1 Kalman filter model (KF) Distribution propagation = = = = - PDF document

6.869 Schedule Computer Vision • Tuesday, May 3: – Particle filters, tracking humans, Exam 2 out Prof. Bill Freeman • Thursday, May 5: – Tracking humans, and how to write conference papers Particle Filter Tracking & give talks, Exam 2 due – Particle filtering • Tuesday, May 10: Readings: F&P Extra Chapter: “Particle Filtering” – Motion microscopy, separating shading and paint (“fun things my group is doing”) • Thursday, May 12: – 5-10 min. student project presentations, projects due. 1 2 1D Kalman filter Kalman filter for computing an on-line average • What Kalman filter parameters and initial conditions should we pick so that the optimal estimate for x at each iteration is just the average of all the observations seen so far? 3 4 Kalman filter model = = σ = σ = d 1 , m 1 , 0 , 1 i i d m i i Initial conditions − = σ − = ∞ x 0 0 0 What happens if the x dynamics are given a non-zero variance? Iteration 0 1 2 y + − y y x 0 0 1 0 i 2 + y + + + y y y y y x 0 1 0 1 2 0 i 2 3 ∞ σ − 1 1 i 2 1 σ + 1 1 i 2 3 5 6 1

Kalman filter model (KF) Distribution propagation = = σ = σ = d 1 , m 1 , 1 , 1 i i d m i i prediction from previous time frame Initial conditions − = σ − = ∞ x 0 0 0 Iteration 0 1 2 y + 2 y − Noise y x 0 0 1 0 added to i 3 y + that + + + y 2 y y 2 y 5 y x 0 1 0 1 2 0 prediction i 3 8 σ − ∞ 5 2 i 3 + σ 2 5 1 Make new measurement at next time frame i 3 8 7 8 [Isard 1998] Distribution propagation Representing non-linear Distributions 9 10 [Isard 1998] Representing non-linear Distributions Representing non-linear Distributions Unimodal parametric models fail to capture real- Mixture models are appealing, but very hard to world densities… propagate analytically! 11 12 2

Representing Distributions using Representing Distributions using Weighted Samples Weighted Samples Rather than a parametric form, use a set of samples Rather than a parametric form, use a set of samples to represent a density: to represent a density: 13 14 Sampled representation of a Representing distributions using weighted samples, another picture probability distribution You can also think of this as a sum of dirac delta functions, each of weight w: ∑ = δ − i i p ( x ) w ( x u ) 15 16 f [Isard 1998] i Marginalizing a sampled density Marginalizing a sampled density If we have a sampled representation of a joint density and we wish to marginalize over one variable: we can simply ignore the corresponding components of the samples (!): 17 18 3

Sampled Bayes Rule Sampled Bayes rule Transforming a Sampled Representation of a Prior into a Sampled Representation of a Posterior: posterior likelihood, prior 19 20 Sampled Prediction Sampled Correction (Bayes rule) = ? Prior � posterior Reweight every sample with the likelihood of the observations, given that sample: yielding a set of samples describing the probability distribution after the correction (update) step: Drop elements to marginalize to get ~= 21 22 Naïve PF Tracking Sample impoverishment • Start with samples from something simple Test with linear case: (Gaussian) • Repeat – Correct kf: x pf: o – Predict But doesn’t work that well because of sample 23 24 impoverishment… 4

Sample impoverishment Resample the prior 10 of the 100 particles, along with the true Kalman In a sampled density representation, the frequency of filter track, with variance: samples can be traded off against weight: s.t. … These new samples are a representation of the same density. I.e., make N draws with replacement from the original set of samples, using the weights as the probability of drawing a sample. 25 26 time A practical particle filter with resampling Resampling concentrates samples 27 28 A variant (predict, then resample, then correct) 29 30 [Isard 1998] 5

A variant (animation) Applications Tracking – hands – bodies – leaves 31 32 [Isard 1998] Contour tracking Head tracking 33 34 [Isard 1998] [Isard 1998] Leaf tracking Hand tracking 35 36 [Isard 1998] [Isard 1998] 6

Mixed state tracking Outline • Sampling densities • Particle filtering [Figures from F&P except as noted] 37 38 [Isard 1998] 7