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

2. Eigenfaces HCI/ComS 575X: Computational Perception
Iowa State University
Copyright © Alexander Stoytchev

3. Lecture Plan A Cool Demo Eigenvectors: Math Background
Eigenvectors: Matlab demo
Eigenfaces: Theory + Notation + Algorithm
Eigenfaces: Matlab demo

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

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

6. Readings for Next Time
Henry A. Rowley, Shumeet Baluja and Takeo Kanade (1997). ``Rotation Invariant Neural Network-Based Face Detection,'' Carnegie Mellon Technical Report, CMU-CS
Paul Viola and Michael Jones (2001). ``Robust Real-time Object Detection'', Second International Workshop on Statistical and Computational Theories of Vision Modeling, Learning, Computing, and Sampling, Vancouver, Canada, July 13, 2001.

7. Review of Eigenvalues and Eigenvectors

8. Review Questions What is a vector? What is a Matrix?
What is the result when a vector is multiplied by a matrix? (Ax = ?)

9. 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:

10. Intuitive Definition
For any matrix there are some vectors such that the matrix multiplication changes only the magnitude of the vector.
These vectors are called eigenvectors.

11. Mathematical Formulation
Ax = x
0 1
1 0 x1 x2
=  
 = 1 /\ 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
[From Ballard (1999)]

16. Finding the Eigenvectors

17. Finding the Eigenvectors
Identity Matrix

18. Finding the Eigenvectors

19. Finding the Eigenvectors
[From Ballard (1999)]

20. Finding the Eigenvectors
Determinant, not matrix
The solution exists if:
Characteristic equation:
i.e.,
[From Ballard (1999)]

21. Finding the Eigenvectors
The solutions to the characteristic equation are the eigenvalues
In this case:
[From Ballard (1999)]

22. Finding the Eigenvectors
Substituting with
We get this system of equations
[From Ballard (1999)]

23. Finding the Eigenvectors
Thus, first eigenvector is:
The corresponding eigenvalue is:
Verification:
[From Ballard (1999)]

24. Finding the Eigenvectors
The second eigenvector is: [-05, 1]T
The corresponding eigenvalue is: 1

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

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

27. Eigenvectors v.s. Clustering

28. [

29. Simple Example

30. Three Sample Data Points
[From Ballard (1999)]

31. The mean is equal to …
[From Ballard (1999)]

32. Subtract the Mean From All Data Points
[From Ballard (1999)]

33. Column-Vector Multiplied by a Row-Vector = Matrix!

34. The Covariance Matrix is Given By
[From Ballard (1999)]

35. One possible coordinate system this set of data points
[From Ballard (1999)]

36. A better one…
[From Ballard (1999)]

37. Eigenvectors of the Covariance Matrix
The eigenvectors of the covariance matrix are important!
They represent the best coordinate system in which once can represent the specific data points. The structure of the data is maximally revealed if these coordinates are used instead of any other ones.

38. Let…

39. Also, Let…

40. Thus the Covariance Matrix can be rewritten as:

41. Why can we drop 1/M?

42. Representing images as vectors

43. A AT

44. AT A

45. What about that Trick with the Dimensions?

46. Matlab Demo

47. Visualizing the Eigenvectors in 3D

49. Recognition Example ? ?

50. Recognition Example ? ?

51. How about this one? ?

52. Eigenfaces

53. Representing images as vectors

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

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68. Matlab Demo

69. THE END