Transcript Slide 1

HCI/ComS 575X:
Computational Perception
Instructor: Alexander Stoytchev
http://www.cs.iastate.edu/~alex/classes/2006_Spring_575X/
The Kalman Filter
(part 1)
February 13, 2006
HCI/ComS 575X: Computational Perception
Iowa State University, SPRING 2006
Copyright © 2006, Alexander Stoytchev
Maybeck, Peter S. (1979)
Chapter 1 in ``Stochastic
models, estimation, and control'',
Mathematics in Science and
Engineering Series, Academic
Press.
Greg Welch & Gary Bishop (2001)
SIGGRAPH 2001 Course: ``An
Introduction to the Kalman Filter''.
Rudolf Emil Kalman
[http://www.cs.unc.edu/~welch/kalman/kalmanBiblio.html]
Application: Lunar Landing
Application: Radar Tracking
Application: Missile Tracking
Application: Sailing
Application: Robot Navigation
Application: Other Tracking
Application: Head Tracking
Face & Hand Tracking
Quick Review of Probability
What is a probability?
Probability of A or B occurring
Probability of both A and B occurring together
(assuming that they are independent of each other)
Conditional Probability
Example: Rolling Dice
[http://www.shodor.org/interactivate/discussions/images/2dicetable.gif]
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]
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)?
Cumulative Probability Density
• In the continuous domain
• Therefore we need something else to
describe a probability distribution
Example of a
Cumulative Density Function
Properties of the cumulative
density functions
Probability Density Function (pdf)
Properties of the probability
density functions
Properties of the probability
density functions
Probability Density Function (pdf)
[http://www.weibull.com/LifeDataWeb/the_probability_density_and_cumulative_distribution_functions.htm]
Relationship b/en pdf and cdf
[http://www.weibull.com/LifeDataWeb/the_probability_density_and_cumulative_distribution_functions.htm]
Summary
• Random Variable:
• Cumulative Density Function:
• Probability Density Function:
Mean (Average)
Mean (Discrete Case)
Expected Value (Discrete Case)
Expected Value (Continuous Case)
Same thing but applied to functions
of the random variable X
• Discrete Case:
• Continuous Case:
Moments
• Let
• Then the k-th moment is given by
Second Moment
Second Moment
Variance
• Let
• Then
A trick for computing the variance
E(X - µ)2 = E(X2 - 2Xµ + µ2)
= E(X2) - 2[E(X)] µ + µ2
= E(X2) – 2 µ2 + µ2
= E(X2) - µ2
A trick for computing the variance
E(X - µ)2 = E(X2 - 2Xµ + µ2)
= E(X2) - 2[E(X)] µ + µ2
σ2
= E(X2) – 2 µ2 + µ2
= E(X2) - µ2
Standard Deviation
• Has the same units as the variable X
• Is Equal to the positive square root of the variance
Example
• Data Set (of 4 numbers):
– 1, 2, 3, 4
• N=4
• Mean
– sum/N = (1+2+3+4)/4 = 10/4 = 2.5
Example
• Data Set: 1, 2, 3, 4
µ = 2.5
N=4
σ2= [ (1-µ)2 + (2-µ)2 + (3-µ)2 + (4-µ)2] / (N - 1)
= [ 1.52 + 0.52 + 0.52 + 1.52] / (4 - 1)
= 5 / 3 = 1.666(6)
Example: Shortcut Way
σ2= [ (12 + 22
+ 3 2 + 4 2) –
(1+2+3+4)2/4] / (4 - 1)
= (30 – 102/4) /3 = 5/3 = 1.666(6)
Gaussian Properties
The Gaussian Function
Gaussian pdf
Properties
• If
• Then
and
pdf for
Properties
Summation and Subtraction
Noise Models
White Noise Properties
Time domain
Frequency domain
The Kalman Filter: Theory
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.
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)]
Why do we need a filter?
• No mathematical model of a real system is
perfect
• Real world disturbances
• Imperfect Sensors
Application: Radar Tracking
Conditional density of position
based on measured value of z1
[Maybeck (1979)]
Conditional density of position
based on measured value of z1
uncertainty
position
measured position
[Maybeck (1979)]
Conditional density of position
based on measurement of z2 alone
[Maybeck (1979)]
Conditional density of position
based on measurement of z2 alone
uncertainty 2
measured position 2
[Maybeck (1979)]
Conditional density of position
based on data z1 and z2
position estimate
[Maybeck (1979)]
Propagation of the conditional density
[Maybeck (1979)]
More about this Next Time
THE END