A real duet - On dual-tree wavelets (and a Matlab toolbox)

True pieces of art elude science (left picture borrowed from Flickr). Anyway, dual-tree wavelets form a real duet in Hilbert transform type harmony. But the true deal comes from some software implementing this possibly complex wavelet decomposition.

Version 1.2 of the M-band dual-tree wavelet Matlab toolbox has been released. It is available from Caroline Chaux webpage or directly from the link:
http://www-syscom.univ-mlv.fr/~chaux/toolbox/TOOLBOX_DTMband1D_v1.2.zip

The toolbox implements several 2-band and M-band wavelets (e.g. Meyer 2, 3 and 4 bands, Haar, Shannon etc.) in the Hilbert transform based dual-tree wavelet framework. Theoretical results, applications and comparisons may be found in the following papers from
http://www-syscom.univ-mlv.fr/~chaux/publications.html

Image Analysis Using a Dual-Tree M-Band Wavelet Transform
IEEE Transactions on Image Processing, Vol. 15, No.8, Aug. 2006, p. 2397-2412

Noise Covariance Properties in Dual-Tree Wavelet Decompositions
IEEE Transactions on Information Theory, Vol. 53, No. 12, Dec. 2007, pp. 4680-4700.

A Nonlinear Stein Based Estimator for Multichannel Image Denoising
IEEE Transactions on Signal Processing, Vol. 56, No. 8, Aug. 2008, pp. 3855-3870.

Cochlear Auxin - Congratulations

Caroline Chaux PhD thesis has been awarded "best thesis in signal and image processing" by the EEA Club, on Tuesday 6th May 2008. The EEA Club is a 40 year old association that gathers teachers and researchers in Electrical and Control Engineering, founding member and part of the European Association for Education in Electrical and Information Engineering (EAEEIE). The official ceremony takes place in Saint-Etienne, France, at the 48ème congrès du Club EEA (www.istase.fr/eea2008/) from May 28 to May 30. Caroline Chaux also received the best student paper award at ICASSP 2005.

The PhD thesis is entitled "Analyse en ondelettes M-bandes en arbre dual ; application à la restauration d’images" or "M-band dual tree wavelet analysis with application to image restoration". Dual tree wavelets represent a special breed of wavelet frames composed of the union of two (M-band) wavelet bases in phase quadrature or forming Hilbert pairs. The sine and cosine functions form a traditional example of Hilbert pairs. Hilbert pairs of wavelets enjoy approximate shift invariance and are low-cost redundant transforms for directional image analysis. They are related to the discrete complex wavelet transform. They have been used for instance in compression, texture analysis, denoising, watermarking... Many others applications are yet to come... and why not on compressed sensing?

A Matlab toolbox for 1-D M-band dual-tree wavelet transforms is made available. Related articles for further reading:

• Nonlinear Stein Based Estimator for Multichannel Image Denoising, IEEE Trans. on Signal Processing, 2008 (to appear),
• Noise covariance properties in Dual-Tree Wavelet Decompositions, IEEE Trans. on Information Theory, 2007
• Image Analysis Using a Dual-Tree M-Band Wavelet Transform, IEEE Trans. on Image Processing, 2006

with applications in the following:

• C. Chaux, P. L. Combettes, J.-C. Pesquet, and V. R. Wajs, "A variational formulation for frame-based inverse problems," Inverse Problems, vol. 23, pp. 1495-1518, June 2007.
• P. L. Combettes and J.-C. Pesquet, "A Douglas-Rachford splitting approach to nonsmooth convex variational signal recovery," IEEE Journal of Selected Topics in Signal Processing, vol. 1, no. 4, pp 564-574, December 2007.
• W. Ayadi, A. Benazza-Benyahia, S. Sevestre-Ghalila, "Quantification of microarchitecture bone junctions based on a dual tree M-band wavelet decomposition", Conf. Proc. IEEE Eng Med Biol Soc. pp. 5634-7; Volume: 2007