HCI/ComS 575X: Computational Perception Instructor: Alexander Stoytchev

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

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 [

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 [

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} [

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

Probability Density Function (pdf) [

Relationship b/en pdf and cdf [

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

Variance Let Then

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

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

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 = ( )/4 = 10/4 = 2.5

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) = [ ] / (4 - 1) = 5 / 3 = 1.666(6)

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

Gaussian Properties

The Gaussian Function

Gaussian pdf

Properties If and Then

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 z 1 [Maybeck (1979)]

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

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

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

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

Propagation of the conditional density [Maybeck (1979)]

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