Hypercomplex Wavelets
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=Implementation= | =Implementation= | ||
[[HornetsEye]] now contains an implementation of the Dual-Tree Hypercomplex Wavelet Transform (DHWT). The implementation makes use of | [[HornetsEye]] now contains an implementation of the Dual-Tree Hypercomplex Wavelet Transform (DHWT). The implementation makes use of | ||
− | Selesnick's [[Complex Wavelet Filters|Hilbert transform pairs of wavelet bases]]. The wavelet transform makes use of [[HornetsEye]]'s [http://www.wedesoft.demon.co.uk/hornetseye-api/files/MultiArray-rb.html MultiArray] class. | + | Selesnick's [[Complex Wavelet Filters|Hilbert transform pairs of wavelet bases]]. The wavelet transform makes use of [[HornetsEye]]'s [http://www.wedesoft.demon.co.uk/hornetseye-api/files/MultiArray-rb.html MultiArray] class. Have a look at the |
+ | [http://www.wedesoft.demon.co.uk/hornetseye-api/files/hypercomplex-txt.html hypercomplex wavelet example] for more information. | ||
=Wavelet Editor= | =Wavelet Editor= | ||
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* [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] | * [http://digitalcommons.shu.ac.uk/mmvl_papers/1/ MMVL paper] | ||
+ | * [http://www.wedesoft.demon.co.uk/hornetseye-api/files/hypercomplex-txt.html Hypercomplex wavelet example] | ||
[[Category:Projects]] | [[Category:Projects]] | ||
[[Category:Nanorobotics]] | [[Category:Nanorobotics]] |
Revision as of 17:06, 2 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
HornetsEye now contains an implementation of the Dual-Tree Hypercomplex Wavelet Transform (DHWT). The implementation makes use of Selesnick's Hilbert transform pairs of wavelet bases. The wavelet transform makes use of HornetsEye's MultiArray class. Have a look at the hypercomplex wavelet example for more information.
Wavelet Editor
An editor for visualising linear combinations of wavelets was implemented. The code requires qt4-qtruby, and HornetsEye. 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