ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
  
within pre-specified areas (Vosselman, 2000); a statistic median 
value of slope angle distribution characterize a terrain slope of 
an investigated local area (Axelsson, 2000); a “good” mixture 
of on- and off-terrain points over a local area is assumed so that 
it facilitates an iterative terrain resampling process as its 
parameters of weighting function are implicitly determined 
(Pfeifer&Kraus, 1998). 
In contrast to this, the other method devised in our current work 
is to fragment the entire terrain surface into a set of piecewise 
segments so that they have “homogeneous” background 
knowledge of the underlying terrain slope as being “plane”. In 
this context, a criterion Ó is explicitly selected in such a way as 
to differentiate on- and off-terrain point from a “plane” terrain. 
This can be universally applied to overall terrain segments 
regardless of terrain surface variances since all terrain segments 
are assumed to be plane terrain surfaces. Hence, a LIDAR 
filtering technique could be converted into a problem to look 
for a set of plane terrain surfaces into which terrain surface 
variation is regularized, rather than to estimate Ó itself. 
To achieve this goal, it is necessary to use a terrain surface 
model y to hypothesize a set of plane terrain surfaces, and a 
criterion óthat is independent of the model y . In this approach, 
the labelling problem in Eq. (1) can be rewritten as follows: 
f 7s fi F(s Iw. 0) fie tonoff) Q) 
where a label / for LIDAR point s, is determined when a 
terrain surface model y and a criterion Ó are given. 
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(b) ö applied on a terrain surface with the mixture of 
variousterrain scales 
Figure 1. Illustration of terrain scale dependency of à when 
given à is explicitly applied on various types of terrain 
scales as a slope angle criterion. 
2.2 Terrain Surface Reconstruction Problem 
Suppose then our terrain surface model y in Eq. (2) can be 
described as a set of piecewise planar surface models {p } as 
follows: 
V - i6): 6, 7 [a, b, c | os =0 G) 
where j is the model index of the piecewise planar surface 
model ¢ ; [a, b, e,] is the parameter vector of the planar 
model 9, ; s is the vector of the LIDAR point s; located 
within à, , which is labelled as on-terrain point when the 
labelling function F is given; F(s, | @,,ö) = {on} . In Eq. (3), the 
planar terrain surface model ÿ, is made of the on-terrain points 
only, which is satisfied with the condition gs” =0 and M is 
the dimension of the model space y created when the entire 
domain S is initialised as the on-terrain point, F(S)= {on}. 
Now, a terrain surface reconstruction problem can be 
reformulated to determine a global optimised solution y” , 
which is obtainable from the searching process of a local 
optimised solution ¢ as follows: 
on 
W(d,k)=(0 Ya 4 (s)) =0 (4) 
where 9; is the locally optimised piecewise planar surface 
model; k is the dimension of the model space y^ , which is less 
than M of Eq. (3); (s; 1" is the vector of on-terrain points 
located within $,. Eq. (4) presents us with several important 
points to be noted in reconstructing the terrain surface. 
e lt considers the dimension & of an optimal terrain surface 
model y^ as a variable to be determined during a terrain 
surface model reconstruction process, rather than being a 
pre-fixed constant as in Baillard & Maitre's work (1999), 
in which global labelling observations are optimized 
within a pre-specified number of flat terrain models having 
almost the same size. 
e A locally determined planar surface model $; is required 
to be comprised of on-terrain points. Thus, the 
determination of & is directly related to the number of on- 
terrain points found, where k increases as on-terrain points 
are iteratively obtained. 
e The methodology to achieve an optimal solution y 
shown in Eq. (4) is based upon a local optimisation 
approach rather than a global minimisation technique. That 
is, a planar surface model is found as a local optimal 
solution @; and thus the global terrain surface model y^ 
is determined as a set of local optimal solutions (9']. 
It is necessary to discuss an optimality criterion of $; used in 
Eq. (4). In Eq. (2), the labelling observation f of the LIDAR 
points S is determined using a criterion ó under the assumption 
that underlying terrain surface is "correctly" reconstructed as 
flat by a planar surface model. However, if its assumption is not 
valid, the labelling error of f becomes large so that real on- 
terrain points are misclassified into off-terrain and vice versa. 
Thus, the optimality is achieved when labelling observation f 
generated supports most properly the prescribed assumption of 
plane terrain surface. 
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