Noise covariance properties in Dual-Tree Wavelet Decompositions

By Chaux & Pesquet & Duval

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 crosscorrelations
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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