Softening the Multiscale Product Method for Adaptive Noise Reduction
Jun Ge and Gagan Mirchandani
(Paper (pdf) / Paper (ps))
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 \cite{Donoho94} pioneered the {\it wavelet shrinkage}. However,
the performance of traditional {\it wavelet shrinkage} is not even as
good as that of a simple {\it multiscale product method} (MPM) \cite{Xu94},
because the wavelet basis representation in the traditional {\it wavelet
shrinkage} is not shift-invariant. We numerically reveal the connection
between the simple MPM \cite{Xu94} and Donoho-Johnstone's hard thresholding
\cite{Donoho94}. Based on the observations and theoretical analysis of
the MPM, we propose a softened version of MPM which is in analogous to
Donoho-Johnstone's soft thresholding \cite{Donoho94}. Thanks to the explicit
detection of singularities and the use of both $\ell_2$ and $\ell_0$
stopping criteria to reduce the false detection, the performance of the
softened MPM is superior to other method 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.