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19 cv mil_temporal_models

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  • 1. Computer vision: models, learning and inference Chapter 19 Temporal models Please send errata to s.prince@cs.ucl.ac.uk
  • 2. GoalTo track object state from frame to frame in a videoDifficulties: • Clutter (data association) • One image may not be enough to fully define state • Relationship between frames may be complicated
  • 3. Structure• Temporal models• Kalman filter• Extended Kalman filter• Unscented Kalman filter• Particle filters• Applications Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 3
  • 4. Temporal Models• Consider an evolving system• Represented by an unknown vector, w• This is termed the state• Examples: – 2D Position of tracked object in image – 3D Pose of tracked object in world – Joint positions of articulated model• OUR GOAL: To compute the marginal posterior distribution over w at time t. Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 4
  • 5. Estimating StateTwo contributions to estimating the state:1. A set of measurements xt, which provide information about the state wt at time t. This is a generative model: the measurements are derived from the state using a known probability relation Pr(xt|w1…wT)2. A time series model, which says something about the expected way that the system will evolve e.g., Pr(wt|w1...wt-1,wt+1…wT) Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 5
  • 6. Assumptions• Only the immediate past matters (Markov) – the probability of the state at time t is conditionally independent of states at times 1...t-2 given the state at time t-1.• Measurements depend on only the current state – the likelihood of the measurements at time t is conditionally independent of all of the other measurements and the states at times 1...t-1, t+1..t given the state at time t. Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 6
  • 7. Graphical ModelWorld statesMeasurements Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 7
  • 8. Recursive EstimationTime 1Time 2 fromTime t temporal model Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 8
  • 9. Computing the prior (time evolution)Each time, the prior is based on the Chapman-KolmogorovequationPrior at time t Temporal model Posterior at time t-1 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 9
  • 10. SummaryAlternate between:Temporal Evolution Temporal modelMeasurement Update Measurement model Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 10
  • 11. Structure• Temporal models• Kalman filter• Extended Kalman filter• Unscented Kalman filter• Particle filters• Applications Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 11
  • 12. Kalman FilterThe Kalman filter is just a special case of this type ofrecursive estimation procedure.Temporal model and measurement model carefullychosen so that if the posterior at time t-1 was Gaussianthen the• prior at time t will be Gaussian• posterior at time t will be GaussianThe Kalman filter equations are rules for updating the means and covariances of these Gaussians Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 12
  • 13. The Kalman FilterPrevious time step PredictionMeasurement likelihood Combination Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 13
  • 14. Kalman Filter DefinitionTime evolution equation State transition matrix Additive Gaussian noiseMeasurement equation Additive Gaussian noise Relates state and measurement Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 14
  • 15. Kalman Filter DefinitionTime evolution equation State transition matrix Additive Gaussian noiseMeasurment equation Additive Gaussian noise Relates state and measurement Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 15
  • 16. Temporal evolutionComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 16
  • 17. Measurement incorporation Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 17
  • 18. Kalman FilterThis is not the usual way these equations are presented.Part of the reason for this is the size of the inverses: is usuallylandscape and so T is inefficientDefine Kalman gain: Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 18
  • 19. Mean TermUsing Matrix inversion relations: Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 19
  • 20. Covariance TermKalman FilterUsing Matrix inversion relations: Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 20
  • 21. Final Kalman Filter EquationInnovation (difference betweenactual and predicted measurements Prior variance minus a term due to information from measurement Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 21
  • 22. Kalman Filter SummaryTime evolution equationMeasurement equationInference Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 22
  • 23. Kalman Filter Example 1Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 23
  • 24. Kalman Filter Example 2Alternates: Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 24
  • 25. Smoothing• Estimates depend only on measurements up to the current point in time.• Sometimes want to estimate state based on future measurements as wellFixed Lag Smoother:This is an on-line scheme in which the optimal estimate for a state at time t -t is calculated based on measurements up to time t, where t is the time lag. i.e. we wish to calculate Pr(wt- |x1 . . .xt ).Fixed Interval Smoother:We have a fixed time interval of measurements and want to calculate the optimal state estimate based on all of these measurements. In other words, instead of calculating Pr(wt |x1 . . .xt ) we now estimate Pr(wt |x1 . . .xT) where T is the total length of the interval. Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 25
  • 26. Fixed lag smoother State evolution equation Estimatedelayed by Measurement equation Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 26
  • 27. Fixed-lag Kalman Smoothing Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 27
  • 28. Fixed interval smoothingBackward set of recursionswhere Equivalent to belief propagation / forward-backward algorithm Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 28
  • 29. Temporal ModelsComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 29
  • 30. Problems with the Kalman filter• Requires linear temporal and measurement equations• Represents result as a normal distribution: what if the posterior is genuinely multi- modal? Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 30
  • 31. Structure• Temporal models• Kalman filter• Extended Kalman filter• Unscented Kalman filter• Particle filters• Applications Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 31
  • 32. Roadmap Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 32
  • 33. Extended Kalman FilterAllows non-linear measurement and temporal equationsKey idea: take Taylor expansion and treat as locally linear Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 33
  • 34. JacobiansBased on Jacobians matrices of derivatives Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 34
  • 35. Extended Kalman Filter Equations Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 35
  • 36. Extended Kalman FilterComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 36
  • 37. Problems with EKFComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 37
  • 38. Structure• Temporal models• Kalman filter• Extended Kalman filter• Unscented Kalman filter• Particle filters• Applications Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 38
  • 39. Unscented Kalman FilterKey ideas:• Approximate distribution as a sum of weighted particles with correct mean and covariance• Pass particles through non-linear function of the form• Compute mean and covariance of transformed variables Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 39
  • 40. Unscented Kalman FilterApproximate with particles:Choose so that Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 40
  • 41. One possible schemeWith: Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 41
  • 42. ReconstitutionComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 42
  • 43. Unscented Kalman Filter Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 43
  • 44. Measurement incorportationMeasurement incorporation works in a similar way:Approximate predicted distribution by set of particlesParticles chosen so that mean and covariance the same Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 44
  • 45. Measurement incorportationPass particles through measurement equationandrecompute mean and variance:Measurement update equations:Kalman gain now computed from particles: Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 45
  • 46. Problems with UKFComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 46
  • 47. Structure• Temporal models• Kalman filter• Extended Kalman filter• Unscented Kalman filter• Particle filters• Applications Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 47
  • 48. Particle filtersKey idea: • Represent probability distribution as a set of weighted particlesAdvantages and disadvantages: + Can represent non-Gaussian multimodal densities + No need for data association - Expensive Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 48
  • 49. Condensation AlgorithmStage 1: Resample from weighted particles according to theirweight to get unweighted particles Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 49
  • 50. Condensation AlgorithmStage 2: Pass unweighted samples through temporal modeland add noise Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 50
  • 51. Condensation AlgorithmStage 3: Weight samples by measurement density Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 51
  • 52. Data AssociationComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 52
  • 53. Structure• Temporal models• Kalman filter• Extended Kalman filter• Unscented Kalman filter• Particle filters• Applications Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 53
  • 54. Tracking pedestriansComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 54
  • 55. Tracking contour in clutter Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 55
  • 56. Simultaneous localization and mapping Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 56