The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
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and tend to cancel each other. Therefore, he proposed the 
modified Laplacian (ML). The expression for the discrete 
approximation of ML is: 
v L./(^ y) = | 2 /(*> y) ~ fix- step, y) - f(x + step, y) \ 
+ \2 /(x, y) - /(x, y - step) - /(x, y + step)\ 
In order to accommodate for possible variations in the size of 
texture elements, Nayar (1994) used a variable spacing (step) 
between the pixels to compute ML. In this paper ‘step’ always 
equals to 1. 
x+N y+N 
SML = Y. for V^/(,;j)>r (2) 
i=x-N j=y-N 
where T is a discrimination threshold value. The parameter 
N determines the window size used to compute the focus 
measure. 
M*) = 2' j (f>(2- J x-k) (4) 
where </>{x) is the scaling function, which is a low-pass filter. 
c jk is also called a discrete approximation at the resolution 
2 j . 
If tp(x) is the wavelet function, the wavelet coefficients are 
obtained by 
<»„ =(/(*),2->(2-'x-4)) (5) 
co j k is called the discrete detail signal at the resolution 2 J . 
As the scaling function ^(x) has the following property: 
v (6) 
c j+]Jk can be obtained by direct computation from c j k 
C j+\,k =Z /î (" _2Â: ) C 7> aIld 
^(p{^) = Y.g{n)(/>{x-n) (7) 
The scalar products (^f(x),2~ (1+x) (p(2~ (j+X) x-k)^j are computed 
with 
1.2 2.2 Stationary wavelet transform 
In this section, we present the basic principles of the SWT 
method. In summary, the SWT method can be described as 
follows (Wang et al.,2003). 
When the high pass and low pass filters are applied to the data 
at each level, the two new sequences have the same length as 
the original sequence without decimation. That is different from 
DWT, where decimation is necessary. 
Supposing a function /(x) is projected at each step j on the 
subset Vj(LL <z V 3 <z V 2 a V, a V 0 ). This projection is defined 
by the scalar product c y . k of /(x) with the scaling function 
<f>{x) which is dilated and translated 
=(/(*), «>,.,«) (3) 
(8) 
Equations (7) and (8) are the multiresolution algorithm of the 
traditional DWT. In this algorithm, a downsampling procedure 
is performed after the filtering process. That is, one point out of 
two is kept during transformation. Therefore, the whole length 
of the function /(x) will reduce by half after the 
transformation. This process continues until the length of the 
function becomes one. 
However, for stationary or redundant wavelet transform, instead 
of downsampling, an upsampling jjrocedure. is carried out 
before performing convolution at each scale. The distance 
between samples increases by a factor of two from scale j to 
the next. c j+i k is obtained by 
‘W=X*(0c, w „ (9) 
/ 
and the discrete wavelet coefficients by 
® w =Xs(')W/ < 10 > 
/ 
The redundancy of this transform facilitates the identification of 
salient features in a signal, especially for recognizing the noises. 
This is the transform for one-dimensional signal. For a two 
dimensional image, we separate the variables x and y and 
have the following wavelets. 
— Vertical wavelet: tp l (x, y) = <j(x)<p(y) 
— Horizontal wavelet: <p 2 (x, y) = tp(x)0(y) 
— Diagonal wavelet: tp 3 (x, y) = tp(x)<p(y) 
Thus, the detail signals are contained in three subimages,