2D Complex Wavelet Decomposition
Instructions
This application shows the DualTree Complex Wavelet Transform (DTCWT) decomposition
of a given input image. The DTCWT 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 preloaded image (the classical
zone plate image, SheppLogan 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.

db1db16: Daubechies wavelet filters.

sym1sym16: Symlet wavelet filters.

coif1coif5: 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. 234253.
 [2] I. W. Selesnick, R. G. Baraniuk, and N. G. Kingsbury: "The DualTree Complex Wavelet Transform," IEEE Signal Processing Magazine, vol 22, no 6, pp 123151, 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 DualTree Complex
Wavelet Transform," submitted to IEEE Int. Conf. Image Processing (ICIP2009), Cairo,
Egypt