o 
reference point 
basis point 
/ ANC 
o 
N [/ 
o o 
o 
  
  
  
  
  
Figure 1: Voronoi diagram of irregular data superimposed 
with regular grid of basis points 
bour approximation the height on a regular grid, in our 
words at a basis point, is linked to the height of the next 
reference point. The region of influence of a reference point 
can then be plotted in a Voronoi diagram (see Figure 1). 
Covering our domain with a grid with fixed mesh-width 
the surface is represented by a height matrix which can 
be processed very efficiently. The height of a sampling 
point (e.g. point on a finer grid for displaying the DTM) 
can be calculated by bilinear interpolation or other meth- 
ods from the heights of the basis points. This means that 
the next neighbour approximation involves 3 data sets, the 
reference points, the basis points and the sampling points. 
Our approach is characterized by the same property. The 
transmission of the height information from a reference 
point to a basis point depends on the spatial distance. In 
our method we analyze the polar distance between two 
points on a unit sphere, defined as the cosine of the angle 
between them. If the points are given in cartesian coor- 
dinates z = (z1, 2,3) it can be calculated by the scalar 
product t = (z,y) — z1y1 4 72y2 4 23ys very easily. This 
distance is equal to 1 if the angle between two points is 
zero and therefore it is suitable to define a weighting func- 
tion. Our SERRE functions are spherical polynomials of 
degree k, B® : [1,1] — IR, k = 1,2,... , r € [0,1) 
given by 
0 for —-1<t<r 
(b). =*t= 
Br = { (i£) fo r<t£1. (1) 
This function is printed in Figure 2 for different values of k 
and fixed r. The spherical representation of this functions 
for a given point y, for example y — (0,0,1), is plotted 
in Figure 3. The function has rotational symmetry about 
the axis through the point y and compact support. Points 
lying on the same latitude have the same spherical distance 
to the North-Pole y. Points with spherical distance greater 
than r = 0.5 (angle > 60 degree) lie outside the defined 
spherical cap around y and get the weight zero. 
The calculation of the height at a basis point can then 
be formulated as analyzing the distance between the ba- 
sis points and the reference points and as weighting the 
heights of the reference points in the spherical cap accord- 
ing to their distance. This can be formulated as summa- 
150 
  
1.01 
0.81 
0.2 
  
  
-1.0 -0.5 0.5 1.0 
  
  
  
Figure 2: B® (t) for k — 1,2,3 and r = 0.5 
  
  
  
  
  
Figure 3: Spherical representation of B? centered at the 
North-Pole for k — 3 and r = 0.5 
tion 
g(z) 2 Y hy BP (s, y) (2) 
where x are the basis points, y are the reference points 
and h, is the height at the reference points. B((a,3)) 
denotes a normalization of the function B(*) defined in (1) 
where the argument is the scalar product (z, y). The choice 
of this normalized function is due to (Cui, et al., 1992) and 
discussed in (Brand, 1994) in more detail. In contrast to 
the next neighbour approximation the height of the basis 
points depends not only on one reference point but also on 
the reference points in a spherical cap of radius r around 
them. For the evaluation of the height at a sampling point 
z formula (2) is used as well, so that our approximation 
g(z) can be calculated according to 
g(x) ER (s, 2) (3) 
with 
e(r) zm A BU (Gy). (4) 
The functional parameter r depends on the distribution 
of reference points and basis points. Its choice is similar 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996 
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