A Bayesian transformation model for wavelet shrinkage.

Wavelet shrinkage estimators, in general, make the additive normal noise assumption and disregard the nonlinear nature of contamination. We develop Bayesian wavelet shrinkage estimators (based on the power transformations in the linear model) to accommodate a broad class of noise models in image processing applications. We intend to admit, under one roof, the widespread additive model, the product models common in imaging (such as in synthetic aperture radar (SAR) imagery), as well as noise that may exist amid these two extremes. Tactful prior elicitation in this model, such as the simultaneous assignment of mixture priors for wavelet coefficients and the transformation, imparts flexibility and ample insight into the underlying structure. The model permits estimation with unknown noise structure for reasonably unimodal and well-behaved (on the tails) distributions, wherein it can outperform common shrinkage estimators. Extensions with multiple transformations and Markov random field priors are also considered for adaptation to local variations in contamination. Modern Markov chain Monte Carlo (MCMC) Bayesian computation has been used for simulations and several examples are reported in the paper.