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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