2-D Complex Wavelet Decomposition

Your browser is not Java enabled. Instead you can download and unzip the software and execute the applets locally through the provided batch scripts: click here. Note that this still requires a local Java installation.

Instructions

This application shows the Dual-Tree Complex Wavelet Transform (DT-CWT) decomposition of a given input image. The DT-CWT improves upon the Discrete Wavelet Transform (DWT), by offering a better directional selectivity (6 orientation subbands) and approximately shift invariance for the complex coefficient magnitudes. This is beneficial in many practical applications.

The input image can either be an pre-loaded image (the classical zone plate image, Shepp-Logan phantom, circle) or loaded from a file. Please be aware that this last feature may not work in the webbrowser due to security restrictions as hard disk access is required. On the left side, you can select:

The wavelet filter used for the first (i.e. finest) scale of the transform.
- Farras: orthogonal wavelet filters designed by dr. Farras Abdelnour.
- db1-db16: Daubechies wavelet filters.
- sym1-sym16: Symlet wavelet filters.
- coif1-coif5: Coiflet wavelet filters.
Important: when Daubechies filters/Symlets/Coiflets are used for the first scale, the filters in the dual tree are typically shifted one sample from the filters in the primary tree (see [2]). Unfortunately this results in a reduced analyticity (and directional selectivity) of the complex basis functions for the first scale. In [3], we propose a filter design technique to solve this problem. The results here make use of this design technique. Also see here for more results.
A wavelet filter for all subsequent scales. A description the wavelet families in this application is given here.

On the right side, you can choose the number of scales of the transform and what you want to display: the real parts, the imaginary parts or the magnitudes of the complex coefficients.

References

[1] 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, pp. 234-253.
[2] I. W. Selesnick, R. G. Baraniuk, and N. G. Kingsbury: "The Dual-Tree Complex Wavelet Transform," IEEE Signal Processing Magazine, vol 22, no 6, pp 123-151, Nov. 2005.
[3] B. Goossens, A. Pizurica and W. Philips, "A Filter Design Technique for Improving the Directional Selectivity of the First Scale of the Dual-Tree Complex Wavelet Transform," submitted to IEEE Int. Conf. Image Processing (ICIP2009), Cairo, Egypt