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 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.
Summary
This paper analyzes the covariance structure induced by Dual-Tree Wavelet (dual-tree) decompositions when decomposing wide-sense stationary processes, with emphasis on how redundancy introduces correlations among coefficients. Readers will learn analytic descriptions of inter- and intra-band covariance for dual-tree coefficients and how these properties impact denoising, detection and statistical modeling in wavelet domains.
Key Takeaways
- Derive analytic covariance expressions for dual-tree wavelet coefficients of wide-sense stationary processes (including white noise).
- Incorporate the predicted covariance into wavelet-domain denoising or Bayesian shrinkage schemes to improve performance over white-noise assumptions.
- Exploit known inter-tree and inter-scale correlation patterns to design decorrelating post-filters or modified thresholding rules.
- Approximate covariance structure (decay with distance/scale and directional dependence) for practical implementation under local stationarity.
- Use covariance models to improve detection and statistical inference in directional, multiscale applications (e.g., imaging or radar).
Who Should Read This
Engineers and researchers in DSP, image processing and statistical signal processing who use wavelet/frame transforms for denoising, detection or modeling and need a rigorous noise-covariance model.
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