Master's Thesis

(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?

In question 15 I want to make a picture how looks like for . I want to take at least two-dimensional vectors, as the inner product is too trivial otherwise. But then is a function on , which is rather difficult to plot. So I want to fix and plot , which is then a function on with values in and therefore easy to plot. As the inner product is , it is in the 2-dimensional case a simple plane-equation ; of course, the inner product is linear and goes through the origin! It cuts the zero-plane in an angle which is dependent on the orientation of the fixed : The zero-line evolves where the vectors are orthogonal to the fixed , i.e., .

This zero-line will now change its angle on the zero-plane while varying the fixed vector . This gives a picture how it behaves as a whole. In Octave, I set

[xx yy]=meshgrid(linspace(-2,2,50));

and then made a plot corresponding to by

mesh(xx+yy)

The resulting 3D-plot shows a boring inclined plane. To better see the location and orientation of the zero-line, I plot the absolute values:

mesh(abs(xx+yy))

To see the orientation even better, I chose

imagesc(abs(xx+yy))
imagesc(abs(xx+2*yy))
imagesc(abs(-xx+4*yy))
corresponding to the further vectors and . This is what I get:

The turning around of that plane now corresponds to the behavior of the mentioned inner product function on . (Beware that the absolute values used in the plots hide the asymmetry of the inner product function!) The values are always real and go into the exponent of the exponential function. then “rotates fastest” as runs along to a fixed , and is constant = 1 if it runs orthogonally; Beware that there’s a difference whether it is constant = 1 or if it reaches that value again and again. So, in the integral of the FT, the exponential function simply covers all frequencies via exhausting all values of , and the FT is then a function in , which takes out a single orientation of those frequencies; As already mentioned some time ago, the term “frequency” isn’t useful anymore in higher dimensions. For two-dimensional signals, i.e. images, a “frequency” can be interpreted as a pattern of lines: The narrower the pattern, the higher the frequency is. The difference to one-dimensional signals is, that those frequencies may now have an angle. It is not enough to just look at a single frequency of that line-pattern, but also to all their orientations! Only with using all orientations of patterns, an image can be analyzed according to the contained frequencies. The 2D-FT of an image therefore tells you what patterns occur in what orientation. To be more precise, you select the frequency by the length and the orientation by its angle , because the inner product as a function in grows faster with growing length of . For me, this is now a milestone in understanding the 2D-FT.

When I started on my way to my Masters degree seriously in January 2006, several questions began to arise in the course of time, and they continue to do so. While there were some very fundamental questions at the beginning (because my last contact to FA and basic analysis had been quite some time ago), they matured to some more advanced questions today. Here, I give a list of them, in their order of appearance. The answers, however, and as far as they have already been provided anyway, did not come up in the same order. Some questions are so general that they are to be understood as a challenge to look into the books and get familiar with it.

This list will continue to grow in separate entries. Also, for the sake of documentation, currently missing answers will only be given separately.

Here I give my personal reading approach to get familiar with the topics which the NuHAG is working on. This is just one way to get started, and I will mention some pros and cons I noticed. There is always some redundancy, but in my eyes that was of great worth.

A good way to get some first impressions is to read A Short Introduction to Gabor Analysis by Thomas Strohmer, which is the introductory chapter of [Feichtinger/Strohmer] and is also available online. You could also read the introductory chapters of [Blatter], which will make clear what the central ideas are in general. After having read that, you also want to understand it all.

I’m starting with the last chapter in Christensen and plan to have finished it on Nov. 5th. I still have some pages left in ch. 11 of Gröchenig as well as the whole ch. 12+13; I plan to have finished them by the mid of November. Then there’s the whole time of the day available for selected chapters of HGFei/Strohmer and—primarily—the GA4IP-stuff.

I also got myself into the MATLAB/Octave toolboxes and went through the diary (i.e. the MATLAB logfile) of HGFei and the examples of P. Søndergaard. (BTW, Søndergaard will be in Vienna at Nov. 8th for a talk at NuHAG.) Tomorrow I’ll visit the IT-staff at the institute to get a mathematics/MATLAB computing account. This term, I’ll visit HGFei’s lecture on Selected Topics in Modern Analysis, which takes place Monday and Thursday 10-11 a.m. Although this hour is “in the middle” of my working time at SOL4-IT, I’ll have to be there regularly, at any cost.