Noise Covariance Properties in Dual-Tree Wavelet Decompositions

Dual-tree wavelet decompositions have recently gained much popularity, mainly due to their ability to provide an accurate directional analysis of images combined with a reduced redundancy. When the decomposition of a random process is performed -- which occurs in particular when an additive noise is corrupting the signal to be analyzed -- it is useful to characterize the statistical properties of the dual-tree wavelet coefficients of this process. As dual-tree decompositions constitute overcomplete frame expansions, correlation structures are introduced among the coefficients, even when a white noise is analyzed. In this paper, we show that it is possible to provide an accurate description of the covariance properties of the dual-tree coefficients of a wide-sense stationary process. The expressions of the (cross-)covariance sequences of the coefficients are derived in the one and two-dimensional cases. Asymptotic results are also provided, allowing to predict the behaviour of the second-order moments for large lag values or at coarse resolution. In addition, the cross-correlations between the primal and dual wavelets, which play a primary role in our theoretical analysis, are calculated for a number of classical wavelet families. Simulation results are finally provided to validate these results

Étude du bruit dans une analyse M-bandes en arbre dual

Dans cet article, nous nous intéressons aux propriétés d'un bruit additif après une analyse en ondelettes M-bandes en arbre dual. Cette décomposition nous permet d'exprimer les liens régissant les coefficients du bruit dans les arbres primal et dual. La connaissance des propriétés statistiques du bruit s'avère en particulier utile à la conception de méthodes efficaces de débruitage spécifiques aux analyses en arbre dual. Notre contribution réside principalement dans le calcul des fonctions d'intercorrélation relatives à cette analyse pour différents types d'ondelettes M-bandes. Nous montrons en particulier que les paires de coefficients issus d'un bruit blanc, regardées point à point dans les décompositions primale et duale, sont décorrélées. Il existe cependant une corrélation significative dans un proche voisinage spatial dépendant du choix des ondelettes M-bandes

2D Dual-Tree M-Band Wavelet Decomposition

We propose a 2D generalization to the M-band case of the dualtree structure (initially proposed by N. Kingsbury and further investigated by I. Selesnick) based on a Hilbert pair of wavelets. We particularly address the construction of the dual basis and the resulting directional analysis. We revisit the necessary pre-processing stage in the M-band case. While several reconstructions are possible because of the redundancy of the representation, we propose a new optimal signal reconstruction technique, which minimizes potential estimation errors. The effectiveness of the proposed M-band decomposition is demonstrated via image denoising comparisons

A new estimator for image denoising using a 2D dual-tree M-band wavelet decomposition

International audienceWe propose a new estimator for image denoising using a 2D dualtree M-band wavelet transform. Our work extends existing blockbased wavelet thresholding methods by exploiting simultaneously coef cients in the two M-band wavelet trees. The contributions of this paper are two-fold. Firstly, we perform a statistical analysis of the noise in the considered redundant decomposition. Secondly, we propose an ef cient method to remove the noise. Our approach relies on an extension of Stein's formula which allows us to take into account the speci c correlations of the noise components. Simulation results are then presented to validate the proposed method