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   -> Volume 6, Issue 1

Preprint: Three preprints about Wavelet Noise Reduction
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Maarten Jansen (

PostPosted: Sat Dec 28, 1996 11:16 am    
Subject: Preprint: Three preprints about Wavelet Noise Reduction
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#3 Preprint: Three preprints about Wavelet Noise Reduction

The following three preprints are available from

1) TITLE: Generalized Cross Validation for wavelet thresholding

AUTHORS: Maarten Jansen, Maurits Malfait and Adhemar Bultheel


Noisy data are often fitted using a smoothing parameter, controling
the importance of two objectives that are opposite to a certain
extent. One of these two is smoothness and the other is closeness to
the input data.

The optimal value of this paramater minimizes the error of the result
(as compared to the unknown, exact data), usually expressed in the
$L_2$ norm.This optimum cannot be found exactly, simply because the
exact data are unknown.In spline theory, the Generalized Cross
Validation (GCV) technique has proven to be an effective (though
rather slow) statistical way for estimating this optimum.

On the other hand, wavelet theory is well suited for signal and image
processing. This paper investigates the possibility of using GCV in a
noise reduction algorithm, based on wavelet-thresholding, where the
threshold can be seen as a kind of smoothing parameter. The GCV
method thus allows choosing the (nearly) optimal threshold, without
knowing the noise variance.

Both an original theoretical argument and practical experiments
are used to show this successful combination.

NOTE: To appear in Signal Processing, 56(1)


2) TITLE: Bayesian Approach To Wavelet-based Image Processing

AUTHORS: Maurits Malfait, Maarten Jansen and Dirk Roose


We present a new method for the reduction of noise in images, using a
wavelet transform. The method relies on two principles. The first is
the characterization of the local function regularity by wavelet
coefficients.The second is an a priori, geometrical model for wavelet
coefficients. Both are combined in a Bayesian framework, to compute
for each wavelet coefficient the probability of being ``sufficiently
clean". The manipulation of the wavelet coefficients is consequently
based on the obtained probabilities.

NOTE: presented at the Joint Statistical Meetings, Chicago,
August 1996


3) TITLE: Multiple wavelet threshold estimation by generalized
cross validation for data with correlated

AUTHORS: Maarten Jansen and Adhemar Bultheel


De-noising algorithms based on wavelet thresholding replace small
wavelet coefficients by zero and keep or shrink the coefficients with
absolute value above the threshold. The optimal threshold minimizes
the error of the result as compared to the unknown, exact data. To
estimate this optimal threshold, we use Generalized Cross
Validation. This procedure does not require an estimation for the
noise energy. Originally, this method assumes uncorrelated noise. In
this paper we describe how we can extend it to images with correlated

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