Tuesday, March 27. 2007
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,seems to be a compact group whenever A is real-valued invertible d×d. d=1 makes
and
a compact group.
- What is a discrete (sub-)group? Why is the dual of a compact group discrete?
- Why is
for
and a real-valued invertible d×d-matrix A?
Maybe becausefor invertible square matrices and
, and clearly
.
- How can I implement the tensor product of separable lattices to get non-separable lattices like the quincunx lattice or a hexagonal lattice?
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