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:

WITS: MIMOlet wavelets

There might be a feeling of complexity in the burgeoning world of wavelets. A relatively general multiresolution framework, entitled MIMOlet (mee-moh-let), has been proposed by Guillaume Flanders et al, with a french-dutch-belgian team originating form Lille university. This scheme includes most of the known forms of discrete filter banks, critical or overcomplete, separable or directional. It borrows its name from MIMO (multi-input, multi-output) systems. More details to be found at this page (broken link at the time of writing). Its extension to the sphere (Booldelill et al.) is in progress, as a generalization to needlets (see for instance here).