Raising questions, 70-73

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:

What are compact groups? What are locally compact groups? What is a compactly generated group? What is an open compact subgroup?
As an example, $\bbR^d/A\bbZ^d$ seems to be a compact group whenever A is real-valued invertible d×d. d=1 makes $A = \alpha\in\bbR\!\setminus\!\{0\}$ and $\bbR/\alpha\bbZ$ a compact group.
What is a discrete (sub-)group? Why is the dual of a compact group discrete?
Why is $\inner{Ak}{(\tp{A})^{-1}\ell} \in\bbZ$ for $k,\ell\in\bbZ^d$ and a real-valued invertible d×d-matrix A?
Maybe because $(\tp{A})^{-1}=\tp{(A^{-1})}$ for invertible square matrices and $\inner{Ax}{y}=\inner{x}{\tp{A}y}$ , and clearly $\inner{k}{\ell} \in\bbZ$ .
How can I implement the tensor product of separable lattices to get non-separable lattices like the quincunx lattice or a hexagonal lattice?

Posted by Stephan Paukner in Master's Thesis at 10:29 | Comments (0) | Trackback (1)

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Logbook of Stephan Paukner on : Images to vectors and back

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&times;q-images where p and q are relativel

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