1. HCI / CprE / ComS 575: Computational Perception
Instructor: Alexander Stoytchev

2. Particle Filters HCI/ComS 575X: Computational Perception
Iowa State University
Copyright © Alexander Stoytchev

3. Sebastian Thrun, Wolfram Burgard and Dieter Fox (2005).
Probabilistic Robotics
MIT Press.

4. F. Dellaert, D. Fox, W. Burgard, and S. Thrun (1999).
"Monte Carlo Localization for Mobile Robots",
IEEE International Conference on Robotics and Automation (ICRA99), May, 1999.

5. A Particle Filter Tutorial for Mobile Robot Localization.
Ioannis Rekleitis (2004).
A Particle Filter Tutorial for Mobile Robot Localization.
Technical Report TR-CIM-04-02, Centre for Intelligent Machines, McGill University, Montreal, Quebec, Canada.

6. Wednesday

7. Next Week
Preliminary Project Presentatons

8. Particle Filters Represent belief by random samples
Estimation of non-Gaussian, nonlinear processes
Monte Carlo filter, Survival of the fittest, Condensation, Bootstrap filter, Particle filter
Filtering: [Rubin, 88], [Gordon et al., 93], [Kitagawa 96]
Computer vision: [Isard and Blake 96, 98]
Dynamic Bayesian Networks: [Kanazawa et al., 95]d

9. Example

10. Using Ceiling Maps for Localization

11. Vision-based Localization
h(x) z
P(z|x)

12. Under a Light
Measurement z:
P(z|x):

13. Next to a Light
Measurement z:
P(z|x):

14. Elsewhere
Measurement z:
P(z|x):

15. Global Localization Using Vision

16. Sample-based Localization (sonar)

18. Example

19. Importance Sampling with Resampling: Landmark Detection Example

20. Distributions

21. Wanted: samples distributed according to p(x| z1, z2, z3)
Distributions
Wanted: samples distributed according to p(x| z1, z2, z3)

22. This is Easy!
We can draw samples from p(x|zl) by adding noise to the detection parameters.

23. Importance Sampling with Resampling
Weighted samples
After resampling

24. Quick review of Kalman Filters

25. Conditional density of position based on measured value of z1
[Maybeck (1979)]

26. Conditional density of position based on measured value of z1
uncertainty
position
measured position
[Maybeck (1979)]

27. Conditional density of position based on measurement of z2 alone
[Maybeck (1979)]

28. Conditional density of position based on measurement of z2 alone
uncertainty 2
measured position 2
[Maybeck (1979)]

29. Conditional density of position based on data z1 and z2
uncertainty estimate
position estimate
[Maybeck (1979)]

30. Propagation of the conditional density
[Maybeck (1979)]

31. Propagation of the conditional density
movement vector
expected position just prior
to taking measurement 3
[Maybeck (1979)]

32. Propagation of the conditional density
movement vector
expected position just prior
to taking measurement 3
[Maybeck (1979)]

33. Propagation of the conditional density
uncertainty 3 z3
σx(t3)
measured position 3

34. Updating the conditional density after the third measurement
position uncertainty z3
σx(t3)
x(t3)
position estimate

36. Some Questions What if we don’t know the start position of the robot?
What if somebody moves the robot without the robot’s knowledge?

37. Robot Odometry Errors

38. Raw range data, position indexed by odometry
[Thrun, Burgard & Fox (2005)]

39. Resulting Occupancy Grid Map
[Thrun, Burgard & Fox (2005)]

40. Basic Idea Behind Particle Filtersx

41. In 2D it looks like this [

42. Robot Pose

43. Odometry Motion Model

44. Sampling From the Odometry Model

45. Motion Model

46. Motion Model

47. Velocity model for different noise parameters

48. Sampling from the velocity model

49. In Class Demo of Particle Filters

50. Example
[Thrun, Burgard & Fox (2005)]

51. Initially we don’t know the location of the robot so we have particles everywhere

52. Next, the robot senses that it is near a door

53. Since there are 3 identical doors the robot can be next any one of them

54. Therefore, we inflate balls (particles) that are next to doors and shrink all others

55. Therefore, we grow balls (particles) that are next to doors and shrink all others

56. Before we continue we have to make all ball to be of equal size
Before we continue we have to make all ball to be of equal size. We need to resample.

57. Before we continue we have to make all ball to be of equal size
Before we continue we have to make all ball to be of equal size. We need to resample.

58. Resampling Rules = = =

59. Resampling Given: Set S of weighted samples.
Wanted : Random sample, where the probability of drawing xi is given by wi.
Typically done n times with replacement to generate new sample set S’.
[From Thrun’s book “Probabilistik Robotics”]

60. Roulette wheel Resamplingwn
Wn-1 w2 w3 w1 wn
Wn-1
Stochastic universal sampling
Systematic resampling
Linear time complexity
Easy to implement, low variance
Roulette wheel
Binary search, n log n
[From Thrun’s book “Probabilistik Robotics”]

61. Also called stochastic universal sampling
Resampling Algorithm
Algorithm systematic_resampling(S,n):
For Generate cdf
Initialize threshold
For Draw samples …
While ( ) Skip until next threshold reached
Insert
Increment threshold
Return S’
Also called stochastic universal sampling
[From Thrun’s book “Probabilistik Robotics”]

62. Next, The robot moves to the right

63. … thus, we have to shift all balls (particles) to the right

64. … thus, we have to shift all balls (particles) to the right

65. … and add some position noise

66. … and add some position noise

67. Next, the robot senses that it is next to one of the three doors

68. Next, the robot senses that it is next to one of the three doors

69. Now we have to resample again

70. The robot moves again

71. … so we must move all balls (particles) to the right again

72. … and add some position noise

73. And so on …

74. Now Let’s Compare that With Some of the Other Methods

75. Grid Localization
[Thrun, Burgard & Fox (2005)]

76. Grid Localization
[Thrun, Burgard & Fox (2005)]

77. Grid Localization
[Thrun, Burgard & Fox (2005)]

78. Grid Localization
[Thrun, Burgard & Fox (2005)]

79. Grid Localization
[Thrun, Burgard & Fox (2005)]

80. Markov Localization
[Thrun, Burgard & Fox (2005)]

81. Kalman Filter
[Thrun, Burgard & Fox (2005)]

82. Particle Filter
[Thrun, Burgard & Fox (2005)]

84. Importance Sampling
Ideally, the particles would represent samples drawn from the distribution p(x|z).
In practice, we usually cannot get p(x|z) in closed form; in any case, it would usually be difficult to draw samples from p(x|z).
We use importance sampling:
Particles are drawn from an importance distribution.
Particles are weighted by importance weights.
[ ]

85. Monte Carlo Samples (Particles)
The posterior distribution p(x|z) may be difficult or impossible to compute in closed form.
An alternative is to represent p(x|z) using Monte Carlo samples (particles):
Each particle has a value and a weight x x
[ ]

86. In 2D it looks like this [

87. Objective-Find p(xk|zk,…,z1)
The objective of the particle filter is to compute the conditional distribution
p(xk|zk,…,z1)
To do this analytically, we would use the Chapman-Kolmogorov equation and Bayes Theorem along with Markov model assumptions.
The particle filter gives us an approximate computational technique.
[ ]

88. Initial State Distributionx0 x0
[ ]

89. State Update x0 x1 = f0 (x0, w0) x1
[ ]

90. Compute Weights p(z1|x1) x1 Before x1 After x1
[ ]

91. Resample x1 x1
[ ]

92. THE END