In: Wagner W„ Szekely, B. (eds.): ISPRS TC VII Symposium - 100 Years ISPRS, Vienna, Austria, July 5-7, 2010, IAPRS, Vol. XXXVIII, Part 7B 
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Figure 9. Discontinuities in overlapping area of both models 
The developed filter is based on the “Multiquadratic Method” 
explained in (Kraus, 2000). Before the filter is processed the 
user has to determine which model has the highest accuracy. As 
in general the user has a priori information about the data sets to 
be fused, this step is done best manually. The most accurate 
model is named Ml, the other one M2. As shown in Figure 8, 
Ml is a subset of M2. In a follow on step a border line in the 
xy-plane between Ml and M2 is calculated (s. Figure 10) and 
for all points defined by the intersections of the border line with 
the models an elevation difference Ah; between both models can 
be computed: 
Ah, 
= P 
Z(Ml)i 
-p 
Z(M2)i 
(10) 
line) and zero at the outer line. This makes clear that transition 
between Ml and M2 is done within the buffer. The smoothed 
elevation difference will be some what in between. The buffer 
size has to be selected very carefully, because the number of 
points, which should be used for the calculation, is strongly 
dependent on the buffer size (s. Figure 10). 
The interpolation of the elevation differences within the buffer 
is carried out by the “Multiquadratic Method”. Looking at 
Figure 11), which shows an arbitrary point P within the buffer 
and Pj intersecting points, the distance PPi in the xy-plane 
between P and Pj is given by 
k(P,P) = V(X-X,) 2 +(Y-Y) 2 (11), 
if X,Y are the coordinates of P and Xj, Yj the coordinates of Pj. 
k(P,P) is also called the core function. The elevation difference 
Ah for point P can be interpolated by 
Ah = Ah(X,Y) = J(X -X,) 2 +(Y -Y,) 2 *m, +... 
, (12) 
+ > /(X -XJ 2 +(Y -Y ) 2 *m„ 
Here is mi a scale factor for distance from point P to the i th tie 
point and n is the number of tie points. In matrix notation one 
can write: 
Figure 10. Definition border line and buffer are 
Ah = k ‘ m (13), 
with 
k T = (k,,k 2 ,k 3 , ,..,k n )and m' =(m l ,m 2 ,m J , ...,m n ). 
For n elevation differences Ahj a linear equation system can be 
setup with Equation (13): 
k(0) 
(symm. 
k(P,P 2 ) 
k(0) 
k(P,P ) 
k(P 2 P) 
v 
k(0) \mj ^Ah ny 
^Ah ^ 
Ah 
which corresponds to 
K • m = dh 
(14) 
(15) 
P3(Ahp3 = 0.0) 
Border line 
Buffer 
□ ALS - Point 
Figure 11. Interpolation setup 
In a next step a so called buffer area is setup about Ml in the 
xy-plane (s. Figure 10). The outer boundary of the buffer is 
displayed by a blue line in Figure 10. The elevation differences 
are set to zero on this line. This means, the elevation differences 
have their maximum (Ahj) at the inner line of the buffer (border 
Now, the actual elevation difference for Ah (X,Y) can be easily 
computed by 
Ah = k T K' 1 dh (16) 
Figure 12 proves the smoothing effect of this filter for the 
transition area between Ml and M2. Here the ALS- and GNSS- 
DTMs already presented in Figure 8 are matched and smoothed 
around the GNSS-DTM.