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   -> Volume 5, Issue 10

Preprint: Wavelet Approaches to Statistical Inverse Problems
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Abramovich Felix (

PostPosted: Tue Nov 12, 1996 4:10 pm    
Subject: Preprint: Wavelet Approaches to Statistical Inverse Problems
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#2 Preprint: Wavelet Approaches to Statistical Inverse Problems

Title: Wavelet Decomposition Approaches to Statistical Inverse Problems

Authors: Felix Abramovich and Bernard W. Silverman

Preprint available on the web:


A wide variety of scientific settings involve indirect noisy
measurements where one faces a linear inverse problem in the presence
of noise. Primary interest is in some function f(t) but the data is
accessible only about some transform (Kf)(t), where K is some linear
operator, and Kf(t) is in addition corrupted by noise. The usual
linear methods for such inverse problems, for example those based on
singular value decompositions, do not perform satisfactorily when the
original function f(t) is spatially inhomogeneous. One alternative
that has been suggested is the wavelet--vaguelette decomposition
method, based on the expansion of the unknown f(t) in wavelet series.

The vaguelette--wavelet decomposition method proposed in this paper is
also based on wavelet expansion. In contrast to wavelet--vaguelette
decomposition, the observed data are expanded directly in wavelet
series. Using exact risk calculations, the performances of the two
wavelet-based methods are compared with one another and with SVD
methods, in the context of the estimation of the derivative of a
function observed subject to noise. A result is proved demonstrating
that, with a suitable universal threshold somewhat larger than that
used for standard denoising problems, both wavelet-based approaches
have an ideal spatial adaptivity property.

Felix Abramovich

Department of Statistics & Operations Research
Tel Aviv University, Tel Aviv 69978, Israel
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