Covariance structure of wavelet coefficients: theory and models in a Bayesian perspective

We present theoretical results on the random wavelet coefficients covariance structure. We use simple properties of the coefficients to derive a recursive way to compute the within- and across-scale covariances. We point out a useful link between the algorithm proposed and the two-dimensional discrete wavelet transform. We then focus on Bayesian wavelet shrinkage for estimating a function from noisy data. A prior distribution is imposed on the coefficients of the unknown function. We show how our findings on the covariance structure make it possible to specify priors that take into account the full correlation between coefficients through a parsimonious number of hyperparameters. We use Markov chain Monte Carlo methods to estimate the parameters and illustrate our method on bench-mark simulated signals.