The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B3b. Beijing 2008 
454 
The active contour model that is used in this paper is proposed 
by Chan and Vese (2001) based on Mamford-Shah image 
segmentation equation. In this model image is segmented to 
regions that the value of all pixels within each region have 
maximum homogeneity and similarity. The most important 
advantage of this active contour model is the ability of them to 
extract objects without obvious edges from images. Another 
advantage of this model is no sensitivity to the noise. 
equations: 
cM) 
c 2 (</>) 
[u 0 H(<f>)dx 
\H{<t>)dx 
^u 0 (\-H{<t>)dx 
¿(1 -H(0)dx 
(6) 
Active contour model based on level set formulation is 
described by this energy function: 
P i (C) + F 2 (C)= £ 
' dx + [ |w 0 -cX dx (1) 
Jbutside(C) 1 U 
Where C is the curve of active contour model, and U Q is 
pixel's value of the image and C } , C 2 are the average of 
pixel's value inside and outside of C. 
In the last step, the equation of curve evolution is: 
— = 
dt 
p.div 
V<p 
M 
V \ ( M 0 C \ ) (w 0 C 2 ) 
= 0 
Finally the above curve evolution equation is solved based on 
finite difference methods and the position of curve is improved 
in the iteration manner. 
3. AUTOMATIC BUILDING EXTRACTION USING 
ACTIVE CONTOUR MODEL 
If the cure C is inside of target object then f (C) » 0 and 
F 2 (C) > 0 and in case that curve C is outside of the target object 
then f ] (C) > 0 and f 2 (C) « 0 and finally if curve C is inside 
and outside of target object then F,(C)>0 and F 2 (C)>0- 
After minimization of function of equation (1), the curve C is 
fitted to the boundary of target object and this relation is 
obtained: 
inf, ft (C) + F 2 (C)} * 0 * (Q) + F 2 (C 0 )} 
For regularization of equation (1), two terms are added to it and 
the equation (2) is obtained: 
F(C,c, ,c 2 ) = ju.(length(C)) p + v.area(inside(C))- 
(2) 
X\ \u n -cYdx + X f lur.-c^dx 
1 inside(.C) 1 u 11 1 iutside(Cy U 
Where /a > 0, v > 0 and X x ,X 2 >0 are constant 
parameters. The equation (2) has a level set formulation that 
was introduced in (Chan et al, 2001). For this purpose they 
defined two additional functions: 
d(z) = —H(z) 
(3) 
H{z) = 
„ z2 ° i(r) = -j-ff(r) 
0, z < 0 dz 
Our active contour model for building extraction is described in 
section 2. Our building extraction method is made up four major 
steps. In the first stage the input image is smoothed using a 
Gaussian kernel. In next step a point inside a building is 
introduced to model as a training data. Then the proposed active 
contour model is implemented on the smoothed image and 
boundaries of building are detected. Finally the accuracy of 
detected buildings is evaluated. Figure 1 demonstrates the 
diagram of steps of proposed model. 
Based on equation (3) the definition of each part of equation (2) 
is: 
length^ = 0} - j}VH(<f>)\dx = \ S(<pp fflx 
area{(/> > 0} = \^H{tf>)dx 
{ >0 |«o ~c\ 2 dx= jjw 0 -c x \ 2 H(<l>)dx (4) 
jj« 0 -c 2 | 2 <&= ^u 0 -c 2 \\\-H(<t>))dx 
In the above equations C2 shows the entire of image domain. 
With regarding the equation (4), the level set definition of 
equation (1) is (Chan et al, 2001): 
F(0, c,,c 2 ) = /*( £ |W/(0|)" dx + v l H(j>)dx + 
K fj«o-c x \ 2 H(<f>)dx + A 2 jjw 0 -c 2 \ 2 (\-H(</>))dx (5) 
Figure 1: diagram of building extraction steps 
Our model is tested on an aerial image from Gachsaran city in 
Iran. The buildings in this image are dense and attached 
together. Figure 2 shows the image of test region. 
In this equation, c and c, are the mean of pixel's value if 
</>>0 and f < 0 respectively and they are obtained from these 
Figure 2: Gachsaran city as the test region