can be written into the following form: 
f(r + hsina, c + hcosa) — f(r, c) 
  
For.) = lim 
h-—0 
h 
second directional derivatives is: 
f. c), = Le c)sina + Te, c) cosa 
If taking f as cubic polynomial of point (r, c), that is 
f(r.c) = Kı + Kor + K,c + Kır? + Ksırc + Ky? 
+ Kır? + Kyrie + Kore? + Kot 
The angle can be calculated by the following formula: 
: K, 
sine ——. ———— 
VKi+ Ki 
Ks 
cosa = 
VK} + Ki 
Second derivatives for any point (r,c) on direction is 
fo(r.c) = (6K,;sin’a + 4K sinacosa + 2K ,cos?a)r 
+ (6K,0cos’a + 4K sinacosa + 2K sin%a)c 
+ (2K sina? + 2K sinacosa + 2K cosa) 
"^" r = psina, c = pcosa, than 
f.(r.c) = 6 (K,sin®a + Kgsin?acosa + Ksinacos?a 
-- K,0cos?a) p + 2(K, sin? + K;sinacosa + Kcos’a) 
= Ap+ B (3) 
When f.(p) = 0 and f.(p) # 0, i. e. when first 
derivatives is not zero and second directional derivatives 
is equal to zero, then, this point is edge point. 
3. Edge Extraction for Two Dimension Image Using Ze- 
ro Crossing of Second Directional Derivatives 
It can be known from the front derived formula 
that to extract edge using zero crossing of second direc- 
tional derivatives not only is related with the order num- 
ber of fitting polynomial, but also is related with the 
window size at the same time. The larger the window, 
the longer the calculation time. In order to beadapted 
for micro — computer processing, 5X 5 window size is 
selected, fitting polynomial comes up order. 
According to weight coefficient and window grey 
value, the coefficient of polynomial can be solved, thus 
first and second derivatives can be further calculated. 
The approximate value of first derivative is 
f (r.c) = K,sina + K,cosa (4) 
Second derivative is 
F(r.c) = Ap+B (5) 
When p = 0, that is center point of window, If sym- 
bles of second derivatives changed among p = 0 and p = 
1 and first derivatives is not equal to 0 and larger than 
threshold, the center point of window is considered as 
edge point. 
For remote sensing image, its grey change is ex- 
tremely complex. Therefore, two problems similar to 
the method existed in actual processing. one is the de- 
termination of threshold, the other is edge which can 
44 
not be continued. 
For different kind of edge, its threshold should be 
different too. For an image. a threshold is obviously 
not suitable. In order to solve this problem, a method 
of manual assistant has been used. 
Great difficulties of image analysis will cause from 
discrete edges. For reasons of edge extraction method 
to be practical, a method of heuristic search of artifical 
intelligence has been used, discrete edge turn to contin- 
uous edge. 
Finally, using editing method, the real satisfied 
edge results can be achieved. 
The main points of edge connection method will be 
described in the following. 
1. Threshold Selection 
In edge extraction using zero crossing of second di- 
rectional derivatives, as the point meets the require- 
ments of second directional derivatives and large then 
initial threshold, then first derivatives of this point 
should be reserved. 
After processing, all the first derivatives of corre- 
sponding edge points can be obtained, then, the original 
image will be shown on screen (e. g. 512 512 pixels), 
and subimage on the screen will be selected, for exam- 
ple, 100X 100 pixels, selecting first derivatives thresh- 
old, the points whose first derivatives is larger than the 
threshold should be superimposed on theoriginal image, 
using visual to check the edge whether is suitable, if it 
is not, then to adjust threshold until optimum results 
can be got. 
Using this method, threshold can be selected flexi- 
bly, so as to meet the needs of partial region of image, 
to achieve the optimum goal of complete image edge ex- 
traction. 
2. Automatic Connection of Discrete Edge 
Owing to the edge change for remote sensing image 
is extremely complicated, after edge extraction, dis- 
crete point often happens, themethod of heuristic search 
of artificial intelligence has been used for automatic con- 
néction. 
A algorithm has been used. The formula of A* al- 
gorithm is: 
f* (n) — g* (n) + h* (n) 
where, g* = K(s,n) , the real cost of an optimum path 
n from node s to node n. h* (n) , the cost of an 
optimum path from node n to object node. 
f*(n), the cost of an optimum path started 
from s through node n. 
À algorithm is used for automatic connection of dis- 
crete edge point, its basic principles in connection are: 
(1). Given a interval threshold between discrete 
points, if interval is smaller than threshold, these dis- 
crete p 
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