Hypercomplex Wavelets
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* [http://www-sigproc.eng.cam.ac.uk/~ngk/ Nick Kingsbury's homepage] | * [http://www-sigproc.eng.cam.ac.uk/~ngk/ Nick Kingsbury's homepage] | ||
* [http://home.comcast.net/~cmdaven/hyprcplx.htm Clyde Davenport's page on commutative hypercomplex mathematics] | * [http://home.comcast.net/~cmdaven/hyprcplx.htm Clyde Davenport's page on commutative hypercomplex mathematics] | ||
+ | * [http://digitalcommons.shu.ac.uk/mmvl_papers/1/ MMVL paper] | ||
[[Category:Projects]] | [[Category:Projects]] | ||
[[Category:Nanorobotics]] | [[Category:Nanorobotics]] |
Revision as of 18:29, 1 October 2007
Contents |
Introduction
Complex wavelets are superior to real-valued wavelets because they are nearly shift-invariant. Complex wavelets yield amplitude-phase information in a similar way as the Fourier transform does. In contrast to the Fourier transform, wavelets allow to analyse the signal locally and thus can be applied to signals with a non-stationary statistic (such as images of a natural scene). In the same way as a one-dimensional signal requires complex numbers to represent the local structure of the signal, two-dimensional signals require hypercomplex numbers. Kingsbury has developed the Dual-Tree Hypercomplex Wavelet Transform (DHWT) which allows to recursively decompose a two-dimensional image.
Implementation
The implementation makes use of Selesnick's Hilbert transform pairs of wavelet bases. The implementation also requires the Ruby-extension HornetsEye which offers fast operations for n-dimensional arrays and hypercomplex numbers as element-types.
The source file can be downloaded here: kingsbury.rb.
Wavelet Editor
An editor for visualising linear combinations of wavelets was implemented. The code requires qt4-qtruby, and the code above. Here is the source code:
- Ruby program: waveletEdit.rb
- Qt4 design: waveletEdit.ui
You need to compile the design file using rbuic4 like this:
rbuic4 waveletEdit.ui > ui_waveletEdit.rb