Hypercomplex Wavelets
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
Under construction ...
Wavelet Editor
An editor for visualising linear combinations of wavelets was implemented.
Under construction ...
