ISPRS, Vol.34, Part 2W2, “Dynamic and Multi-Dimensional GIS", Bangkok, May 23-25, 2001
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operator is invariable in different directions, it means you can
get the same detected edges when you rotate the operator
The Laplace operator not only response to image edges, but
also to corner points, end points of a line and isolated points.
To restrain noise, the LOG operator smoothes an image at
first, then performs the difference. Though the image noise is
partly restrained, the detected result is influenced. From
above discussion, we can find the operators which are relate
to direction are not effective to detecting edges when the
image is complicated and there are abundant edges in the
image. In addition, Due to noise in an image the detected
result is not perfect if the operators are directly used.
Therefore in this paper, Our objective is to introduce a new
method to detect image edge so as to get a better result.
2.The Á Trous Wavelet Decomposition
2.1Wave!et Decomposition
Wavelet analysis has been successfully used in image
processing field. Wavelet transform is a new method which
can decompose an image into different resolution images.
Suppose function ijj (x) e L 2 (R)(L 2 (R) is quadratically
integrable space), and ijj(x) satisfies:
|2
+ r v(e>)‘
C r = 2 n J—j—r—dco +co
(1)
or \i//{x)dx = 0
(2)
Where ijj(u)) is the Fourier transform result of ijj(x).
When ijj(x) satisfies equation (1 ) or ( 2 ) and can rapidly
converge, ip(x) is called basic wavelet.
ijj(x) flexes a and shifts b then we have:
VaA X )
I 4 (x-b
M 2 v / —
^ a
(3)
Where a, beR, a*0
and ijja.b(x) is called wavelet.
Wavelet transform of distribution function f(x) can be defined as
following
+°° i r 1
wf(a,b)= ^f(x)-\a\iy/\a A (x-b)\lx (4)
can’t be used in computation by program. To do it by computer,
we must make the continuous wavelet transform discrete. There
are many methods to perform discrete wavelet transform, such
as pyramidal algorithm which make use of orthogonal basis to
decompose an image (or a signal), but the dimension of the result
image is changed, that is not advantageous in some applications
like pattern recognition and image fusion etc.
To get the result image of the same dimension as the original one,
we adopt the algorithm: ä trous algorithm. The discrete approach
of the wavelet transform can be done with the special version of
the so-called ä trous algorithm (with holes). The algorithm can
decompose an image (or a signal) into an approximate signal and
a detail signal at a scale, the detail signal is called a wavelet
plane, which is the same as the original image in dimension.
Suppose that the sampled image data C 0 (k) are the scalar
products at pixels k of the function f(x) with a scaling function cj)(x)
which corresponds to a low pass filter. The first filtering is then
performed by a twice magnified scale leading to the C,(k) set.
The image difference C 0 (k)-Ci(k) which is called first wavelet
plane contains the information between those two scales and is
the discrete set associated with the wavelet transform
corresponding to 4>(x). The associated wavelet is therefore ijj(x).
(5)
The distance between samples increasing by a factor 2 from the
scale (i-1) (¡>0) to the next one, Ci(x) is given by
C,(*) = 2>(/)-C M (* + 2 w /) (6)
and the discrete wavelet transform w¡(x) by:
w,(x) = C l _ l (x)-C l (x)
(7)
Where Wi(x) is the wavelet coefficient and Cj(x) is approximate
signal at the i scale, h(l) is a low pass filter.
The coefficients h(l) is derived from the scaling function <£(x):
l \l) i
(8)
The algorithm can be used to rebuild the data frame because the
last smoothed array C np is added to all the differences w (
up
c 0 (x) = c„ p w+X w J W
7=1
(9)
Where f(x) eL 2 (R),beR, y/ (x)is complex conjugate of ip(x).
2.2 The A Trous Wavelet Decomposition Method
Continuous wavelet transform definition is introduced above. It If the linear interpolation for the scaling function i>(x) (see figure
