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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
the points into left and right) until no significant change 
of the intersection line between the surface pairs can be 
recognised. In order to reduce the influence of points with 
à increasing orthogonal distance to the breakline a weight 
function is used. The weight function (in this case a bell 
curve) can be parameterised in a way that points near the 
breakline get a high weight, whereas the weight of points 
far away from the breakline is decreased. Points with a 
certain distance to the breakline should have no influence 
to the run of the surface patches and therefore they should 
get the weight zero. An example of such a weight function 
(we use an individual weight function for the left and right 
points) can be seen in figure 3. Additionally, this weight 
function reduces the weight of points in a small buffer zone 
around the breakline itself. This is done to reduce the in- 
fluence of points very close to the breakline, which are 
usually affected by small distance measurement errors due 
to a too big footprint size and furthermore has a positive 
influence on the iterative refinement of the approximation 
of the breakline. 
  
T T T^ XT 
weight [0:1] 
  
ae etre rer ete cr enr ete” 
40 5 5 5 12 15 
distance [m] 
Figure 3: Weight functions defining an individual weight 
for cach point depending on the distance to the break- 
line; Green (negative distance values): Weight function for 
points belonging to the support of the left surface patch; 
Blue (positive distance values): Weight function for points 
of the right surface patch; Additionally points in a small 
buffer zone around the breakline (distance zero) get a lower 
weight. 
As a result of the modelling procedure one representative 
point on the intersection line and the direction of the in- 
tersection line (tangent of the breakline) are stored per 
patch pair. An additional interesting result is the inter- 
section angle between the surface patches along the whole 
breakline, which allows a further analysis. Due to the fact 
that the surface determination is performed within an ad- 
justment procedure, accuracies for the estimation of the 
unknowns (plane parameters) can be computed. This al- 
lows the computation of further quality measures (e.g. the 
accuracy of the intersection line) with the help of error 
propagation. 
3.2 Robust Modelling 
As mentioned before the breakline should be modelled 
with the help of unclassified originally acquired ALS point 
clouds in order to exclude preprocessing errors. Therefore 
we have to adapt the basic concept in a way that the influ- 
ence of off-terrain-points (e.g. due to reflections of the laser 
1099 
beam on the vegetation or measurement errors caused by 
multipath reflection ("Long Ranges")) is reduced as much 
as possible. This is performed with the help of robust esti- 
mation and is based on the robust interpolation technique 
for ALS data presented by (Kraus and Pfeifer 1998). 
The robust estimation of the surface patches is forced with 
the help of à second weight function. Depending on the 
vertical distance of the point to the intermediate surfaces, 
this weight function assigns a low weight to potential off- 
terrain points (mainly above, but also to points signifi- 
cantly below the terrain) and a high weight to terrain 
points. The robust estimation is initiated automatically 
if off-terrain points (gross errors in the adjustment) are 
detected and adjusts itself within a fully automatic proce- 
dure using an individual self-adapting weight function for 
every iteration step (cf. figure 4). Step by step the weight 
of off-terrain points is reduced. A practical example, which 
presents the capability of the robust modelling procedure, 
an be seen in figure 5. 
Weight function for robust adjustment (3 iterations) 
  
  
  
  
! T Tp T T 
09 3 
08 "i 
= | 
2 06 
= 
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O4F 
23+- 4 
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91 iT-3 
ü i + i i i = il 
10 -8 6 E 2 0 2 4 & 8 10 
residual [m] 
Figure 4: Example for automatically determined weight 
functions per robust iteration step (IT). The function 
adapts itself step by step depending on the data and al- 
lows a separation into terrain and off-terrain points. Posi- 
tive residuals belong to points above the terrain, whereas 
points below the terrain have negative residuals. Espe- 
cially the weight of off-terrain points above the terrain 
(manly caused by vegetation) is reduced to a high degree. 
  
Figure 5: 3D modelled breakline determined on the basis 
of unclassified ALS data.