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,