Image Compression SVD Defuses the Bomb

Where Mathematics and Classical Art come together to combat terrorism

Dusty Wilson Western Washington University Graduate Student Instructor

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

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

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

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

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

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

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

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

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

For this, we need powerful and effective image compression.

Image Compression An application of the Singular Value Decomposition Theorem

The Way It Works. S V D Image Compression

ImageArray

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.

Image R G B

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

The Singular Value Decomposition Theorem

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

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

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

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

A diagonal matrix is ordered if where

An Example of The Singular Value Decomposition Theorem

Let R be where R has rank 2

The singular value decom- position of R is:

Which we recognize as

With fiddling, this is equal to

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

This can be seen in the following Mathematica animation

How does SVD apply to Image Compression ?

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

By the SVD Theorem

So SVD says: But Image Compression???

Approximate R with and continue on to

In particular, R r = R We have that

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

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

This is illustrated in the following example Consider approximations of RR

Again

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

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

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

10 iterations gives 71.3%

100% 5.9% Original Approximation

So what are the drawbacks? Computing Power -

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

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

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

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

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) to confirm that the world is safe once more for the American people. (You are also free to call and chat;-)

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

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

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

Dusty Wilson Graduate Student Instructor Western Washington University (360)

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