ISPRS, Vol.34, Part 2W2, “Dynamic and Multi-Dimensional GIS”, Bangkok, May 23-25, 2001 
376 
1) is chosen 
(p(x) = 1 - |x| 
if 
x e [- l,l] 
(/)(x) = 0 
f 
x <£ [— l,l] 
Figl : linear interpolation 
By computation, we can get h(-1) = 1/4, h(0) = 1/2, h(1) = 1/4. 
) = \^ x+ ')+- 0 
C+i (x ) = — C, (x - 2 ' ) +—C, (x ) + — C, (x + 2 1 ) 
Therefore we have 
The figure 2 shows the wavelet associated to the scaling 
function. 
Fig2: wavelet ijj(x) 
The above £ trous algorithm is easily extensible to the two 
dimensions space. This leads to convolution with a mask of 3x3 
pixels for the wavelet connected to linear interpolation. The 
coefficients of the mask are 
At each scale j, we obtain a set of Wj(x) (we also call it wavelet 
plan) which has the same number of pixels as the image. 
If a B 3 -spline for the scaling function is chosen, the coefficients of 
convolution mask in one dimension are (1/16,1/4,3/8,1/4,1/16), 
and in two dimensions 
1 
1 
3 
1 
1 
256 
64 
128 
64 
256 
1 
1 
3 
1 
1 
64 
16 
32 
16 
64 
3 
3 
9 
3 
3 
128 
32 
64 
32 
128 
1 
1 
3 
1 
1 
64 
16 
32 
16 
64 
1 
1 
3 
1 
1 
256 
64 
128 
64 
256 
3.Detecting Edges Using the A Trous Algorithm 
From above analysis, it is found that an image can be 
decomposed into several wavelet planes at different scales, 
some of high frequency information is included in the 
wavelet planes. Therefore we can detect edges of a remote 
sensing image by using the following procedure 
1) Initialize i = 0, and input an original image fi(x,y); 
2) Convolute the image with the low pass filter h(x,y) 
fi+i(x,y) = fi(x,y) * h(x,y) 
3) Get a wavelet plan 
w i+ i(x,y) = fi(x,y)-f i+1 (x,y) 
4) If i < n (the n is defined the decomposition number) 
then i = i+1, and return to the step 2); 
5) Repeat the steps 2), 3), 4) until i = n. 
In order to process the image borders, the mirror symmetry 
method is adopted, namely, 
In the row direction: 
f(-U) = f(U) 
f(i+k, j) = f(i-k, j) 
Where i < N, k = 1,2 N is the total rows of the image. 
In the column direction: 
f(i. -j) = ffl, j) 
f(i, j+k) = f(i, j-k) 
Where j ^ N, k = 1,2,..., N is the total columns of the image. 
In the experiment, the n is set to 3, we add the three wavelet 
planes w1, w2 and w3 to get an resultant image which contains 
abundant high frequency information and little low frequency 
information. 
4.Discussing and Conclusions