The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B4. Beijing 2008 
434 
d(x) is the distance between x and C(p, t=0), and the plus 
(minus) sign is chosen if the point x is outside (inside) the initial 
curve C (p, t=0).thus we have build the initial level set function. 
The equation for the evolving function which contains 
the embedded motion of C (p, t) as the level set {0 = 0) is given 
out by equation^ = V ;; ГС . V(k) is the speed function and 
we refer to this as a Hamilton-Jacobi “type” equation. 
There are many advantages to this Eulerian Hamilton-Jacobi 
formulation, the most important are that it avoid the evolution 
of the parameters of the curve equation, makes curve topology 
changes(split or merge) become very natural and as evolving 
function 0$C. ?> always remains a function, it can be easily 
carried out by using numerical approximation methods. 
2.2 C-V model 
In 2001, Chan and Vese proposed an simplified model based on 
Munford-Shah segmentation model. Suppose I(x,y) is the image, 
there are a few homogenous regions in an image, Q is the edge 
of region Rj ,the DN value of very region is a constant, that is to 
every region Rj , I(Ri)=constant(Ci). What the C-V model to do 
is create an minimize energy function to find the optimal 
segmentation, which makes the distinction between segmented 
image 
. _ f вднешкОД ifll Inside Cl 
w t£p ¡f Ri sutiri» Ci 
and I(x,y) is the smallest. 
We use an simple example that given by Chan and Vese in their 
paper to explain this idea(Tony F.Chan, Luminita A. Vese, 
2001)(Fg.l). Define the evolving curve C in O, as the boundary 
of an open subset of П. inside(C) denotes the region , 
and outside(C) denotes the region Q\<k). The image u 0 is 
divided into two regions by curve C: Uq and Ug. The average 
DN value of very region is C| and C 2 .Then consider the 
following “fitting” term: 
F- CO ~ F; it) = I lii-Cx V-) - ?,J : ¿жф 
In this simple case, it is obviously that C 0 , the boundary of the 
object, is the minimize of the fitting term 
WWMQ+ F-CCJ9 * 0 * F a CC«H F,(C„) For 
instance, if the curve C is outside the object, then F5 (CJ > 0 
and F e (Q*o. If the curve C is inside the object, then 
F S <CJ*0 and F; C If the curve c is both inside and 
outside the object, then MC»& and F; CO > ^.Finally, 
the fitting energy is minimized if C=C 0 , if the curve C is on the 
boundary of the object. 
Based on this idea, Chan and vese create the energy function 
F(C- defined by 
FCffc, sv. C) = u>L«n$&.0 v ■ 
+¿4 * [ h*c0s-> r 5 - 
Where 
}t ^ 0, V 2 0, 2 € are fixed parameters. 
L$ngtfi(C) is the length of the curve C 
AmClnsJdeCC)) is the area of the region inside C 
LtngthfjC) Ar«&(insld«(0) are regularizing terms. 
By solving the minimization problem iwig^(<B| t Vy, O, 
we can get the final curve C. 
Figure 1 .all possible cases in the position of the curve 
3. EXPERIMENTAL PROCEDURES 
The experiment procedure of this paper is directly and in a 
semi-automatic method. At first, the program will get initial 
edge by Human-Computer Interaction way. Usually, initial edge 
was simply given by using circular or rectangular, because it 
would be very easy to build signed distance function. But when 
we use this way to process remote sensing image, it will cost 
very long time to make the curve move to the real edge. Though 
the algorithm of building signed distance function will become 
a little more complicated and time costing, because the initial 
edge is much more accurate, the total time will be shorter. 
After getting the initial curve, the program will build signed 
distance function and evolve the curve automatically. 
We provide two ways to stop the evolving of the curve. The 
first way is that the program judges if the curve has moved to 
the right place and determines to continue or stop by itself. The 
second way is to stop the curve by human sending commands, 
this way is more effective because we use our experience to 
determine if the curve has moved to the right place.