Automatic Smoothing With Wavelets for a Wide Class of Distributions

This work considers a unifying l1-penalized likelihood approach to regularize the maximum likelihood estimation by adding an l1 penalty of the wavelet coefficients to give an explicit formula for the universal smoothing parameter to denoise Poisson measurements.

Abstract

AbstractWavelet-based denoising techniques are well suited to estimate spatially inhomogeneous signals. Waveshrink (Donoho and Johnstone) assumes independent Gaussian errors and equispaced sampling of the signal. Various articles have relaxed some of these assumptions, but a systematic generalization to distributions such as Poisson, binomial, or Bernoulli is missing. We consider a unifying l1-penalized likelihood approach to regularize the maximum likelihood estimation by adding an l1 penalty of the wavelet coefficients. Our approach works for all types of wavelets and for a range of noise distributions. We develop both an algorithm to solve the estimation problem and rules to select the smoothing parameter automatically. In particular, using results from Poisson processes, we give an explicit formula for the universal smoothing parameter to denoise Poisson measurements. Simulations show that the procedure is an improvement over other methods. An astronomy example is given.Key Words: Convex programmingInterior point methodKarush–kuhn–tucker conditionsPenalized likelihoodRegularizationSignal denoisingUniversal rule