Monte Carlo Hidden Markov Models - People | MIT...
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![Page 1: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/1.jpg)
Stanford CS223B Computer Vision, Winter 2007
Lecture 12 Tracking Motion
Professors Sebastian Thrun and Jana Košecká
CAs: Vaibhav Vaish and David Stavens
![Page 2: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/2.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Overview
The Tracking Problem Bayes Filters Particle Filters Kalman Filters Using Kalman Filters
![Page 3: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/3.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
The Tracking Problem
Can we estimate the position of the object? Can we estimate its velocity? Can we predict future positions?
Image 4Image 1 Image 2 Image 3
![Page 4: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/4.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
The Tracking Problem
Given Sequence of Images Find center of moving object Camera might be moving or stationary
We assume: We can find object in individual images.
The Problem: Track across multiple images.
Is a fundamental problem in computer vision
![Page 5: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/5.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Methods
Bayes Filter
Particle Filter
Uncented Kalman Filter
Kalman Filter
Extended Kalman Filter
![Page 6: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/6.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Further Reading…
![Page 7: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/7.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Example: Moving Object
![Page 8: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/8.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Kalman Filter Tracking
![Page 9: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/9.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Particle Filter Tracking
![Page 10: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/10.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Mixture of KF / PF (Unscented PF)
![Page 11: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/11.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Overview
The Tracking Problem Bayes Filters Particle Filters Kalman Filters Using Kalman Filters
![Page 12: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/12.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
SlowDown!
Example of Bayesian Inference
?
Environment priorp(staircase) = 0.1
Bayesian inferencep(staircase | image) p(image | staircasse) p(staircase) p(im | stair) p(stair) + p(im | no stair) p(no stair) = 0.7 • 0.1 / (0.7 • 0.1 + 0.2 • 0.9) = 0.28
Sensor modelp(image | staircase) = 0.7p(image | no staircase) = 0.2p(staircase)
= 0.28
Cost modelcost(fast walk | staircase) = $1,000cost(fast walk | no staircase) = $0cost(slow+sense) = $1
Decision TheoryE[cost(fast walk)] = $1,000 • 0.28 = $280E[cost(slow+sense)] = $1
=
![Page 13: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/13.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Bayes Filter Definition
Environment state xt
Measurement zt
Can we calculate p(xt | z1, z2, …, zt) ?
![Page 14: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/14.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Bayes Filters Illustrated
![Page 15: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/15.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Bayes Filters: Essential Steps
Belief: Bel(xt)
Measurement update: Bel(xt) ← Bel(xt) p(zt|xt)
Time update: Bel(xt+1) ← Bel(xt) ⊗ p(xt+1|ut,xt)
∝
![Page 16: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/16.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
x = statet = timez = observationu = actionη = constant
Bayes Filters
),|()( 00 tttt uzxpxBel =
),,,|(),,,,|( 011011 zzuxpzzuxzp ttttttt −−−−= η
),,,|()|( 011 zzuxpxzp ttttt −−= η
1011011 ),,|(),,,|()|( −−−−−∫= tttttttt dxzuxpzuxxpxzp η
1111 )(),|()|( −−−−∫= ttttttt dxxBeluxxpxzpη
),,,,|( 011 uzuzxp tttt −−=
Markov
Bayes
Markov
1021111 ),,|(),|()|( −−−−−−∫= ttttttttt dxuuzxpuxxpxzp η
![Page 17: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/17.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Bayes Filtersx = statet = timez = observationu = action
1111 )(),|()|()( −−−−∫= tttttttt dxxBeluxxpxzpxBel η
![Page 18: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/18.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Bayes Filters Illustrated
![Page 19: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/19.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Bayes Filters
Initial Estimate of State Iterate
– Receive measurement, update your belief (uncertainty shrinks)
– Predict, update your belief (uncertainty grows)
![Page 20: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/20.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Methods
Bayes Filter
Particle Filter
Uncented Kalman Filter
Kalman Filter
Extended Kalman Filter
![Page 21: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/21.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Overview
The Tracking Problem Bayes Filters Particle Filters Kalman Filters Using Kalman Filters
![Page 22: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/22.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Particle Filters: Basic Idea
)(xp
tx
)|()( ...1 tttt zxpXxp ≈∈ (equality for )∞↑n
set of n particles Xt
![Page 23: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/23.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Particle Filter Explained
![Page 24: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/24.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
11...11...111...1...1 ),|(),|()|(),|( −−−−−∫= tttttttttttt dxuzxpxuxpxzpuzxp η
Basic Particle Filter Algorithm
),|()( ...1...1 ttttt uzxpXxp ≈∈
Initialization:
X0 ← n particles x0 [i] ~ p(x0)
particleFilters(Xt −1 ){ for i=1 to n xt
[i] ~ p(xt | xt −1
[i]) (prediction) wt
[i] = p(zt | xt
[i]) (importance weights)
endfor for i=1 to n include xt
[i] in Xt with probability ∝ wt[i] (resampling)
}
![Page 25: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/25.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Weight samples: w = f / g
Importance Sampling
![Page 26: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/26.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Particle Filter
By Frank Dellaert
![Page 27: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/27.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Case Study: Track moving objects from Helicopter
1. Harris Corners
2. Optical Flow (with clustering)
3. Motion likelihood function
4. Particle Filter
5. Centroid Extraction
David Stavens, Andrew Lookingbill, David Lieb, CS223b Winter 2004
![Page 28: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/28.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
1. Harris Corner Extraction
David Stavens, Andrew Lookingbill, David Lieb, CS223b Winter 2004
![Page 29: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/29.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
2. Optical Flow + Motion Detection
David Stavens, Andrew Lookingbill, David Lieb, CS223b Winter 2004
![Page 30: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/30.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
3. Motion Likelihood Function
David Stavens, Andrew Lookingbill, David Lieb, CS223b Winter 2004
![Page 31: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/31.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
4. Particle Filters
David Stavens, Andrew Lookingbill, David Lieb, CS223b Winter 2004
![Page 32: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/32.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
5. Extract Centroid
David Stavens, Andrew Lookingbill, David Lieb, CS223b Winter 2004
![Page 33: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/33.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
More Particle Filter Tracking
David Stavens, Andrew Lookingbill, David Lieb, CS223b Winter 2004
![Page 34: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/34.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Some Robotics Examples
Tracking Hands, People Mobile Robot localization People localization Car localization Mapping
![Page 35: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/35.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Examples Particle Filter
Siu Chi Chan McGill University
![Page 36: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/36.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Another Example
Mike Isard and Andrew Blake
![Page 37: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/37.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Tracking Fast moving Objects
K. Toyama, A.Blake
![Page 38: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/38.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Particle Filters: IllustrationWith: Wolfram Burgard, Dieter Fox, Frank Dellaert
),|()( ...1...1 ttttt uzxpXxp ≈∈
![Page 39: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/39.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Particle Filters (1)
![Page 40: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/40.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Particle Filters (2)
![Page 41: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/41.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Particles = Robustness
![Page 42: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/42.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Tracking People from Moving Platform
• robot location (particles)• people location (particles)• laser measurements (wall) With Michael Montemerlo
![Page 43: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/43.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Tracking People from Moving Platform
With Michael Montemerlo
• robot location (particles)• people location (particles)• laser measurements (wall)
![Page 44: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/44.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Particle Filters for Tracking Cars
With Anya Petrovskaya
![Page 45: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/45.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Mapping Environments (SFM)
With Dirk Haehnel
![Page 46: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/46.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Overview
The Tracking Problem Bayes Filters Particle Filters Kalman Filters Using Kalman Filters
![Page 47: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/47.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Tracking with KFs: Gaussians!
updateinitial estimte
x
y
x
y
prediction
x
y
measurement
x
y
![Page 48: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/48.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Kalman Filters
),(~)( ΣµNxp
−Σ−−∝ − )()(
2
1exp)( 1 µµ xxxp T
![Page 49: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/49.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Kalman Filters
prior
Measurementevidence
posterior
)()|()|( xpxzpzxp ∝
dxxpxxpxp )()|'()'( ∫=
![Page 50: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/50.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
A Quiz
prior
Measurementevidence
)()|()|( xpxzpzxp ∝
posterior?
![Page 51: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/51.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Gaussians
2
2)(
2
1
2
2
1)(
:),(~)(
σµ
σπ
σµ
−−=
x
exp
Nxp
-σ σ
µUnivariate
)()(2
1
2/12/
1
)2(
1)(
:)(~)(
μxΣμx
Σx
Σμx
−−− −
=t
ep
,Νp
dπ
µ
Multivariate
![Page 52: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/52.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
),(~),(~ 22
2
σµσµabaNY
baXY
NX+⇒
+=
Properties of Univariate Gaussians
++
++
⋅⇒
−− 22
21
222
21
21
122
21
22
212222
2111 1
,~)()(),(~
),(~
σσµ
σσσµ
σσσ
σµσµ
NXpXpNX
NX
![Page 53: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/53.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Measurement Update Derived( ) ( )
−−−−⋅=⋅⇒
22
22
21
21
212222
2111
2
1
2
1expconst.)()(
),(~
),(~
σµ
σµ
σµσµ xx
XpXpNX
NX
( ) ( )) new(for 0
2
1
2
12
2
22
1
12
2
22
21
21 µ
σµ
σµ
σµ
σµ =−+−=
−−−−
∂∂ xxxx
x
( ) ( ) 0212
221 =−+− σµµσµµ
212
221
22
21 )( σµσµσσµ +=+
22
21
212
221
σσσµσµµ
++=
( ) ( ) 22
212
2
22
21
21
2
2
2
1
2
1 −− +=
−−−−
∂∂ σσ
σµ
σµ xx
x
22
21
1−− +
=σσ
σ
![Page 54: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/54.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
We stay in the “Gaussian world” as long as we start with Gaussians and perform only linear transformations.
),(~),(~ TAABANY
BAXY
NXΣ+⇒
+=Σ
µµ
Properties Multivariate GaussiansEssentially the same as in the 1-D case, but with more general notation
( )112
1121
12112
12121
222
111 )(,)()(~)()(),(~
),(~ −−−−− Σ+ΣΣΣ+Σ+ΣΣ+Σ⋅⇒
ΣΣ
µµµµ
NXpXpNX
NX
![Page 55: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/55.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Linear Kalman Filter
tttttt uBxAx ε++= −1
tttt xCz δ+=
Estimates the state x of a discrete-time controlled process that is governed by the linear stochastic difference equation
with a measurement
![Page 56: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/56.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Components of a Kalman Filter
tε
Matrix (n × n) that describes how the state evolves from t to t+1 without controls or noise.
tA
Matrix (n × i) that describes how the control ut changes the state from t to t+1.tB
Matrix (k × n) that describes how to map the state xt to an observation zt.tC
tδ
Random variables representing the process and measurement noise that are assumed to be independent and normally distributed with covariance Rt and Qt respectively.
![Page 57: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/57.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Kalman Filter Algorithm
1. Algorithm Kalman_filter( µt-1, Σt-1, ut, zt):
2. Prediction:3. 4.
5. Correction:6. 7. 8.
9. Return µt, Σt
ttttt uBA += −1µµ
tTtttt RAA +Σ=Σ −1
tttt CKI Σ−=Σ )(
1)( −+ΣΣ= tTttt
Tttt QCCCK
)( tttttt CzK µµµ −+=
![Page 58: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/58.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Kalman Filter Updates in 1D
old belief
measurement
belief
new belief
belief
mea
sure
men
t
![Page 59: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/59.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Kalman Filter Updates in 1D
1)(with )(
)()( −+ΣΣ=
Σ−=Σ−+=
= tTttt
Tttt
tttt
ttttttt QCCCK
CKI
CzKxbel
µµµ
2,
2
2
22 with )1(
)()(
tobst
tt
ttt
tttttt K
K
zKxbel
σσσ
σσµµµ
+=
−=−+=
=
old belief
new belief
mea
sure
men
t
![Page 60: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/60.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Kalman Filter Updates in 1D
+Σ=Σ+=
=−
−
tTtttt
tttttt RAA
uBAxbel
1
1)(µµ
+=+=
= −2
,2221)(
tactttt
tttttt a
ubaxbel
σσσµµ
old belief
new belief
mea
sure
men
t
new belief
newest belief
![Page 61: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/61.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Kalman Filter Updates
belief
latest belief
belief
measurement
belief
![Page 62: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/62.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
The Prediction-Correction-Cycle
+Σ=Σ+=
=−
−
tTtttt
tttttt RAA
uBAxbel
1
1)(µµ
+=+=
= −2
,2221)(
tactttt
tttttt a
ubaxbel σσσ
µµ
Prediction
![Page 63: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/63.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
The Prediction-Correction-Cycle
1)(,)(
)()( −+ΣΣ=
Σ−=Σ−+=
= tTttt
Tttt
tttt
ttttttt QCCCK
CKI
CzKxbel
µµµ
2,
2
2
22 ,)1(
)()(
tobst
tt
ttt
tttttt K
K
zKxbel
σσσ
σσµµµ
+=
−=−+=
=
Correction
![Page 64: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/64.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
The Prediction-Correction-Cycle
1)(,)(
)()( −+ΣΣ=
Σ−=Σ−+=
= tTttt
Tttt
tttt
ttttttt QCCCK
CKI
CzKxbel
µµµ
2,
2
2
22 ,)1(
)()(
tobst
tt
ttt
tttttt K
K
zKxbel
σσσ
σσµµµ
+=
−=−+=
=
+Σ=Σ+=
=−
−
tTtttt
tttttt RAA
uBAxbel
1
1)(µµ
+=+=
= −2
,2221)(
tactttt
tttttt a
ubaxbel σσσ
µµ
Correction
Prediction
![Page 65: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/65.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Kalman Filter Summary
Highly efficient: Polynomial in measurement dimensionality k and state dimensionality n: O(k2.376 + n2)
Optimal for linear Gaussian systems!
Most robotics systems are nonlinear!
![Page 66: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/66.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Overview
The Tracking Problem Bayes Filters Particle Filters Kalman Filters Using Kalman Filters
![Page 67: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/67.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Let’s Apply KFs to Tracking Problem
Image 4Image 1 Image 2 Image 3
![Page 68: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/68.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Kalman Filter with 2-Dim Linear Model
Linear Change (Motion)
Linear Measurement Model
),0( QNDCxz ++=What is A, B, Q?
=
2
2
0
0
σσ
Q
=
0
0D
=
10
01C
=
0
0B
=
10
01A
),0(' RNBAxx ++=What is C, D, R?
=
2
2
0
0
ρρ
R
![Page 69: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/69.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Kalman Filter Algorithm
1. Algorithm Kalman_filter( µt-1, Σt-1, ut, zt):
2. Prediction:3. 4.
5. Correction:6. 7. 8.
9. Return µt, Σt
ttttt uBA += −1µµt
Ttttt RAA +Σ=Σ −1
1)( −+ΣΣ= tTttt
Tttt QCCCK
)( DCzK tttttt −−+= µµµtttt CKI Σ−=Σ )(
=
2
2
0
0
σσ
Q
=
0
0D
=
10
01C
=
0
0B
=
10
01A
=
2
2
0
0
ρρ
R
![Page 70: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/70.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Kalman Filter in Detail
Measurements Change Prediction Measurement Update
),0( RNxz +=
),0(' QNxx +=
( ))'('''''
'''1
111
µµµ −Σ+=+Σ=Σ
−
−−−
zR
R
µµ =+Σ=Σ '' Q
updateinitial position
x
y
x
y
prediction
x
y
measurement
x
y
![Page 71: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/71.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Can We Do Better?
![Page 72: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/72.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Kalman, Better!
initial position prediction measurement
x
x
x
x
x
x
next
pre
dici
ton
x
x
update
x
x
![Page 73: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/73.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
We Can Estimate Velocity!
past measurements
prediction
![Page 74: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/74.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Kalman Filter For 2D Tracking
Linear Measurement model (now with 4 state variables)
Linear Change
=
2
2
000
000
0000
0000
q
=
2
2
0
0
r
rR),0(
0010
0001RN
y
x
y
x
y
x
obs
obs +
=
),0(
1000
0100
010
001
'
'
'
'
QN
y
x
y
x
t
t
y
x
y
x
+
∆
∆
=
![Page 75: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/75.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Putting It Together Again
Measurements
Change
Prediction
Measurement Update
),0(0010
0001RNxz +
=
),0(
1000
0100
010
001
' QNxt
t
x +
∆
∆
=
( ))(''
'1
111
µµµ AzRA
ARAT
T
−Σ+=+Σ=Σ
−
−−−
µµ C
CQC T
=+Σ=Σ
'
'
![Page 76: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/76.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Summary Kalman Filter Estimates state of a system
– Position– Velocity– Many other continuous state variables possible
KF maintains– Mean vector for the state– Covariance matrix of state uncertainty
Implements– Time update = prediction– Measurement update
Standard Kalman filter is linear-Gaussian– Linear system dynamics, linear sensor model– Additive Gaussian noise (independent)– Nonlinear extensions: extended KF, unscented KF: linearize
More info: – CS226– Probabilistic Robotics (Thrun/Burgard/Fox, MIT Press)
![Page 77: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/77.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Summary Kalman Filter Estimates state of a system
– Position– Velocity– Many other continuous state variables possible
KF maintains– Mean vector for the state– Covariance matrix of state uncertainty
Implements– Time update = prediction– Measurement update
Standard Kalman filter is linear-Gaussian– Linear system dynamics, linear sensor model– Additive Gaussian noise (independent)– Nonlinear extensions: extended KF, unscented KF: linearize
More info: – CS226– Probabilistic Robotics (Thrun/Burgard/Fox, MIT Press)
Particle
and discrete
set of particles(example states)
= predictive sampling= resampling, importance weights
fully nonlinear
easy to implement
![Page 78: Monte Carlo Hidden Markov Models - People | MIT CSAILpeople.csail.mit.edu/rplatt/thrun_tracking_lecture.pdfStanford CS223B Computer Vision, Winter 2007 Lecture 12 Tracking Motion Professors](https://reader034.fdocuments.in/reader034/viewer/2022052016/602e371ec84a1268d8048916/html5/thumbnails/78.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Summary
The Tracking Problem Bayes Filters Particle Filters Kalman Filters Using Kalman Filters