Hypercomplex Wavelets
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signal locally and thus can be applied to signals with a non-stationary statistic (such as images of a natural scene). In the same | 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 | way as a one-dimensional signal requires complex numbers to represent the local structure of the signal, two-dimensional signals | ||
| − | require hypercomplex numbers. [http://www-sigproc.eng.cam.ac.uk/~ngk/ Kingsbury] has developed the '''Dual-Tree Complex Wavelet Transform''' which allows to recursively decompose a two-dimensional image. | + | require hypercomplex numbers. [http://www-sigproc.eng.cam.ac.uk/~ngk/ Kingsbury] has developed the '''Dual-Tree Complex Wavelet Transform''' which allows to recursively decompose a two-dimensional image. Analogous to one-dimensional analysis requiring complex values, Kingsbury's wavelets require hypercomplex values (4-valued complex number). |
=Implementation= | =Implementation= | ||
Revision as of 17:44, 13 November 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 Complex Wavelet Transform which allows to recursively decompose a two-dimensional image. Analogous to one-dimensional analysis requiring complex values, Kingsbury's wavelets require hypercomplex values (4-valued complex number).
Implementation
HornetsEye now contains an implementation of the Dual-Tree Complex Wavelet Transform. The implementation makes use of 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
See Also
External Links
- Ivan Selesnick's homepage
- Nick Kingsbury's homepage
- N G Kingsbury: Complex wavelets for shift invariant analysis and filtering of signals, Journal of Applied and Computational Harmonic Analysis, vol 10, no 3, May 2001
- Thomas Bülow: Hypercomplex Spectral Signal Representations for Image Processing and Analysis, PhD thesis, 1999
- Clyde Davenport's page on commutative hypercomplex mathematics
- J. Wedekind, B. Amavasai, K. Dutton: Steerable Filters Generated With The Hypercomplex Dual-Tree Wavelet Transform (to be published in ICSPC07 proceedings)
- Hypercomplex wavelet example
