or edge 
ooth the 
. depicts 
osen for 
1e of the 
ansform 
at large 
(1) 
1 g(x) 
(3) 
986), 1s 
faster in 
r, 5. Let 
(4) 
1 with a 
(5) 
er) The 
e in the 
rapidly 
points is 
window, 
if there 
Yunham Dong 
  
As the mean calculation is adapted according to the existence of edge crossings, the algorithm does two things together: 
smoothing uniform areas and sharpening edges. If the algorithm is used recursively, the final results can converge to the 
ideal situations. Figure 2 shows iteration results for signals with a local oscillation and an edge crossing, respectively. 
(a) (b) 
Figure 2. Convergence of the algorithm: (a) for local oscillations and (b) for edge crossings. 
2.2 Two Dimensional Algorithm 
Theoretically the above one-dimensional algorithm can be easily extended to two dimensions by considering edge 
detection in the x and y directions. That is, for a two dimensional function, f(x,y) edge crossings can be found by 
evaluating edge detecting functions in the x and y directions together: 
qs (x, y) = f(x, y)**g  (x)6(y) (6) 
q; (x, y) 9 f (x, y) **6(x)g (y) (7) 
where ** denotes two-dimensional convolution, and §(-) is the Diract delta function. This implies that for a direction 
à = Xcos0 + ysin 6 , the directional edge detection function is 
q; (X, y) * 4; (x, y) cos0 4 q? (x, y)sinO (8) 
This method works well for most images. However, it is found that due to speckle in SAR images, the discontinuity of 
the discrete function f(x,y) is very serious and the correlation between neighboring pexels is low, so the information 
given by q;(x,y) and gq} (x,y) is not sufficiently accurate to indicate edge information in (8). Therefore, the 
directional edge detecting function is defined as, 
q; 7 f(*g,G) (9) 
where f(r) is one dimensional function whose elements consists of pixels along the 7 direction. It is found that for 
images having high correlation among neighboring pixels, results of (8) and (9) are the same. However, the result of (9) 
is much better than that of (8) for images having low correlation among neighboring pixels, such as found in SAR 
images. 
Based on the above analysis, the proposed filtering algorithm for two-dimensional SAR image consists of two steps: 
1. Apply the one-dimensional filtering algorithm to an image in four directions, namely, vertical (zero-degree), 
vertical (90-degree), 45-degree diagonal and 135-degree diagonal directions, respectively, to obtain four 
directionally filtered images. 
2 The final filtered image is the mean of the four directionally filtered images, i.e., the final value for each pixel is 
the mean of the corresponding pixels in the four diretionally fitltered images. 
Each pixel in an image is, therefore, considered and filtered in four (eight) directions as depicted in Figure 3. 
  
Figure 3. Each pixel in an image is filtered in four (eight) directions. 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part Bl. Amsterdam 2000. 91