1. Image Compression SVD Defuses the Bomb

2. Where Mathematics and Classical Art come together to combat terrorism

3. Dusty Wilson Western Washington University Graduate Student Instructor

5. In the midst of the political unrest in Russia, many nuclear warheads have mysteriously disappeared Russian Nuclear Facilities

6. Some missiles are believed in the hands of anti-American terrorist groups

7. In the interest of national security, the CIA has worked with the Russians to neutralize the terrorist threat.

8. Extreme political paranoia in Russia led to fitting their missiles with safety mechanisms.

9. Deactivation codes - 11 digit phone numbers - were encrypted in classical art.

10. Creation of Adam O 15 = 5 mod(10) 5 o

11. We hold the key to decode what is locked in the images.

12. However, the images are stored remotely and we must access them before we can decode the key

13. There is a ten minute window for us to access the images and deactivate the terrorist missiles.

14. For this, we need powerful and effective image compression.

15. Image Compression An application of the Singular Value Decomposition Theorem

16. The Way It Works. S V D Image Compression

17. ImageArray

18. Every color has a red, green, and blue (RGB) component. where each r i,j is the i,j th component of the matrix R. Define G and B in a similar manner.

19. Image R G B

20. R is real valued and has rank r. For simplicity, consider R

21. The Singular Value Decomposition Theorem

22. Any real valued matrix is the product of orthonormal and diagonal matrices. where the rank of R is r, U r and V r are orthonormal, and Σ r is an ordered diagonal matrix

23. Orthonormal and Diagonal Matrices What are they, and how hard do they bite?

24. Orthonormal rotates by θ. For example, the orthonormal matrix Visualize orthonormal matrices as rotations.

25. Diagonal Visualize diagonal transformations as a component-wise stretches mapsinto For example:

26. A diagonal matrix is ordered if where

27. An Example of The Singular Value Decomposition Theorem

28. Let R be where R has rank 2

29. The singular value decom- position of R is:

30. Which we recognize as

31. With fiddling, this is equal to

32. We visualize: 1.) A rotation by 2.) A diagonal stretch 3.) A rotation by

33. This can be seen in the following Mathematica animation

34. How does SVD apply to Image Compression ?

35. R Recall that R was the red component of an imageImageG B R

36. By the SVD Theorem

37. So SVD says: But Image Compression???

38. Approximate R with and continue on to

39. In particular, R r = R We have that

40. Choose the k such that R k approximates R to the desired degree

41. We measure the quality of an approximation by our satisfaction with the compressed image

42. This is illustrated in the following example Consider approximations of RR

43. 2 1 4 6 8 10 3 5 7 9

44. 2 1 4 6 8 10 3 5 7 9 Again

45. How much does SVD compress an image? What would we be satisfied with?

46. Each iteration costs 0.59% in storage 10 iterations = 5.9%

47. The  determine what percent of the original image is gained at each iteration. This can be seen in the following graph.

48. 10 iterations gives 71.3%

49. 100% 5.9% Original Approximation

50. So what are the drawbacks? Computing Power -

51. SVD requires more power than any groups other than the government, military, and some research institutes have.

52. For Example, the SVD of this image required 28 seconds On a Pentium II operating at 300mhz with 128mb of RAM on MATLAB v6

53. More dramatically, SVD on this image took 6 min 31 sec On a Pentium II operating at 300mhz with 128mb of RAM on MATLAB v6

54. Now that the power and limitations of SVD are evident, it is time to save our nation from terrorists

56. Solutions (1) American Gothic (3) Mona Lisa (6) Last Supper (0) Old Guitar Player (6) Praying Hands (5) Washington Crossing the Delaware (0) Sunday Afternoon On La Grande Jatte (4) David (8) The Thinker (3) The Scream (6) Waterfall Feel free to call 1 (360) 650- 4836 to confirm that the world is safe once more for the American people. (You are also free to call and chat;-)

57. Now that the world is safe from terrorists and I have explained singular value decomposition...

58. Special thanks to Charlene Wilson my wonderful wife Dr. John Woll my academic advisor

59. Additional thanks to Peter Caldwell Alison Haukaas Audrey Kalinowski Doug Ronne Vika Savalei Nathaniel Wilson Dean J Alan Ross Travel Fund Travel Fund Dr. Tjalling Ypma Math Dept. Staff WWU Graduate School Staff School Staff

60. Dusty Wilson Graduate Student Instructor Western Washington University (360) 756-8309 dwwiv@yahoo.com

62. Historical Notes SVD was independently discovered and rediscovered a number of times including: Eugenio Beltrami (1835-1899) in 1873 M.E. Camille Jordan (1838-1922) in 1875 James J. Sylvester (1814-1897) in 1889 L. Autonne in 1913 C. Eckart and G. Young in 1936