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rhombic grid with the interval of d1as shown in figure 1. 
It is easy to derive the following equation: 
disi. di (12) 
7 
The second step is to interpolate the height of the center 
point(.) from the heights of four neighboring rhombic 
lattices. In the same manner, we can get expression 
4 
1 
hl2=— > hli+A2 (13) 
12 
j=1 
After this step, the rhombic grid is changed into a square 
grid with the d2 interval as shown in figure 1: 
dl = (14) 
1 
dl 
V2 
Figure 1, midpoint subdivision scheme 
A constrain of such kind of subdivision is not to change 
the previous computing points in the later subdivision 
level. So in every interpolation step, the heights of grid 
are known,and the only problem is to determine the 
random displacement value( A ). 
Many existing fractal model used for modeling a virtual 
natural surface is based on the statistical criteria 
opposed by visual acceptability, so the choice of A is 
only considered with respect to the basic requirements 
for approximating the fBf, such as 
A = scale x ER x gauss (15) 
where,scale is displacement factor, and gauss-N(0. 1). 
For virtual terrain generation, the choices of scale and H 
may depend on the tests or experiences. However, for 
real data,the parameters scale and H should be coincide 
with the c and H extracted from DEM. 
With regard as a “shape preserved” fractal interpolation 
for real data reported by Yokoya et al.(1989),the A 
model was as follows. 
International Archives of Photogrammetry and Remote Sensing. Vol. XX 
Ai = di > ox Jic x gauss (16) 
where the i is the iteration level of interpolation. 
In fact, this is a approximation to fBm in 1-dimensional 
case(voss,1985). Apparently it is not stationary for 
midpoint displacement in more than 1-dimensional 
space. A local stationary midpoint displacement model 
should be as follows(Qing Zhu,1995): 
Aiz di. x ox el x gauss (17) 
Compare Eqs.(16) with Eqs.(17), it is easy to see that 
the Yokoya's model is intended to smooth the real relief . 
In order to control the different details efficiently in 
different subareas, it is important to use the results of 
adaptive analysis. On the other hand, because the 
midpoint displacement subdivision is usually 
accomplished in object space, it is difficult to relate the 
depth of recursion to the last screen coordinates. 
However, it is possible to relate it to the intervals in world 
coordinate system. 
4. RESULTS OF EXPERIMENT 
Figure 2 shows the contour map of a studied area which 
is a square grid DEM with the interval as 10m and size 
of 70x70 points. 
      
dm rite 
Figure 2: the contour map of studied DEM 
Because it is impossible to consider all the Ax and Ay to 
estimate the H and co by using Eqs.(9), in order to 
extract the isotropic fractal features of an area, it is a 
good choice to compute the average height difference 
1009 
XI, Part B4. Vienna 1996