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Computer Vision
Prof. Bill Freeman
Particle Filter Tracking – Particle filtering
Readings: F&P Extra Chapter: “Particle Filtering”
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Schedule
• Tuesday, May 3: – Particle filters, tracking humans, Exam 2 out
• Thursday, May 5: – Tracking humans, and how to write conference papers
& give talks, Exam 2 due
• Tuesday, May 10:– Motion microscopy, separating shading and paint (“fun
things my group is doing”)
• Thursday, May 12: – 5-10 min. student project presentations, projects due.
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1D Kalman filter
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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?
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Iteration 0 1 2
∞== −−00 0 σx
+
−
+
−
i
i
i
i
x
x
σ
σ
0
∞
1,0,1,1 ====ii mdii md σσ
Kalman filter model
Initial conditions
0y
1
0y
1
210 yy +
21
210 yy +
21
3210 yyy ++
31
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What happens if the x dynamics are given a non-zero variance?
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Iteration 0 1 2
∞== −−00 0 σx
+
−
+
−
i
i
i
i
x
x
σ
σ
0
∞
0y
1,1,1,1 ====ii mdii md σσ
Kalman filter model
Initial conditions
1
0y
2
32 10 yy +
32
35
852 210 yyy ++
85
32 10 yy +
(KF) Distribution propagation
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prediction from previous time frame
Noise added to
that prediction
Make new measurement at next time frame
[Isard 1998]
Distribution propagation
9[Isard 1998]
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Representing non-linear Distributions
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Representing non-linear Distributions
Unimodal parametric models fail to capture real-world densities…
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Representing non-linear Distributions
Mixture models are appealing, but very hard to propagate analytically!
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Representing Distributions using Weighted Samples
Rather than a parametric form, use a set of samples to represent a density:
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Representing Distributions using Weighted Samples
Rather than a parametric form, use a set of samples to represent a density:
Representing distributions using weighted samples, another picture
15[Isard 1998]
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Sampled representation of a probability distribution
You can also think of this as a sum of dirac delta functions, each of weight w:
∑ −=i
iif uxwxp )()( δ
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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 (!):
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Marginalizing a sampled density
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Sampled Bayes Rule
Transforming a Sampled Representation of a Prior into a Sampled Representation of a Posterior:
posterior likelihood, prior
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Sampled Bayes rule
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Sampled Prediction= ?
~=Drop elements to marginalize to get
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Sampled Correction (Bayes rule)
Prior posteriorReweight 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:
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Naïve PF Tracking
• Start with samples from something simple (Gaussian)
• Repeat
But doesn’t work that well because of sample impoverishment…
– Predict
– Correct
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Sample impoverishment
Test with linear case:
kf: xpf: o
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Sample impoverishment
10 of the 100 particles, along with the true Kalmanfilter track, with variance:
time
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Resample the prior
In a sampled density representation, the frequency of samples can be traded off against weight:
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.
s.t.
…
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Resampling concentrates samples
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A practical particle filter with resampling
A variant (predict, then resample, then correct)
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A variant (animation)
[Isard 1998]
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Applications
Tracking– hands– bodies– leaves
Contour tracking
33[Isard 1998]
Head tracking
34[Isard 1998]
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Leaf tracking
[Isard 1998]
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Hand tracking
[Isard 1998]
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Mixed state tracking
[Isard 1998]
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Outline
• Sampling densities• Particle filtering
[Figures from F&P except as noted]
