M-band 2D dual-tree (Hilbert) wavelet multicomponent image denoising

The toolbox implements a parametric nonlinear estimator that generalizes several wavelet shrinkage denoising methods. Dedicated to additive Gaussian noise, it adopts a multivariate statistical approach to take into account both the spatial and the inter-component correlations existing between the different wavelet subbands, using a Stein Unbiased Risk Estimator (SURE) principle, which derives optimal parameters. The wavelet choice is a slightly redundant multi-band geometrical dual-wavelet frame. Experiments on multispectral remote sensing images outperform conventional wavelet denoising techniques (including curvelets). Since they are based on MIMO filter banks (multi-input, multi-ooutput), in a mullti-band  fashion,, we can called they MIMOlets quite safely. The dual-tree wavelet consists in two directional wavelet trees, diisplayed below for a 4-band filter:

4-band directional dual-tree wavelets

The set of wavelet functions implements:

• several dual-tree M-band wavelet transforms from: Image analysis using a dual-tree M-band wavelet transform, IEEE TRANSACTIONS ON IMAGE PROCESSING, 2006, 
• a neighborhood choice from: Noise covariance properties in dual-tree wavelet decompositions, IEEE TRANSACTIONS ON INFORMATION THEORY, 2007,  
• the non-linear Stein estimator: A nonlinear Stein-based estimator for multichannel image denoising, IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2008,
• as a complement, relative merits of different directional 2D wavelets are detailed in: A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity, SIGNAL PROCESSING, 2011, 

The demonstration script is Init_Demo.m, and the functions for M-band dual-tree wavelets are provided in the directory TOOLBOX_DTMband_solo. For instance, the clean multispectral image (port of Tunis, only one channel):

The (very) noisy version:

The denoised one:

Marseille: from the Vieux Port with hole

Following a quick trip to Marseille (my first one) for a discussion on directiona/geometric wavelets (slides) at the séminaire signal - apprentissage in LATP (Laboratoire d’Analyse Topologie et Probabilités), a rough pinhole view of the "Vieux Port", on the bridge to the brand new museum MUCEM. Have to get back for a broader overview of the city.

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