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

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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

• f all the observations seen so far?

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Iteration 0 1 2

∞ = =

− −

σ x

+ − + − i i i i

x x σ σ

∞

1 , , 1 , 1 = = = =

i i

m d i i

m d σ σ

Kalman filter model Initial conditions

y

1

y

1

2

1

y y +

2 1

2

1

y y +

2 1

3

2 1

y y y + +

3 1

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What happens if the x dynamics are given a non-zero variance?

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Iteration 0 1 2

∞ = =

− −

σ x

+ − + − i i i i

x x σ σ

∞

y 1 , 1 , 1 , 1 = = = =

i i

m d i i

m d σ σ

Kalman filter model Initial conditions 1

y

2

3 2

1

y y +

3 2 3 5

8 5 2

2 1

y y y + +

8 5

3 2

1

y y +

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(KF) Distribution propagation

[Isard 1998]

prediction from previous time frame Noise added to that prediction Make new measurement at next time frame

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Distribution propagation

[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:

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[Isard 1998]

Representing distributions using weighted samples, another picture

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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 i i f

u x w x p ) ( ) ( 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 posterior Reweight every sample with the likelihood of the

• bservations, 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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Test with linear case:

Sample impoverishment

kf: x pf: o

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10 of the 100 particles, along with the true Kalman filter track, with variance:

Sample impoverishment

time

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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

• riginal set of samples, using the weights as the

probability of drawing a sample.

Resample the prior

s.t. …

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Resampling concentrates samples

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A practical particle filter with resampling

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A variant (predict, then resample, then correct)

[Isard 1998]

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A variant (animation)

[Isard 1998]

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Tracking

– hands – bodies – leaves

Applications

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Contour tracking

[Isard 1998]

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Head tracking

[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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• Sampling densities
• Particle filtering

[Figures from F&P except as noted]

Outline