Wavelet shrinkage: unification of basic thresholding functions and thresholds

This work addresses the unification of some basic functions and thresholds used in nonparametric 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 regularisation 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. ∗TELECOM Bretagne, [email protected] †TELECOM Bretagne, [email protected] ‡TELECOM Bretagne, [email protected]