The author also uses the fractal Brownian function to 
represent a real terrain surface(DEM).In the following,the 
definition of fBf and its characteristics will be 
introduced,especially the physical meanings of the two 
features H and o,which will be extracted from real DEM 
and characterize the DEM,will be discussed. 
The fBf f(x) is a real-valued random function such 
that,for all x and Ax, 
Pr Gt 0- fo 7 - F(z) (1) 
Axl 
  
  
  
  
where, x is a vector quantities in 2-dimensional 
Euclidean space,and F(z) is a cumulative distribution 
function (Pentland,1984).The parameter H( € [0,1]) is an 
indicator of the surface complexity(Polidori,1991),and 
the relationship between H and fractal dimension of the 
surface is as follows: 
D=3-H (2) 
The smaller H, the larger D and the more irregular the 
surface.On the contrary, the larger H ,the smaller D and 
the simpler the surface. 
If let f(x,y) denote the DEM,we have the following 
expression: 
E||f Gc Ax y A) - f. yy] 
(3) 
eet) 
F() - N(o. o) (4) 
— 
On 
— 
E(g) - 
20 
2% 
9 
Let C=E( 
N 
) (6) 
and Ax Ay. zl (7) 
then Ere yh - fel) o (8) 
From Eqs. (8) ,we can see that C equals the height up or 
down through a unit distance, it is obviously the slope. 
Therefore, C represents the average slope of the whole 
fractal surface with respect to all the Ax and Ay . 
The greater the o,the larger the slope, or vice versa. This 
is the physical meaning of the fractal feature. 
Because c represents the total character of the surface 
relief, it can be used to distinguish different kinds of 
shapes, which may be have the similar fractal 
dimensions such as broken level terrain and smooth 
mountainous terrain. 
Based on Eqs.(3) we can derive the following equation: 
log E(| f(x + Ax, y+ Ay) = f(x, ))) 
~Hlogy| Ax’ + Ay’ -logC 
since both H and C are constant,Eqs.(9) implies that a 
plot of E(f Gc Ax, y-* Ay) - f Gr, y) as a function of 
2 2 ; : : 
Ax +Ây on a log-log scale lies on a straight line 
and its slope is H,and the intersection between the line 
and function axis indicates the logC. 
For practical applications,such as extract H and o from a 
real DEM,there are a few problems should be dealed 
with carefully. For example,because of the errors caused 
by various sources, a real DEM possesses a largest and 
| 2 
a smallest limit of scale( Ax * Ay ) between 
which the surface has a constant fractal features and 
can be described by a single fBf.Then it is important to 
determine the range of the scales.Otherwise,an 
important fact is that fractal features are not constant 
over all areas of the real DEM but vary smoothly from 
position to position(Yokaya et al.,1989).So in order to 
(9) 
  
. describe the DEM more completely,it is also necessary 
to extract some subareas’fractal features which is called 
adaptive fractal analysis. In our experiences, the 
dimension of the subarea may be two times the largest 
scale or more larger. 
The fractal features H and c with respect to global area 
or local area extracted from real DEM can be used for 
many purposes such as terrain assessment and 
classification, DEM quality assessment, remote sensing 
information ^ classification precise engineering 
calculations and the “shape preserved" terrain 
interpolation, etc. 
3. ADAPTIVE LOCAL STATIONARY FRACTAL 
SUBDIVISION MODEL 
This is a kind of fast recursive subdivision technique, i.e. 
the midpoint displacement scheme(Fourier et al., 1991). 
As illustrated in Figure 1,let 01,02...,and On denote the 
initial grid DEM point (o) with the interval as dO. Each 
subdivision level include two interpolation steps, the first 
step is to interpolate the height h11 of the center point(x) 
from the heights hOi(i is from 1 to 4)of four neighboring 
grid points. We have the relationship between h11 and 
hOi as follows: 
4 
4x 
MI- hOi+Al (10) 
i=l 
where A1— NO, VA (11) 
After this step, the initial square grid is changed to a 
1008 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996