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

2. Eigenfaces March 10, 2006 HCI/ComS 575X: Computational Perception
Iowa State University, SPRING 2006
Copyright © 2006, Alexander Stoytchev

3. Lecture Plan Update on Project Proposals HW3: Sample Solutions
HW4: next time
Eigenvectors: Math Background
Eigenvectors: Matlab demo
Eigenfaces: Theory + Notation + Algorithm
Eigenfaces: Matlab demo

4. Project Proposals

5. HW3: Some of the better results (according to the TA)
Wen-Chieh Chang (did all 3)
In no specific order:
Dongheng Li
Ethan Slattery
Jace Otting
Matt Swanson
Krista M. Thompson

6. HW4 preview

7. M. Turk and A. Pentland (1991).
``Eigenfaces for recognition''. Journal of Cognitive Neuroscience, 3(1).

8. ``An Introduction to Natural Computation (Complex Adaptive Systems)'',
Dana H. Ballard (1999).
``An Introduction to Natural Computation (Complex Adaptive Systems)'',
Chapter 4, pp 70-94, MIT Press.

9. Review of Eigenvalues and Eigenvectors

10. Systems of LinearEquations
Ax = b
mn matrix
n-vector (column)
m-vector (column) 
2 1
–3 4 x1 x2 = 1 4
2x1 + x2 = 1
–3x1 + 4x2 = 4
Example:

11. Eigenvalues Ax = x 0 1 1 0 x1 x2 =    = 1 /\ x1 = x2
eigenvector1
eigenvector2
A symmetric   real
A positive definite   > 0
A positive semi-definite    0 x1 x2 Ax
x an eigenvector 

12. On-line Java Applets
webfeatsII/Lite_Applets/contents.html
general/115a.3.02f/EigenMap.html

13. Java Applet Example [

14. Java Applet Example [

15. Finding the Eigenvectors

16. Finding the Eigenvectors

17. Finding the Eigenvectors
The solution exists if:
Characteristic equation:

18. Finding the Eigenvectors
The solutions to the characteristic equation are the eigenvalues
In this case:

19. Finding the Eigenvectors
Substituting with
We get this system of equations

20. Finding the Eigenvectors
Thus, first eigenvector is:
The corresponding eigenvalue is:
Verification:

21. Eigenvectors and Eigenvalues in Matlab
>> [V, D]=eig(A)
V =
D =

22. Eigenvectors and Eigenvalues in Matlab
>> L = [ V(:,1), [0; 0], V(:,2)]'
L =
>> line(L(:,1), L(:,2))
>> axis([-1, 1, -1, 1])

23. Eigenvectors v.s. Clustering

24. [

25. Matlab Demo

26. Visualizing the Eigenvectors in 3D

28. Recognition Example ? ?

29. Recognition Example ? ?

30. How about this one? ?

31. Eigenfaces

32. Representing images as vectors

33. The Next set of slides come from:
Prof. Ramani Duraiswami’s class
CMSC: 426 Computer Vision

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47. Matlab Demo

48. THE END