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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B2. Istanbul 2004 
an open set containing 4. Here, x(U ) (or more generally, 
x( A) ) is called the map trace of X. 
Definition of Monge Patch: A Monge patch is a patch 
x :U- » R^ ofthe form 
X(u, v) = (u, v, h(u, v)) (5) 
5 
where {J is an open set in Rad h:U-—-R isa 
differentiable function. 
Definition of Gaussian curvature K and Mean curvature. H : 
For a Monge patch, the Gaussian curvature K and Mean 
curvature 77 are 
2 
vof isi hs 
m: 2 242 
{1h +h) 
2 2 
zs Bh, 2h A Für hy, 
> 2483/2 
20 3 5 +R) 
h 
"uv o? vy 
K uy 
(6) 
H 
  
where h, : h, ; 5 ; 
h(u,v). 
Here, then, we generalize the À to a map from an open set in 
R^w R’ $ jte i= , and think that any arbitrary 
point in R represents a color in CIE-Lab color-space. As 
defined in section 2,the distance between two colors can be 
measured by Eq. (1). Based on the definition of differentiation 
in R', we defined partial derivatives of A^" (u, V) as 
: ; : = s color ; : 
derivatives of the function À when all but the variables of 
interest are held fixed during the differentiation (Wilhelm 
Klinggenberg, 1978). 
Definition of Mean Curvature Of Color-image: 
~7, color 
peor a Oh 
H 
  
m 
OH (7) 
lin d'{color(u + Au, v)— color(u, v)) 
zm s 
Au-»0 Au 
Where color(u + Au, v) and color(u,v) are the i, jin 
Eg. (1) respectively. Assume hU is a "nice" two- 
dimensional function, so h, ; h. ; n p he exist. With 
Hv 
these we can calculate the Mean curvature. /7 as Mean 
Curvature Of Color-image. 
At last, in our experiment, we used the Mean curvature /7 as 
the property of the color-image texture. After finishing the work 
of section 2 and 3,we got many potential rectangles; we think 
these potential rectangles as the patches in which we calculate 
the curvatures and histogram of curvatures, the latter represents 
the probability density of local image’s curvature. 
5. RECOGNITION USING FUZZY INTEGRALS 
After the complex pre-proceedings, we got such datum: No.1 
is the rectangles, No.2 is the histogram of curvatures. As to 
rectangles, we use these features of them: height, width, size; as 
to the histogram of curvatures, the features that we employ here 
are: high mean absolute deviation and low mean absolute 
deviation. High mean absolute deviation is the mean absolute 
619 
I. are Partial derivatives. of 
deviation calculated with the whole histogram of curvatures. 
Low mean absolute deviation is the minimum onec calculated by 
1.0 mpl) 
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1.0 m(@2) 
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X. 
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$5, ó T4 5; 
Figure 3. Fuzzy densities to all possible properties 
partial histogram of curvatures, which includes the 70% of total 
pixels. Besides the two mean absolute deviations, the average 
color of the rectangle is also a property used here. 
Based on these data (properties). we need to decide whether 
or not this rectangle is a building. In order to achieve a reliable 
decision we must combine information from all of them. This