The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
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Let (p jk =<f>(2 J X-k) and y/ jk = y/(2 J X - k) be sets 
of dilated scaled and wavelet functions, respectively. Both 
functions can be constructed from the higher level scaling 
functions (Burrus,1998): 
(1) 
k 
y/(2 J X x) = Y,g j+ \{k)()){2 J x-k) 
(2) 
k 
Where h(k) and g(k) are low and high pass filters respectively. 
Any function f(x) can be represented by given scaling and 
wavelet coefficients with respect to the corresponding functions 
as: 
f(x) = Ya C j-\ (JM?* X x-k) + Yé d j-l( k )'/'j,k( 2J l x-k) 
k k 
(3) 
For orthonormal scaling and wavelet functions the scaling 
(approximation) and wavelet (detail) coefficients can be 
represented in terms of their values in a previous scale as 
follows: 
Cj_i (k) = ^ h(m - 2 k)cj (m) (4) 
m 
dj-1 (*) = Z ~ 2k ^ d i ( w ) (5) 
Recalling that the scaling function application on the signal is a 
low pass filter and the wavelet function application on the 
signal is a high pass filter, it can be concluded that obtaining the 
approximation and detail coefficient constitutes a single step in 
an iterative filter bank that results in multiple level 
decomposition of the signal. This iterative filter bank forms the 
basis of the discrete wavelet transform. A reverse operation can 
also be used to completely reconstruct the signal. In image 
analysis, a generalized form of the one-dimensional discrete 
wavelet transform can be used, by applying tensor product 
between the two sets of coefficients in the x and y directions. A 
scaling and wavelet transform can be defined as follows: 
<P(x,y)= <f>(x) (/)(y) (6) 
Vl(x, y)= </> (x) If/ (y) (7) 
W2(x,y)= \j/ (x)(j)(y) (8) 
T3(x,y)= \f/ (x) \j/ (y) (9) 
Where <P(x, y), Wl(x, y), W2(x, y), and W3(x, y) are the scaling, 
vertical, horizontal, and vertical wavelet functions, respectively. 
2.2 Quality Analysis) 
Another cloud free image of the area could be used as a base for 
assessing the quality of the developed algorithm. Two metrics 
suggested in this analysis, which are the root mean square error 
(RMSE) and the entropy. The RMSE can be expressed as 
follows: 
RMSE = tJEx(\(u(m, n) - v{m, n)| 2 ) 
Where u(m,n) and v(n,m) represent the tested and reference 
images respectively. The increase in the RMSE value indicates 
more differences between the tested and the reference images. 
The second metric is the image entropy, which represent the 
amount of information in the image and can be expressed as 
follows: 
Ent = Y J P(d i )\og 2 p{d i ) 
where d t is the number of gray levels possible and p(dj is the 
probability of occurrence of a particular gray level. 
2.3 Developed Algorithm 
Modification of the wavelet detail coefficients is an efficient 
way to perform an adaptive image enhancement. A schematic 
diagram of the developed algorithm for image enhancement in 
shadow areas, which depends on enhancing the image details 
content, is shown in Figure 1. 
A contrast shift is applied to the image so that the gray 
values in the shadow areas have the same range as the rest 
of the image. 
1. The Biorthogonal wavelet family (Biro2,2) is used 
due to its linear phase property. The image is 
decomposed into multiple levels using the Bior2,2 
discrete wavelet transform. The resulting 
approximation and detail coefficients are denoted Cj 
and dj respectively, where j refers to the wavelet 
decomposition level. It should be noted here that the 
detail coefficient d'. refer to the horizontal, vertical, 
and diagonal coefficients mentioned in Equations 7, 
8, and 9. The universal threshold is used with fixed 
values in all wavelets levels and directions (vertical,