19 cv mil_temporal_models

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19 cv mil_temporal_models

  1. 1. Computer vision: models, learning and inference Chapter 19 Temporal models Please send errata to s.prince@cs.ucl.ac.uk
  2. 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. 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. 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. 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. 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. 7. Graphical ModelWorld statesMeasurements Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 7
  8. 8. Recursive EstimationTime 1Time 2 fromTime t temporal model Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 8
  9. 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. 10. SummaryAlternate between:Temporal Evolution Temporal modelMeasurement Update Measurement model Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 10
  11. 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. 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. 13. The Kalman FilterPrevious time step PredictionMeasurement likelihood Combination Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 13
  14. 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. 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. 16. Temporal evolutionComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 16
  17. 17. Measurement incorporation Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 17
  18. 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. 19. Mean TermUsing Matrix inversion relations: Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 19
  20. 20. Covariance TermKalman FilterUsing Matrix inversion relations: Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 20
  21. 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. 22. Kalman Filter SummaryTime evolution equationMeasurement equationInference Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 22
  23. 23. Kalman Filter Example 1Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 23
  24. 24. Kalman Filter Example 2Alternates: Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 24
  25. 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. 26. Fixed lag smoother State evolution equation Estimatedelayed by Measurement equation Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 26
  27. 27. Fixed-lag Kalman Smoothing Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 27
  28. 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. 29. Temporal ModelsComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 29
  30. 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. 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. 32. Roadmap Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 32
  33. 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. 34. JacobiansBased on Jacobians matrices of derivatives Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 34
  35. 35. Extended Kalman Filter Equations Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 35
  36. 36. Extended Kalman FilterComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 36
  37. 37. Problems with EKFComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 37
  38. 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. 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. 40. Unscented Kalman FilterApproximate with particles:Choose so that Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 40
  41. 41. One possible schemeWith: Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 41
  42. 42. ReconstitutionComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 42
  43. 43. Unscented Kalman Filter Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 43
  44. 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. 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. 46. Problems with UKFComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 46
  47. 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. 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. 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. 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. 51. Condensation AlgorithmStage 3: Weight samples by measurement density Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 51
  52. 52. Data AssociationComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 52
  53. 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. 54. Tracking pedestriansComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 54
  55. 55. Tracking contour in clutter Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 55
  56. 56. Simultaneous localization and mapping Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 56

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