Wavelet shrinkage : from sparsity and robust testing to smooth adaptation

International audienceWavelet transforms are said to be sparse in that they represent smooth andpiecewise regular signals by coefficients that are mostly small except for afew that are significantly large. The WaveShrink or wavelet shrinkage estimators introduced by Donoho and Johnstone in their seminal work exploit this sparsity to estimate or reconstruct a deterministic function from the observation of its samples corrupted by independent and additive white Gaussian noise AWGN. After a brief survey on wavelet shrinkage, this chapter presents several theoretical results, from which we derive new adaptable WaveShrink estimators that overcome the limitations of standard ones, have an explicit close form and can apply to any wavelet transform (orthogonal, redundant, multi-wavelets, complex wavelets, among others) and to a large class of estimation problems. These WaveShrink estimators do not induce additional computational cost that could depend on the application and let the user choose freely the wavelet transform suitable for a given application, in contrast to all parametric methods and also in contrast to some non-parametric methods. In addition, our estimators can be adapted to each decomposition level thanks to known properties of the wavelet transform. The two theoretical results on which our WaveShrink estimators rely are, fist, a new measure of sparsity for sequences of noisy random signals and, second, the construction of a family of smooth shrinkage functions, the so-called Smooth Sigmoid Based Shrinkage (SSBS) functions. The measure of sparsity is based on recent results in non-parametric statistics for the detection of signals with unknown distributions and unknown probabilities of presence in independent AWGN. The SSBS functions allow for a flexible control of the shrinkage thanks to parameters that directly relate to the attenuation wanted for the small, median and large coefficients. The relevance of the approach is illustrated in image denoising, a typical application field for WaveShrink estimators