Abstract
The paper deals with the use of wavelet transform for signal and image de-noising
employing a selected method of thresholding of appropriate decomposition coef-
¯cients. The proposed technique is based upon the analysis of wavelet transform
and it includes description of global modi¯cation of its values. The whole method
is veri¯ed for simulated signals and applied to processing of biomedical signals
representing EEG signals and MR images corrupted by additional random noise.
1 Introduction
The wavelet transform (WT) is a powerful tool of signal processing for its multiresolutional
possibilities. Unlike the Fourier transform, the WT is suitable for application to non-stationary
signals with transitory phenomena, whose frequency response varies in time [2].
The wavelet coe±cients represent a measure of similarity in the frequency content between
a signal and a chosen wavelet function [2]. These coe±cients are computed as a convolution
of the signal and the scaled wavelet function, which can be interpreted as a dilated band-pass
¯lter because of its band-pass like spectrum [5].
The scale is inversely proportional to radian frequency. Consequently, low frequencies correspond
to high scales and a dilated wavelet function. By wavelet analysis at high scales, we extract
global information from a signal called approximations. Whereas at low scales, we extract ¯ne
information from a signal called details.
Signals are usually band-limited, which is equivalent to having ¯nite energy, and therefore we
need to use just a constrained interval of scales. However, the continuous wavelet transform
provides us with lots of redundant information.
The discrete wavelet transform (DWT) requires less space utilising the space-saving coding based
on the fact that wavelet families are orthogonal or biorthogonal bases, and thus do not produce
redundant analysis. The DWT corresponds to its continuous version sampled usually on a dyadic
grid, which means that the scales and translations are powers of two [5].
In practise, the DWT is computed by passing a signal successively through a high-pass and a low-
pass ¯lter. For each decomposition level, the high-pass ¯lter hd forming the wavelet function
produces the approximations A. The complementary low-pass ¯lter ld representing the scaling
function produces the details D [3]. This computational algorithm shown in Fig. 1a is called
the subband coding.
The resolution is altered by the ¯ltering process, and the scale is changed by either upsampling
or downsampling by 2. This is described by the following two equations [4]
D1[n] =
1X
k=¡1
hd[k] x[2n ¡ k] (1)
A1[n] =
1X
k=¡1
ld[k] x[2n ¡ k] (2)
where n and k denote discrete time coe±cients, x the decomposed signal.

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