Noise Covariance Properties in Dual-Tree Wavelet Decompositions

Abstract : 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.
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IEEE Transactions on Information Theory, Institute of Electrical and Electronics Engineers, 2007, 53 (12), pp.4680 - 4700. <10.1109/TIT.2007.909104>
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Caroline Chaux, Jean-Christophe Pesquet, Laurent Duval. Noise Covariance Properties in Dual-Tree Wavelet Decompositions. IEEE Transactions on Information Theory, Institute of Electrical and Electronics Engineers, 2007, 53 (12), pp.4680 - 4700. <10.1109/TIT.2007.909104>. <hal-00617268>

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