Revision as of 13:48, 30 September 2007

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.

Under construction ...

Wavelet Editor

An editor for visualising linear combinations of wavelets was implemented. The code requires qt4-qtruby, HornetsEye, and the code above. See here for more information. 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

Under construction ...

See Also

• HornetsEye
• Complex Wavelet Filters

External Links

• Ivan Selesnick's homepage
• Nick Kingsbury's homepage
• Clyde Davenport's page on commutative hypercomplex mathematics