Analyse en ondelettes M-bandes en arbre dual; application à la restauration d'images

Abstract : We propose in this thesis a new type of wavelet transform, the dual-tree M-band wavelet decomposition, that has some unique geometrical features. This type of wavelet decomposition provides a local, multi-scale, directional analysis of images and can be applied to image restoration. There is a popular need for tools that improve the representation of geometric information like textures and edges, and yet preserve them during processing. The work that is presented here is an extension to the M-band case of the previously obtained results by N. Kingsbury and I. Selesnick for the dyadic case. The dual-tree decompositions are shown to be quasi shift-invariant and offer directional selectivity. Firstly, we address the conditions satisfied by the primal and dual filter banks for them to be used in the analysis and synthesis of the processed data. Secondly, we investigate the pre-processing stage that has to be applied to the discrete data. Due to the decomposition redundancy of the dual-tree decomposition (typically 2 in the real case and 4 in the complex case), several reconstructions are possible. An optimal pseudo-inverse based frame reconstruction is proposed in order to overcome this problem. These new transforms have also been generalized to the biorthogonal and complex cases. A study of the statistical properties of the M-band dual-tree coefficients of a widesense stationary random process was also necessary so that these analysis tools could be applied to image denoising. We first calculated the second order statistics of these coefficients and then investigated the influence of the pre-processing stage on correlation calculations. Asymptotic results on the correlations of a pair of primal/dual coefficients have been obtained. Cross-correlations between primal and dual wavelets play a major role in our study. We were able to propose closed-form expressions for some usual wavelet families and numerical simulations allowed us to validate our theoretical results as well as to evaluate the area of influence of the correlations. The effectiveness of our decompositions are demonstrated for image denoising and deconvolution applications. Concerning denoising problems, we are interested in examining two primary objectives: firstly, for mono-channel images, we show that M-band dual-tree wavelet decompositions bring a significant quality gain (both objective and subjective) in comparison with classical, as well as dyadic dual-tree wavelet decompositions. We then consider image denoising for the multichannel case, for which we build a new multivariate estimator based on Stein's principle. The estimator that is proposed allows an arbitrary neighborhood like spatial, inter-component and inter-scale. The problem of deconvolution was addressed under the context of variational methods. An iterative algorithm based on recently developed convex analysis tools is proposed. The followed approach allows to solve inverse problems associated with numerous probabilistic models and is applicable in the dual-tree M-band wavelet framework as well as for any frame-based representation.
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Caroline Chaux. Analyse en ondelettes M-bandes en arbre dual; application à la restauration d'images. Traitement du signal et de l'image [eess.SP]. Université de Marne la Vallée, 2006. Français. ⟨tel-00714292⟩

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