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

The Kalman Filter (part 3) HCI/ComS 575X: Computational Perception Iowa State University, SPRING 2006 Copyright © 2006, Alexander Stoytchev February 20, 2006

Brown and Hwang (1992) “Introduction to Random Signals and Applied Kalman Filtering” Ch 5: The Discrete Kalman Filter

Arthur Gelb, Joseph Kasper, Raymond Nash, Charles Price, Arthur Sutherland (1974) Applied Optimal Estimation MIT Press.

Let’s Start With a Demo Matlab Program Written by John Burnett

A Simple Recursive Example Problem Statement: Given the measurement sequence: z 1, z 2, …, z n find the mean [Brown and Hwang (1992)]

First Approach 1. Make the first measurement z 1 Store z 1 and estimate the mean as µ 1 =z 1 2. Make the second measurement z 2 Store z 1 along with z 2 and estimate the mean as µ 2 = (z 1 +z 2 )/2 [Brown and Hwang (1992)]

First Approach (cont’d) 3. Make the third measurement z 3 Store z 3 along with z 1 and z 2 and estimate the mean as µ 3 = (z 1 +z 2 +z 3 )/3 [Brown and Hwang (1992)]

First Approach (cont’d) n. Make the n-th measurement z n Store z n along with z 1, z 2,…, z n-1 and estimate the mean as µ n = (z 1 + z 2 + … + z n )/n [Brown and Hwang (1992)]

Second Approach 1. Make the first measurement z 1 Compute the mean estimate as µ 1 =z 1 Store µ 1 and discard z 1 [Brown and Hwang (1992)]

Second Approach (cont’d) 2.Make the second measurement z 2 Compute the estimate of the mean as a weighted sum of the previous estimate µ 1 and the current measurement z 2: µ 2 = 1/2 µ 1 +1/2 z 2 Store µ 2 and discard z 2 and µ 1 [Brown and Hwang (1992)]

Second Approach (cont’d) 3.Make the third measurement z 3 Compute the estimate of the mean as a weighted sum of the previous estimate µ 2 and the current measurement z 3: µ 3 = 2/3 µ 2 +1/3 z 3 Store µ 3 and discard z 3 and µ 2 [Brown and Hwang (1992)]

Second Approach (cont’d) n.Make the n-th measurement z n Compute the estimate of the mean as a weighted sum of the previous estimate µ n-1 and the current measurement z n: µ n = (n-1)/n µ n-1 +1/n z n Store µ n and discard z n and µ n-1 [Brown and Hwang (1992)]

Comparison Batch MethodRecursive Method

Analysis The second procedure gives the same result as the first procedure. It uses the result for the previous step to help obtain an estimate at the current step. The difference is that it does not need to keep the sequence in memory. [Brown and Hwang (1992)]

Second Approach (rewrite the general formula) µ n = (n-1)/n µ n-1 +1/n z n

Second Approach (rewrite the general formula) µ n = (n-1)/n µ n-1 +1/n z n µ n = µ n-1 + 1/n (z n - µ n-1 )

Second Approach (rewrite the general formula) µ n = (n-1)/n µ n-1 +1/n z n µ n = µ n-1 + 1/n (z n - µ n-1 ) Old Estimate Difference Between New Reading and Old Estimate Gain Factor

Second Approach (rewrite the general formula)

How should we combine the two measurements? [Maybeck (1979)] σZ1σZ1 σZ2σZ2

Calculating the new mean Scaling Factor 1 Scaling Factor 2

Calculating the new mean Scaling Factor 1 Scaling Factor 2

Calculating the new mean Scaling Factor 1 Scaling Factor 2 Why is this not z 1 ?

Calculating the new variance [Maybeck (1979)] σZ1σZ1 σZ2σZ2

Calculating the new variance Scaling Factor 1 Scaling Factor 2

The scaling factors must be squared! Scaling Factor 1 Scaling Factor 2

The scaling factors must be squared! Scaling Factor 1 Scaling Factor 2

The new variance is

What makes these scaling factors special? Are there other ways to combine the two measurements? They minimize the error between the prediction and the true value of X. They are optimal in the least-squares sense.

Minimize the error

What is the minimum value? [

What is the minimum value? [

Finding the Minimum Value Y= 9x x + 50 dY/dx = 18 x -50 =0 The minimum is obtained when x=50/18= (7) The minimum value is Y(x min ) = 9*(50/18) 2 -50*(50/18) +50 = (4)

Start with two measurements v 1 and v 2 represent zero mean noise

Formula for the estimation error The new estimate is The error is

Expected value of the error If the estimate is unbiased this should hold

Find the Mean Square Error = ?

Mean Square Error

Minimize the mean square error

Finding S 1 Therefore

Finding S 2

Finally we get what we wanted

Finding the new variance

Formula for the new variance

New Topic: Particle Filters

Michael Isard and Andrew Blake (1998) ``CONDENSATION -- conditional density propagation for visual tracking'', International Journal of Computer Vision, 29, 1,

Movies from the CONDENSATION Web Page

More about this next time

THE END