Master's Thesis

For the first time it really seems that I can keep my time plan. I try to reach page 32 today, and if I manage to keep writing one page per day, I will reach the half of the average thesis page number this week, and will indeed manage to be finished with the beginning of October. The theses at my institute indeed range over a number of about 70 pages.

I met with HGFei on Friday, and he said he can fully support my will to be finished soon. There won’t be a serious research or new results in my thesis (except for how the Gabor transform can be computed more easily for separable 2D atoms), but only a summary and visualization of the known results in the 2D case, where of course will be no surprises. Serious research is reserved for dissertations anyway. He wants the results to be suitable for talks or demos.

I proceeded with Chapter 4 about digital images and 2D frequencies rather than with Chapter 3 about finite discrete GA, as it isn’t that mathematical and I wanted to read more on the corresponding things before. I’ll soon have it finished, and maybe I’ll manage to finish the finite discrete GA in the first September week, what is 2 weeks from today.

I already got a little time panic, as I noticed that I wouldn’t make it within September. I’ll need the whole month of September, too. A problem is that Doris wanted to go on a two-week holiday. As she has to take a course anyway, we delayed it to the beginning of October. After that I only want to have to do cosmetic stuff, and no content anymore. I want to have it printed and handed in by the mid of October. And already at the mid or the end of November, there should be the Master exam. I want to be finished by my 30th birthday. This. Must. Work. I can’t afford needing more time, as January 2008 is the last possibility, otherwise I’d have to pay €6,000 back.

On the content: I looked at multi-window Gabor frames, and M. Dörfler’s PhD thesis is a good source. Maybe I can apply her results 1:1 to images, as natural images also dominate in the lower frequencies. The lower frequencies are “area-producing”, already determining the image to a very high degree. Therefore small windows are needed densely in the lower frequency area, resulting in a dense covering here, incorporating many different orientations, yielding only a rough coverage of the image space. The higher frequencies are “contrast-producing” or “border-producing”. Frequency coverage doesn’t have to be that dense here, the exact frequencies are not that interesting, but their location is important. So, a multi-window Gabor system could be of similar type in 2D as it had been taken for 1D music signals by Dörfler.

But multi-window systems shall appear rather late in my thesis. First there are a lot more other things to mention. I made a rough table of contents, giving me a good lead:

• Gabor Frames in as Section 2.3
• Chapter 3: Finite-dimensional discrete Gabor Analysis
• TF-matrices, stuff, frames in
• Gabor matrix
• GA on finite groups and general lattices
• Dual atom on general sampling sets
• Chapter 4: Frequency behavior of digital images (nix Gabor, nix shifts)
• Digital representation of images, RGB, matrix, values
• Understanding 2D frequencies, tensor product of 1D freq
• FFT2, low frequencies dominate, point to multi-windows
• Chapter 5: Image representation by Gabor expansion (Gabor stuff, experiments)
• Atoms can be separable or non-separable
• 2D PF-shifts of an atom, 4D position-freq space
• Separable atom: 2D-dual is tensor of 1D-duals
• Huge frame matrices, applying TF-matrices to image matrices for sep-atoms, proof
• Separable atom, separable lattice, separable dual
• Separable atom, non-separable lattices (1 dim only, both dims), dual
• Non-separable atom, isomorphism 1D-2D, various lattices, dual
• Unassigned
• Multi-window Gabor frames, similar to music signals
• Biological vision

Chapters 3 and 4 should be possible within August, and the rest will be in September.

It is currently completely open if there really will be some “real world” applications of GA to image processing, like serious deblurring, denoising or compression. I’d have to compare it to existing methods, actually. Currently I only plan to compare the various duals and do some thresholding of the Gabor coefficients. If there really will be some multi-window stuff, then maybe only for separable atoms, as I don’t know how I could check the frame quality otherwise.