Wavelet shrinkage: unification of basic thresholding functions and thresholds

This work addresses the unification of some basic functions and thresholds used in non-parametric estimation of signals by shrinkage in the wavelet domain. The soft and hard thresholding functions are presented as degenerate smooth sigmoid-based shrinkage functions. The shrinkage achieved by this new family of sigmoid-based functions is then shown to be equivalent to a regularization of wavelet coefficients associated with a class of penalty functions. Some sigmoid-based penalty functions are calculated, and their properties are discussed. The unification also concerns the universal and the minimax thresholds used to calibrate standard soft and hard thresholding functions: these thresholds pertain to a wide class of thresholds, called the detection thresholds. These thresholds depend on two parameters describing the sparsity degree for the wavelet representation of a signal. It is also shown that the non-degenerate sigmoid shrinkage adjusted with the new detection thresholds is as performant as the best up-to-date parametric and computationally expensive method. This justifies the relevance of sigmoid shrinkage for noise reduction in large databases or large size images.