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id
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