Entries from Stephan Paukner

Given an image im of size p×q with gcd(p,q)=1.

• Find a number s that is not divided by the divisors of N=pq.
• Index vector, method 1:
• idx = mod(1:s:N*s, N);
• nil = find(idx==0); idx(nil)=N;
• Index vector, method 2 (preferred):
• idx = mod((1:s:N*s)-1, N)+1;
• imv(idx) = im(:);

Reconstruction from vector of length N=pq:

• rim = reshape(imv(idx), p, q);

Let’s try this with our zebra. In my previous tries I simply took p=479 and q=480 to have gcd(479,480)=1, but 479 is a prime number and I therefore have no possibility to divide it by some step size. For GA I have to lay some kind of lattice over the image. So there is actually a second condition forbidding p and q to be prime. I take p=473 instead, it has 11 and 43 as divisors.

As one can see, N has many divisors, but e.g. 19 is not one of them. We can’t take e.g. 9 although it doesn’t occur either, as it is divided by 3, which is a divisor of N. Taking such a wrong step size will lead one back to the first pixel too early and one can never span the whole image.

And now the other way round:

Indeed looks like our zebra!

By the way, another try of comparing the FFT of the vector with the FFT2 of the image shows again that one cannot get the FFT2 by doing an FFT of the vector. This is because the line-patterns in the image might not be preserved under that transform, as neighboring pixels might not become neighbors in the resulting vector.

I found a quicker way to compute the index matrix I mentioned: So nothing with 58 seconds anymore.

Regarding the previous questions I noticed that GA on LCA groups is too general for my concerns and that the study of GA on finite abelian groups is enough.

In a certain paper HGFei published a result that for p×q-images where p and q are relatively prime there is an isomorphism to a vector of length N=pq. The theory considers the groups and where . One explicit mapping from the matrix indices to the corresponding vector indices is given by

where α and β are integers chosen such that

My 480×480 zebra-image doesn’t have a height and width that are relatively prime, so I simply drop one pixel of height:

Just for a first test I set α=p and β=q and define a primitive mapping function. (I’ll have to find a quicker transform, as in my tests the switching takes too long. But this could be due to the actually large test image. In addition, I currently only manage to do the backward step by storing the indices in an index matrix.)

I have to return N instead of 0 to have a correct index value. Although the mapping is actually an isomorphism I didn’t know how to go back to the tuple (j,k), so I store the indices in an index matrix while building the image vector:

58 seconds!? However, the image looks like this as a vector:

Die Köchin befindet sich in Schmackhaft.
Der Konditor befindet sich in Nahrhaft.
Der Forscher befindet sich in Rätselhaft.
Der Skeptiker befindet sich in Zweifelhaft.
Das kleine Mädchen befindet sich in Schreckhaft.

We had the fashion of the sixties and the seventies coming back. After that, everyone was already fearfully awaiting the comeback of the eighties. And here we are! Doris always says that this comeback can only be intended for people who were too young in the eighties and therefore had no chance yet to feel the terrifying clothing style of that time on their own skin.

I noticed that I’ll have to find a representation of images as vectors, and not as matrices as usual. Because otherwise I won’t be able to describe linear operations as matrices on vectors. To be able to do things like an SVD or a look at eigenvalues or eigenvectors, I’ll have to evaluate matrices on vectors. For building lattices or Gabor systems I’ll have to try to use the algorithms which have been developed for 1D-signals. This leads me to Gabor Analysis on locally compact abelian groups, but group theory is a little bit away from ordinary signal analysis. Reading first intros, I asked myself:

70. What are compact groups? What are locally compact groups? What is a compactly generated group? What is an open compact subgroup?
    As an example, seems to be a compact group whenever A is real-valued invertible d×d. d=1 makes and a compact group.
71. What is a discrete (sub-)group? Why is the dual of a compact group discrete?
72. Why is for and a real-valued invertible d×d-matrix A?
    Maybe because for invertible square matrices and , and clearly .
73. How can I implement the tensor product of separable lattices to get non-separable lattices like the quincunx lattice or a hexagonal lattice?

Ich kenne jemanden, der/die darauf spekuliert, dass das regelmäßige Zahlen der Strafen fürs Schwarzfahren in Summe billiger kommt als die Jahreskarte der Wiener Linien. Diese kostet bei Teilzahlung €417/Jahr, und einmal beim Schwarzfahren erwischt werden kostet €60. Damit darf man höchstens sechs Mal im Jahr erwischt werden. Ich bin aber seit Jänner 2007 schon vier Mal kontrolliert worden. Wenn der Trend so weitergeht, bin ich bis Jahresende 16 Mal kontrolliert worden und hätte €960 hingeblättert. Die erstgenannte Spekulation kann also nur funktionieren wenn er/sie nur selten und/oder nur zu “exotischen” Uhrzeiten fährt.