Estimation of the local dominant edge orientation using the Dual-Tree Complex wavelet transform

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Instructions

From the output of the Dual-Tree Complex Wavelet Transform, the local dominant edge orientation can be computed, based on the same principles as in Freeman & Adelson [4]. A brief discussion on this topic for the Steerable Pyramid transform can also be found here. Depending on the selected color map, each color (or gray shade) corresponds to an orientation angle.

The input image can either be an internally preloaded image (the classical zone plate image, Shepp-Logan phantom, circle) or loaded from a file. Please be aware that this last feature does 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. 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 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," in Proc. of the IEEE Int. Conf. Image Processing (ICIP2009), Nov. 7-11, Cairo, Egypt, p. 3805-3808
[4] W. T. Freeman and E. H. Adelson, "The design and use of steerable filters", IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, no. 9, pp. 891--906, 1991.