%0 Generic %1 chaux2011noise %A Chaux, Caroline %A Pesquet, Jean-Christophe %A Duval, Laurent %D 2011 %K covariance noise properties tree %R 10.1109/TIT.2007.909104 %T Noise Covariance Properties in Dual-Tree Wavelet Decompositions %U http://arxiv.org/abs/1108.5395 %X 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.

@misc{chaux2011noise, 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.}, added-at = {2013-12-23T04:31:43.000+0100}, author = {Chaux, Caroline and Pesquet, Jean-Christophe and Duval, Laurent}, biburl = {https://www.bibsonomy.org/bibtex/2deb6bd8f1fc7425fe10d03798d6668bd/aeu_research}, description = {Noise Covariance Properties in Dual-Tree Wavelet Decompositions}, doi = {10.1109/TIT.2007.909104}, interhash = {48e21394810745f6ef740753bfd4867e}, intrahash = {deb6bd8f1fc7425fe10d03798d6668bd}, keywords = {covariance noise properties tree}, note = {cite arxiv:1108.5395}, timestamp = {2013-12-23T04:31:43.000+0100}, title = {Noise Covariance Properties in Dual-Tree Wavelet Decompositions}, url = {http://arxiv.org/abs/1108.5395}, year = 2011 }