International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
method presented above. The subdivision can be continued
until the area or a side of the current triangle reaches a
predefined limit. So the original triangle is transformed into a
fractal triangle whose irregular surface consists of many small
triangular facets.
As pointed out by Fournier and Goodchild (1995) , the
presented methods for rendering curves and surfaces are
satisfactory approximations of fractional Brownian motion.
They allow us to create realistic surfaces in faster time than
with exact calculations. Another advantage of these approaches
is the possibility of computing surfaces to arbitrary levels of
detail without increasing size of the database.
Figure 1 illustrates the behavior of fractal curves created using
fractional Brownian motion, different values of H, and a
constant vertical scale factor. The curves were rendered using
the fractal poly line subdivision method.
B= 9
H-7
Hs I n pe Sa TEN,
H=3 es anat rir reel rater
Figure 1. Stochastic curves rendered for different values of the
parameter H (Fournier and Goodchild 1995).
3. METODOLGY
This section describes the methodology used to analyze surface
fitting on TIN models.
The first step for modeling surfaces is the definition of the input
sample set that will be used to construct the surface. This
sample set must be representative of the phenomenon to be
modeled.
The next step involves the use of the sample set to construct the
basic structure of the DTM model. Here the input samples were
transformed on the vertices of the triangles of a TIN model.
Our case study was in two parts of Hamedan city in Iran (rough,
smooth) that sample set has been extracted from a map at a
scale of 1:25000 and then TIN model was constructed (Figure
2.3)
552
[3800
[38085
| 3.808
i
| 38075
[Ue
2600 3.8065
2400 1 3.806
2200 : R
T. 3.8055
2000 i
1800 7| ] m | 1 T T | 13805
3,39 3395 34 340 34 3415 — 342 3425 343
x10.
Figure 2. TIN model of one rough land in Hamedan, Iran.
Units are in meter.
.
x10
T 3.0165
3816
38155
i 3815
: 3.8145
i | 3.014
i | 3.8135
20007] NES : [73813
10 |
wr TY I’ TT 738125
343 3435 — 34 245 345 3485 3.46 3.465 347
x10
Figure 3. TIN model of one smooth land in Hamedan, Iran.
Units are in meter.
After this stage we use fractal concept for interpolation. This
method used to estimate the z values of the rectangular grid,
which was based on the polygon subdivision approach
presented in Felgueiras and Goodchiled (1995). The method
begins finding the current triangle 7: of the original TIN
model, that contains the grid point P. (x;, y;, Z;). Then the
triangle T is subdivided recursively in four smaller triangles
by connecting the midpoints of its sides. The z values of these
midpoints are defined by a fractal poly line subdivision
approach described in section 2. A new triangle 7. that
contains the point P,, is chosen among the four smaller
triangles. The subdivisions continue until the point P is within
a defined proximity criterion of one of the vertices of the
triangle T. When the proximity is reached, one can define
Z; equal to the Z of this vertex. In this paper, sample points of
the map were condensed with fractals up to contour interval of
5 meters (Figure 4, 5)
Intern
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