Marc Raimondo, University of Sydney. 

The WaveD method for wavelet deconvolution with noisy eigen-values

Abstract: Over the last decade there has been a lot of interest in
wavelet-vaguelette methods for the recovery of noisy signals or images
in motion blur.  Non-linear wavelet estimators are known to have good
adaptive properties and to outperform linear approximations over a
wide range of signals and images, see e.g.  the recent WaveD method of
Johnstone, Kerkyacharian, Picard and Raimondo, JRSS(B) (2004).  The
wavelet-vaguelette methods rely on the complete knowledge of a
convolution operator's eigen-values.  This is an unlikely situation in
practice, however.  A more realistic scenario, such as would arise
when passing the Fourier basis as an input signal through a
Linear-Time-Invariant system, is to imagine that one also observes a
set of noisy eigen-values.  In this talk we define a version of the
WaveD estimator which is near-optimal when used with noisy
eigen-values.  A key feature of our method includes a data-driven
method for choosing the fine resolution level in WaveD
estimation. Asymptotic theory is illustrated with a wide range of
finite sample examples.

This is a joint work with Laurent Cavalier, Universit\'e Aix-Marseille 1.