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
- 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