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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
to localize each coefficient spatially, what is impossible in the 
case of Fourier transform. 
3.2 Mallat’s Multiresolution Analysis 
The procedure for determining the image wavelet expansions 
(two-dimensional signals) with the help of multi-level 
decomposition utilizing one-dimensional filters, separately 
applied to the rows and columns of the image, was given by 
(Mallat, 1998). There are four components in the wavelet 
expansion of the image: so-called coarse component (LL) and 
three details, named as vertical- (LH), horizontal- (HL) and 
diagonal (HH) detail. The characteristic feature of wavelet 
transformation is the possibility to continue applying it to the 
chosen component. This is the coarse detail that is expanded 
most often. 
3.3 The dependence of noise and wavelet coefficients 
distribution 
Simonceli noticed that wavelet detail coefficients distribution 
has a sharp maximum in zero and has a good symmetry, 
whereas the flattening of histogram is correlated with the 
presence of noise in the image (Simonceli, 1996, 1999). The 
kurtosis, which is the fourth moment divided by the square of 
the variance, was employed as a parameter describing the 
histogram shape. 
Figure 1. The histograms of wavelets detail components 
It was claimed (Pyka, 2005) that the estimation of the shape of 
coefficients distribution should be made for all three detail 
components, but it is enough to limit research to wavelet 
decomposition on one level of resolution. Further 
decomposition of coarse component (LL) does not give more 
information on noise, because each subsequent coarse 
component is the effect of the smoothing of the preceding one, 
what decreases the noise content. In Figure 1 the typical 
histograms of three wavelets components for a image without 
noise and for the same image with white noise are shown. 
3.4 The rule of image preservation of energy through its 
wavelet transform 
The wavelet decomposition preserves the image energy (Mallat, 
1998). In case of decomposition on first level of resolution we 
can write: 
E(I) = E(LL, ) + E{LH X ) + E(HL X ) + E(HH X ) (1) 
where E(I) = energy of image / 
E(LL\) = energy of coarse component LL on first level 
of decomposition 
E(LHy), E(HL\), E(HH0 = energy of details 
components LH,HL,HH (on first level of 
decomposition) 
Further decomposition of coarse component allows writing: 
£(ZZ, ) = E(LL 2 ) + E(LH 2 ) + E(HL 2 ) + E(HH 2 ) (2) 
where E(LLi) EfLHj), E(HLi), E(HHi) = energy of 
components on second level of decomposition 
The general form of equation (1) and (2) is shown below: 
E(I) = E(LL r ) + £ [E(LH r ) + E(HL r ) + E(HH R ] < 3 ) 
where R = the number of level of decomposition 
3.5 The equation of relative variance preservation by 
wavelet decomposition 
The equation (3) is also true when we use variance instead of 
energy (Pyka, 2005): 
y (I) = V(LL r ) + X [V(LH r ) + V(HL r ) + V(HH r ] ( 4 ) 
where V(I\) = variance of image 
V(LL r = variance of coarse components on level R of 
decomposition, 
V(LHf V(HLf V(HHj = variance of detail 
components on level r of decomposition 
The image variance is dependent of pixels value scaling. If we 
linearly transform the image pixels value (called also 
brightness or DN) then the wavelet components undergo the 
same transformation. It is disadvantage of rule given by 
equation (4). When we divide either side of equation (4) by V(I) 
we receive the following equation: 
The equation (5) shows that the sum of relative variance of 
wavelet transform components equals 1 and that is true for any 
levels of decomposition. It is worth to note that the rule is 
independent of pixels value scaling. For images without noise 
the following rule should be true: 
I LH 
j HL 
I 13,9 
112,5 
1 1 
Image without noise 
1 LH 
A HL I 
■ 3.0 
Jti 
HH 
18.0 
The same image with white noise