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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004 
  
2002), and land cover classification of forest (Blanco and 
Garcia 1997). 
In this paper we use fractal for optimum sampling in generating 
a digital terrain model (DTM ). 
2. CONCEPT OF FRACTAL DIMENSION 
Recent advances in the area of fractal geometry have allowed us 
to model natural objects dimensionality. For example, the 
length of a coastline can vary depending on scale, ranging from 
an apparently infinitely high length to a very short distance if 
we highly generalize the shape. It is interesting that fractal 
geometry can give us measure of the dimensionality of objects 
that are different from Euclidian geometry. The fractal 
dimension tells us how densely a phenomenon occupies the 
space in which it is located. It is independent from the 
measurement units used or alteration of the space by stretching 
or condensing. 
The fractal dimension of many entities can be obtained by the 
Equation 1 or 2 (Laurini and Thompson 2002): 
_ logn 
(log( 1) (1) 
Or 
_logn 
de log) @ 
Where n -number of pieces in the repetitor 
r =self-similarity ratio 
d=fractal dimension. 
Alternatively, s, the scaling factor, the inverse of the self- 
similarity ratio, can be thought of as the number of pieces that 
an entity is split into. In the case of the snowflake example 
already mentioned, n=4, r=1/3, giving d=1.2619. 
We can imagine a continuum where a value of d close to 0 
would mean an entity is close to a point, a value of 1 means a 
line, and if it is near 2, it is an area. 
Similarly, a smooth line will have a dimensionality of 1, but an 
irregular line has a higher value, certainly greater than 1. For 
coastlines the mean fractal dimension is d=1.2, wherease for 
A 
terrain, d is about 2.3 
Brownian motion is the most popular model used to perform 
fractal interpolations from a set of samples. Brownian motion, 
first observed by Robert Brown in 1827, is the motion of small 
particles caused by continual bombardment by other 
neighboring particles. Brown found that the distribution of the 
particle position is always Gaussian with a variance dependent 
only on the length of the time of the movement observation 
(Laurini and Thompson 2002). 
The Fractional Brownian motion (FBm), derived from 
Brownian motion, can be used to simulate topographic surfaces. 
FBm provides a method of generating irregular, self-similar 
surfaces that resemble topography and that have a known 
fractional dimension. 
551 
The FBm functions can be characterized by variograms 
(graphic that plots the phenomenon variation against the spatial 
distance between two points) of the form (Felgueiras and 
Goodchild 1995): 
Ez, -z, =K*(d,)™ © 
where E=statistical expectation 
Z;, Z ; =heights of the surface at the points i and j 
d j —spatial distance between these points 
K-constant of proportionality 
H-parameter in the range 0 to 1 
K is also related to a vertical scale factor S that controls the 
roughness of the surface. H describes the relative smoothness at 
different scales and has a relation with the fractal dimension D 
as formulated in Equation 4 (Felgueiras and Goodchild 1995): 
D-3-—" (4) 
When H is 0.5 we get the pure Brownian motion. The smaller 
H, the larger D and the more irregular is the surface. On the 
contrary, the larger H, the smaller D and the smoother the 
surface. 
Fournier et al (1982) presented recursive procedures to render 
curves and surfaces based on stochastic models. They described 
two methods to construct two-dimensional fractal surface 
primitives. The first one is based on a subdivision of polygons 
to create fractal polygons while the second approach is based on 
the definition of stochastic parametric surfaces. : 
The subdivision of polygons is based on the fractal poly line 
subdivision method. A fractal poly line subdivision is a 
recursive procedure that interpolates intermediate points of a 
poly line. The algorithm recursively subdivides the closest 
extreme intervals and generates a scalar value at the midpoint 
which is proportional to the current standard deviation oc times 
the scale or roughness factor S. So, the Z,, value of the middle 
point between two consecutive points, 7 and / , of a poly line is 
determined by the Equation 5 (Felgueiras and Goodchild 1995): 
Zn 12, +2,M2+5 56, * MOI) (5) 
Where Ovaries according to Equation 5 and N(0,1) is a 
Gaussian random variable with zero mean and unit variance. 
The subdivision of polygons method is suitable to create 
stochastic surfaces based on TIN digital models. Each triangle 
of the TIN model can be subdivided into four smaller triangles 
by connecting the midpoints of the triangles. The z value of the 
midpoints is calculated by the fractal poly line subdivision