1. HCI/ComS 575X: Computational Perception Instructor: Alexander Stoytchev http://www.cs.iastate.edu/~alex/classes/2006_Spring_575X/

2. The Kalman Filter (part 1) HCI/ComS 575X: Computational Perception Iowa State University, SPRING 2006 Copyright © 2006, Alexander Stoytchev February 13, 2006

3. Maybeck, Peter S. (1979) Chapter 1 in ``Stochastic models, estimation, and control'', Mathematics in Science and Engineering Series, Academic Press.

4. Greg Welch & Gary Bishop (2001) SIGGRAPH 2001 Course: ``An Introduction to the Kalman Filter''.

5. Rudolf Emil Kalman [http://www.cs.unc.edu/~welch/kalman/kalmanBiblio.html]

6. Application: Lunar Landing

8. Application: Radar Tracking

9. Application: Missile Tracking

10. Application: Sailing

11. Application: Robot Navigation

12. Application: Other Tracking

13. Application: Head Tracking

14. Face & Hand Tracking

15. Quick Review of Probability

16. What is a probability?

17. Probability of A or B occurring

18. Probability of both A and B occurring together (assuming that they are independent of each other)

19. Conditional Probability

20. Example: Rolling Dice [http://www.shodor.org/interactivate/discussions/images/2dicetable.gif]

21. Example: Rolling Dice Event A={The sum of the numbers the dice show is 7 or 9} Event B={The second die shows 2 or 3} [http://www.shodor.org/interactivate/discussions/images/2dicetable.gif]

22. Example: Rolling Dice What is P(A)? What is P(B)? What is P(A or B)? What is P(A and B)? What is P(A given B)?

23. Cumulative Probability Density In the continuous domain Therefore we need something else to describe a probability distribution

24. Example of a Cumulative Density Function

25. Properties of the cumulative density functions

26. Probability Density Function (pdf)

27. Properties of the probability density functions

29. Probability Density Function (pdf) [http://www.weibull.com/LifeDataWeb/the_probability_density_and_cumulative_distribution_functions.htm]

30. Relationship b/en pdf and cdf [http://www.weibull.com/LifeDataWeb/the_probability_density_and_cumulative_distribution_functions.htm]

31. Summary Random Variable: Cumulative Density Function: Probability Density Function:

32. Mean (Average)

33. Mean (Discrete Case)

34. Expected Value (Discrete Case)

35. Expected Value (Continuous Case)

36. Same thing but applied to functions of the random variable X Discrete Case: Continuous Case:

37. Moments Let Then the k -th moment is given by

38. Second Moment

40. Variance Let Then

41. A trick for computing the variance E(X - µ) 2 = E(X 2 - 2Xµ + µ 2 ) = E(X 2 ) - 2[E(X)] µ + µ 2 = E(X 2 ) – 2 µ 2 + µ 2 = E(X 2 ) - µ 2

42. A trick for computing the variance E(X - µ) 2 = E(X 2 - 2Xµ + µ 2 ) = E(X 2 ) - 2[E(X)] µ + µ 2 = E(X 2 ) – 2 µ 2 + µ 2 = E(X 2 ) - µ 2 σ2σ2

43. Standard Deviation Has the same units as the variable X Is Equal to the positive square root of the variance

44. Example Data Set (of 4 numbers): –1, 2, 3, 4 N=4 Mean – sum/N = (1+2+3+4)/4 = 10/4 = 2.5

45. Example Data Set: 1, 2, 3, 4 µ = 2.5 N=4 σ 2 = [ (1-µ) 2 + (2-µ) 2 + (3-µ) 2 + (4-µ) 2 ] / (N - 1) = [ 1.5 2 + 0.5 2 + 0.5 2 + 1.5 2 ] / (4 - 1) = 5 / 3 = 1.666(6)

46. Example: Shortcut Way σ 2 = [ (1 2 + 2 2 + 3 2 + 4 2 ) – (1+2+3+4) 2 /4] / (4 - 1) = (30 – 10 2 /4) /3 = 5/3 = 1.666(6)

47. Gaussian Properties

48. The Gaussian Function

49. Gaussian pdf

50. Properties If and Then

51. pdf for

52. Properties

53. Summation and Subtraction

54. Noise Models

55. White Noise Properties Time domain Frequency domain

56. The Kalman Filter: Theory

57. Definition A Kalman filter is simply an optimal recursive data processing algorithm Under some assumptions the Kalman filter is optimal with respect to virtually any criterion that makes sense.

58. Definition “The Kalman filter incorporates all information that can be provided to it. It processes all available measurements, regardless of their precision, to estimate the current value of the variables of interest.” [Maybeck (1979)]

59. Why do we need a filter? No mathematical model of a real system is perfect Real world disturbances Imperfect Sensors

60. Application: Radar Tracking

61. Conditional density of position based on measured value of z 1 [Maybeck (1979)]

62. Conditional density of position based on measured value of z 1 [Maybeck (1979)] position measured position uncertainty

63. Conditional density of position based on measurement of z 2 alone [Maybeck (1979)]

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

65. Conditional density of position based on data z 1 and z 2 [Maybeck (1979)] position estimate

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

67. More about this Next Time

68. THE END