR) • Bootstrap filters • Condensation trackers • Interacting particle approximations • Survival of the fittest
Approaches to Particle Filters METAPHORS
Particle filters • Sequential and Monte Carlo properties • Representing belief by sets of samples or particles • are nonnegative weights called importance factors • Updating procedure is sequential importance sampling with re-sampling
Tracking in 1D: the blue trajectory is the target.The best of10 particles is in red.
Short, more formal, Introduction to Particle Filters and Monte CarloLocalization
Particle filtering ideas • Recursive Bayesian filter by Monte Carlo sampling • The idea: represent the posterior density by a set of random particles with associated weights. • Compute estimates based on these samples and weights • Posterior density • Sample space
Particle filtering ideas • Particle filters are based onrecursive generation of random measures that approximatethe distributions of the unknowns. • Random measures: particles and importance weights. • As new observationsbecome available, the particles and the weights are propagated by exploiting Bayes theorem. • Posterior density • Sample space
Recall “law of total probability” and “Bayes’ rule” Mathematical tools needed for Particle Filters
Recursive Bayesian estimation (I) • Recursive filter: • System model: • Measurement model: • Information available:
Recursive Bayesian estimation (II) • Seek: • i = 0: filtering. • i > 0: prediction. • i<0: smoothing. • Prediction: • since:
Recursive Bayesian estimation (III) • Update: • where: • since:
System state dynamics • Observation dynamics • We are interested in: Belief or posterior density Bayes Filters (second pass) • Estimating system state from noisy observations
From above, constructing two steps of Bayes Filters • Predict: • Update:
Assumptions: Markov Process • Predict: • Update:
Bayes Filter • How to use it? What else to know? • Motion Model • Perceptual Model • Start from:
Example 1: theoretical PDF
Step 0: initialization • Step 1: updating • Example 1: theoretical PDF
Step 0: initialization • Each particle has the same weight • Step 1: updating weights. Weights are proportional to p(z|x) Example 2: Particle Filter
Step 3: updating • Step 4: predicting • Step 2: predicting • Example 1 (continue) • 1
Step 3: updating weights. Weights are proportional to p(z|x) • Step 4: predicting. • Predict the new locations of particles. • Step 2: predicting. • Predict the new locations of particles. Example 2: Particle Filter • Particles are more concentrated in the region where the person is more likely to be
Compare Particle Filter with Bayes Filter with Known Distribution • Updating • Example 1 • Example 2 • Predicting • Example 1 • Example 2
Classical approximations • Analytical methods: • Extended Kalman filter, • Gaussian sums… (Alspach et al. 1971) • Perform poorly in numerous cases of interest • Numerical methods: • point masses approximations, • splines. (Bucy 1971, de Figueiro 1974…) • Very complex to implement, not flexible.
Mobile Robot Localization • Each particle is a potential pose of the robot • Proposal distribution is the motion model of the robot (prediction step) • The observation model is used to compute the importance weight (correction step)
Monte Carlo Localization • Each particle is a potential pose of the robot • Proposal distribution is the motion model of the robot (prediction step) • The observation model is used to compute the importance weight (correction step)