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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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Zn
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I
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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