Minimax description length for signal denoising and optimized representation

Approaches to wavelet-based denoising (or signal enhancement) have generally relied on the assumption of normally distributed perturbations. To relax this assumption, which is often violated in practice, we derive a robust wavelet thresholding technique based on the minimax description length (MMDL) principle. We first determine the least favorable distribution in the epsilon-contaminated normal family as the member that maximizes the entropy. We show that this distribution, and the best estimate based upon it, namely the maximum-likelihood estimate, together constitute a saddle point. The MMDL approach results in a thresholding scheme that is resistant to heavy tailed noise. We further extend this framework and propose a novel approach to selecting an adapted or best basis (BB) that results in optimal signal reconstruction. Finally, we address the practical case where the underlying signal is known to be bounded, and derive a two-sided thresholding technique that is resistant to outliers and has bounded error.