# Basic introduction to Gabor Transformation

It is very challenging to use wordpress Latex plugin to write mathematical article in wordpress. It is not impossible but it is very challenging and realized it in my attempt to write the explanation of Schrodinger equation.

Looking for how to apply Gabor transformation/filters in credit risk management. My attempt to explain the basics of Gabor transformation and step wise proof that Gabor functions holds the minimum uncertainty in the joint time frequency domains.  Please click the following link for the details.

# Theory (conjecture) of Object Symmetry

Like NDPL, it is another incomplete work of Praba Siva. Being curious is a strength and also it is a weakness. Strength is, you will learn something new all the time and the weakness is, once you know the basics, you are not motivated to go to end of it. I belong to that category. Once up on a time, I used to spend 15 hours a day thinking about the Praba’s theory (conjecture) of object symmetry. I still remember, at Toronto airport, after having one or two molson, I had thought about how to mathematically prove it by writing numerous equation in a napkin. I believe that napkin is still hanging in my home office.

Let me state the theory (conjecture) . That is, what is Praba’s theory (conjecture) of object symmetry. Theory (conjecture) of object symmetry states that every objects (signals) has a perfect symmetry with respect to a finite set of line of axes.

It may not be clear & obvious what the theory states. To simplify the statement, look into the natural objects like stone, trees, flowers, insects, humans, you split them with respect to different set of line of axes, you would see a perfect symmetry. Most of the man made object obey this theory obviously and all of them should obey mathematically. In another simpler words, take a tree and break it into smaller pieces such way that each smaller piece has a perfect symmetry with respect to different line of axes. Image is a signal. It applies to the other types of signals like voice. What ever language you speak, English or Tamil, that is a signal which can be found a perfect symmetry by finding the smaller line of axes. The bigger (or major) line of axes more the application you could find. The smaller the line of axes, lesser practical applications since it is not worth it.

I spent almost 8-12 months proving this theory using Harmonic analysis. I could not do it. It must be because I do not fully understand the concept of harmonic analysis.

I assumed that the conjecture (theory with out proof) is true and took an empirical approach. I applied it for pose estimation problem. That is, for improving the car safety, identification of the driver position is very important. If the driver is sleeping nor paying full attention to driving, the safety system should warn them that they are not fully paying attention to avoid any potential accidents.

The national high way traffic safety administration (http://www.nhtsa.dot.gov) concluded that the car/truck driver position can be classified into 6 different set of position. If the driver is not focusing on one position (ie. position 3, looking forward ) for more than significant time (15 seconds), then the system should alarm the driver. I applied the theory (conjecture) of object symmetry principle to this problem and compared it with other solutions. Other solutions available (like Eigen faces, Neural network, Gabor Jets) are not practical since it takes at least 30 seconds to classify each frame of the video. Where as, by applying the theory of object symmetry, it took only 1-2 seconds per image.

Empirically, the theory is proven and it works. I studied cognitive psychology for a brief time to see how the human mind does the discrimination naturally and simply. There are some theory in cognitive psychology believes that human brain does discrimination fast by identifying the symmetry and corresponding line of axes. I also understood, when a human being is considered as beautiful, they have a symmetry with major line of axes. ie. take a women who is considered as most beautiful. You could see her face (other parts) have a symmetry with major line of axes.

I believe this theory could be applied to credit risk management, signal processing, scene analysis, image analysis, astronomy, etc.

I lost interest since I know it works and now I’m interested in something I do not know. This could easily be an another Ph.d thesis in the field of computer engineering or applied mathematics. As I understand Princeton Mathematics department works on the symmetry problems.

If any one working on this subject and want to work together to mathematically prove this.. I’m all for it. I can provide all the work I have done so far.