Softening the multiscale product method for adaptive noise reduction

The goal of denoising is to remove the noise while preserving the important features as much as possible. By exploring the power of parsimonious wavelet basis representation and statistical decision methods, Donoho and Johnstone [1994] pioneered the wavelet shrinkage. However, the performance of traditional wavelet shrinkage is not even as good as that of a simple multiscale product method (MPM) [Y. Xu, et al., 1994], because the wavelet basis representation in the traditional wavelet shrinkage is not shift-invariant. We numerically reveal the connection between the simple MPM [Y. Xu, et al., 1994] and Donoho-Johnstone´s hard thresholding [1994]. Based on the observations and an analysis of the MPM, we propose a softened version of MPM which is in analogous to Donoho-Johnstone´s soft thresholding [1994]. Thanks to the explicit detection of singularities and the use of both l₂ and l₀ stopping criteria to reduce the false detection, the performance of the softened MPM is superior to other methods with redundant wavelet representations for the functions of one-dimensional piecewise linear class. Combined with the local variance analysis discussed elsewhere, we extend the new method to two-dimensional image denoising.