Noise Covariance Properties in Dual-Tree Wavelet Decompositions

TL;DR

This paper characterizes the covariance properties of dual-tree wavelet coefficients when analyzing stationary processes, especially under noisy conditions, providing theoretical formulas and validation through simulations.

Contribution

It offers an accurate description of covariance structures of dual-tree wavelet coefficients for stationary processes, including asymptotic behaviors and cross-correlations for classical wavelet families.

Findings

Derived covariance expressions for 1D and 2D cases.

Provided asymptotic behavior of second-order moments.

Validated theoretical results with simulations.

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.