The Complex Wavelet Browser

This tool allows you to inspect many properties of the complex wavelets (spatial localization, frequency localization, directional sensitivity).

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.

• The wavelet filter used for the first (i.e. finest) scale of the transform (also see here)
  • 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 [4], we propose a filter design technique to solve this problem. The results here make use of this design technique. Without this modification (choose e.g. Farras filters), several artifacts appear in the results, such sudden transitions at +/- 45° and orientation aliasing. With our proposed solution, these effects are greatly reduced.
• A wavelet filter for all other scales. Currently supported are the 6-tap Q-Shift wavelet filter from Nick Kingsbury [2] and the Thiran-based wavelet filters from Ivan Selesnick [3].

• Psi_g is the real part of the complex wavelet, Psi_h is the imaginary part.
• Notes: support for the Q-Shift design method from Kingsbury is currently in progress.

References

• [1]  I. Daubechies, "Ten Lectures on Wavelets," CBMS-NSF Lecture Notes nr. 61, SIAM, 1992.
• [2]  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.
• [3]  I. W. Selesnick. "The design of approximate Hilbert transform pairs of wavelet bases,"
  IEEE Trans. on Signal Processing, 50(5):1144-1152, May 2002.
• [4]  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