Master's Thesis - 9

While being employed 40 hours/week I started to repeat basics in functional analysis in January 2006. In April I started to do some general reading on the subject of time-frequency analysis. I wanted to have the topic of my Master’s thesis set until May, but it took me until August to file it with the working title “Gabor Analysis for Image Processing”. The finish of my thesis had originally been targeted for Christmas 2006, but it soon was clear that it would also take the whole spring of 2007.

With May 2006 I reduced my working times to 26 hours/week, and I quit my occupation in January 2007 as I was granted a 6-month scholarship. Now I had time to do numerical experiments and to actually write my thesis. The scholarship ended by July 2007, and I hoped to have my thesis finished by September. The final work, entitled Foundations of Gabor Analysis for Image Processing, was printed in the mid of November and graded an A. I had my master exam on December 18th and finished my studies with distinction.

Some statistics: It took me 8 months to read up on time-frequency analysis (while being employed). I was in the official status of a graduand for 16 months. I authored my thesis within 9 months.

Today I had my Master exam. We were starting shortly after 11h, and at 11:56h I already had a talk with HGFei in his room. Although I had a few confusions during the “two” exams (functional analysis and measure theory), I ended up with two A’s, so it obviously wasn’t too critical. A funny thing was that K. Sigmund didn’t know what to ask me initially, because the last time he lectured measure theory was about six years ago. Nevertheless, I finished my studies with distinction and earned the academic degree Magister rerum naturae, and thus:

PROJECT ACCOMPLISHED.

HGFei graded my Master’s thesis with an A (“Sehr gut”, i.e. excellent). (Rumor has it that this doesn’t happen too often.) I thus officially make my Foundations of Gabor Analysis for Image Processing available in the web. The PDF contains hyperlinks, what resulted in a few imperfect pagebreaks, unfortunately. Due to a small supplement before the print the pure content raised to 95 pages, 107 pages including abstract and appendix, and 123 pages all in all.

By the way, a professionally printed and bound version of one’s thesis looks so good and so much different from one’s own test prints! The cover of mine is held in claret leather with a golden imprint of my name. The work took Janetschek more than one week, but the result was worth it. I got six pieces for €32 each.

I’m finished! “Foundations of Gabor Analysis for Image Processing” became the final title, 94 pages of pure content, 105 pages including abstract and appendix, and 121 pages all in all. Currently I’m waiting for the print shop to deliver a bid. Maybe I can hand it in for appraisal during the next week. I won’t publish it in the web before it is appraised.

I can now start preparing for the Master exam. Don’t expect to hear much from me during the next weeks.

I broke the sound barrier of 70 pages yesterday. I wrote some rather mathematical pages about that tensor products of frames for Hilbert spaces are frames for the (tensor) product space, and that the frame coefficients of a 2D product frame can be computed by applying the individual frame matrices to the left and right of an image. Next, I want to show Gabor thresholding for a separable 2D window on a fully separable 4D lattice. I should also mention that computational speed can be increased when using the STFT code.

The next section will look at separable windows on partially non-separable 4D lattices that are still given as a product Λ=Λ₁×Λ₂. Not much will change here, I can show how a dual looks like and do again some thresholding. I want to start with that two days from now.

Then I’ll come to non-separable windows and show how this case can be reduced to the 1D case if the lattice is fully separable and width and height of the image are relatively prime. I’ll again show a dual and some thresholding. This should be done over the weekend.

Then there will be non-separable windows on partially and truly non-separable lattices. I’ll just use Prinz’ code to show a dual and demonstrate thresholding. I don’t know whether I will show computations on a truly non-separable lattice. Maybe I’ll just mention the idea of image segmentation or down-/upsampling.

So, maybe I can start doing a roundup at Oct. 15th and hand my thesis out to HGFei during that week, as my holidays are starting at Oct. 20th and I’ll return at Nov. 3rd. In the week after my holidays I have time for cleaning it up (minor corrections/additions, preface, appendix), and in the week between Nov. 12th and Nov. 16th I want to hand it out to the print shop. The Master exam is to be scheduled in the mid of December, maybe in the week before the winter family days. This. Must. Work.

I finished Chapter 3 today, it spans 13 pages, and so I start with Chapter 5 on page 59 now. According to my time plan, I wanted page 59 to be done on Sunday, so I’m just 3 days behind. In the third chapter I mentioned factorizations of the Gabor matrices and showed what general sampling sets and the dual windows on those lattices look like, and that the duals might have a significant imaginary part. Chapter 5 will now explain Gabor expansions of images and will proceed with increasing difficulty:

• Separable atom, fully separable 4D PF-lattice (into 1D subgroups)
• Separable atom, partially non-separable PF-lattice (i.e., product of non-separable 2D lattices)
• Non-separable atom, fully separable 4D PF-lattice
• Non-separable atom on general 4D PF-lattices (only mentioned shortly)

I expect to write at least 20 pages for that chapter, they might become 30, and so I should have finished everything by the mid of October, where I want to go on a 2-week holiday while HGFei reads through my thesis. After that I just want to do cosmetic changes (minor corrections, appendix, preface) and have it printed by the beginning of November. About one month later I want to take the Master exam.

HGFei currently wants me to deal with the topic of down-/upsampling, but I’m not sure yet to what extent I’ll follow that.

I managed to finish Chapter 4 today and will continue with Chapter 3 about finite discrete GA tomorrow. Sure, this might not be done within one week as hoped, and one could think that I’m out of my time plan, but I included the sections about the 4D STFT of 2D signals and (non-)separable 2D windows, what I originally had intended for Chapter 5. The chapter spans 21 pages, and so I’m effectively on page 45 of my thesis.