Revision as of 22:59, 18 October 2007

High- and low-frequency decomposition using the hypercomplex wavelet transform. The approximate shift-invariance leads to reduced aliasing

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 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

• HornetsEye
• Complex Wavelet Filters

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
• PhD thesis by Thomas Bülow
• 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