Master's Thesis - 6

I tell a little about what I did with Octave during the last days. This is the first part and it's about creating a video of a modulated Gaussian. The second part is about doing FFT's of time-frequency shifted Gaussians.

I figured out how to merge images into a video stream. I rely on Linux tools rather than trying to do it from within Octave, because I don't know whether libraries for creating videos are somehow available there. However, I wrote an M-file which creates images of a modulated 2D-Gaussian. The 2D-frequency changes by going on a spiral in the complex plane starting from the origin. This yields a modulation which is rotating while the line-pattern gets narrower:

I merged the bunch of 900 images to a video by using MEncoder, the open-source encoder from the MPlayer suite. I did it this way:

I know, the usage of a proprietary container format is a contradiction to using open-source tools, but I only want to care about such details later.

I won't publish the M-file here, as it's too basic.

HGFei is leaving Austria in summer 2007 to go to Australia until Christmas 2007. I’d have to have my thesis finished by the end of June 2007. He asked me to give a schedule to be able to finish it by the end of the current term. Of course, regarding my scholarship, I should have finished it by July 2007 anyway. But I currently don’t know what the schedule could look like. I don’t know what my thesis could or should look like in the end, and therefore I don’t know what path to take to go there, and therefore I don’t really know my next steps. He said I shouldn’t be too ambitious and should produce a spate of demos now. But I don’t actually know what these should demonstrate. Maybe that kind of panic is somehow natural, but I’m not comfortable with that. Sure, I have time now for doing my thesis, and I really have to make use of it. I still have plenty of literature to read.

What I had done in February was starting to write the introductory chapter with LaTeX. That was a step what had to be done anyway, I can concentrate mostly on the content now. I wrote some pages about the foundations of TF-analysis, including some more side notes that are usually not mentioned in the other Master’s theses at my institute. As one of the next things I want to get familiar with the computing language of MATLAB/Octave. During the last days I did some experiments with FFT2, and I might write down some results here in the blog. Now I know how to do different things in Octave, I produced some pictures and rendered them to videos. This might provide some munition for doing 2D-Gabor Analysis.

Now I should look how to finally do image processing using Gabor Analysis methods. In our last conversation at the end of December 2006, HGFei told me some results on how to use the 1D-algorithms for higher dimensions, because their tools actually only provide algorithms for 1D-signals. I’d be the first to actually use that in the 2D-case. To understand how this is obtained I’d have to be familiar with the basics of GA on groups—a rather abstract approach. And mapping 1D to 2D implies that I can do it in the 1D case also. At least, if I can’t provide a schedule and don’t know how to be done by the end of this term, I know what my next steps are.

He also mentioned that I simply should do some basic experminents such as thresholding, i.e., doing reconstructions by leaving out coefficients with low contribution; this is a leering to the task of image compression.

Tomorrow, the summer term is starting, and I’ll attend HGFei’s project seminar on numerical harmonic analysis, where we’ll have to approach different problems using MATLAB.

(Q34+44) If an operator S can be written as S = TT, then we abbreviate S^1/2 := T. In the mapping sequence x → Tx → TTx, clearly S^1/2 “lies between” S⁰ and S¹. We know that (AB)⁻¹ = B⁻¹A⁻¹, so far existant, and so S⁻¹ = T⁻¹T⁻¹. So we can abbreviate S^−1/2 := T⁻¹, and (S^1/2)⁻¹ = S^−1/2. It is now clear that S^−1/2SS^−1/2 = T⁻¹TTT⁻¹ = Id.

It remains to show that S^−1/2 is a limit of polynomials in S⁻¹ in the strong operator topology.

I stop working at SOL4-IT at the end of January 2007. From that time on I can work 100% on my Master thesis! But I don’t have to live from my savings: I’ll apply for a final degree scholarship (Studienabschluss-Stipendium). For 6 months I will (or could) be promoted with €1000 every month, with an additional (unpaid) timeframe of 6 months until I’d have to pay the amount back if I hadn’t finished by that time. I’ll not be allowed to hold any job during that phase. That gives me everything I need for a scientific work.

(Q26) Meanwhile, I know that a convolution of one function with another function means that will be “-smeared”, and vice versa. I visualized some convolutions using Octave. I was prepared to implement for two vectors, but then stepped over HGFei’s conv2fei which makes the surprising use of the identity ; I wouldn’t have come to the idea to do it this way, you only have to multiply two FFT’s and then do the inverse FFT.

First I convolved a square with a small Gaussian—this is nothing less than a Gaussian blur:

Note that the blur would already occur even if the second object were solid! Look at the square convolved with itself:

Then I watched how these bars convolve with each other:

Notice how the blurness shrinks when the two objects can’t overlap much.

66. What are Fourier multipliers?
67. What are Frechet spaces? Why are and not Banach, but only Frechet?
68. Is there a difference between and in HGFei’s notation?
69. Why is ? (There’s a relation to Q2.)

(Q15) I’m gonna show you some cool Gabor atoms on . First, the function , as described previously, indeed looks like this:

See how the value of the frequency changes with the length of , and the direction with the orientation of , just as described previously.
In the STFT, these frequencies get reduced locally by an “envelope function”. One could take the Gaussian window to achieve this:

And now these are the modulated Gaussians, whose set of translates across forms the building blocks for Gabor analysis on :

63. What is ?
64. What are singular matrices?
65. What is the condition number of a matrix?

(Continued from questions 1-62)