ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
  
To obtain these local optima, we adopt the hypothesis-test 
approach. A local terrain is hypothesized as a plane terrain 
surface by a number of planar surface model candidates (9^). 
According to Eq. (2), corresponding labelling observations 
{f°} can be generated when ó is given. We can then try to 
measure the closeness between a model candidate @° and its 
observation /* to test the hypothesis of a plane terrain surface 
reconstructed by ¢° . Such closeness measurement can be 
described in Bayes estimate framework as follows: 
  
6 cpm TP) © 
where @; is a planar surface model candidate generated for a 
local terrain; f? is the observation when given ¢; and ó ; 
P( F 6,0 ) is the conditional probability density function of 
the observation f7; P(g;) 1s the prior probability of the model 
¢; . Thus, an optimal solution $; can be found as maximizing 
Eq. (5). 
3. OVERALL STRATEGY 
In this section, we discuss several important concepts used for 
implementing our terrain surface reconstruction algorithm; 
firstly, we define a criterion selected to differentiate the on- and 
off-terrain points and describe its role in the current framework; 
secondly, we explain how to measure the closeness between the 
plane surface model and its labelling observation used in Eq. 
(5); finally, we describe overall strategy to reconstruct real 
terrain surface as a set of piecewise planar surface models. 
3.1 Plane Terrain Prior 
Since it is assumed a priori that the underlying local area is 
flattened by a planar surface model, our criterion to differentiate 
on- and off- terrain points can be applied to the entire terrain 
surface model with the same meaning in such a way as to 
classify a LIDAR dataset into on- and off-terrain points when 
the underlying area is projected into a horizontal flat terrain. 
To this end, we select a constant 0, as the criterion, which is 
vertical height measured relative to a local terrain surface model 
ÿ, . Once a local terrain surface is reconstructed by the planar 
surface model 9, , this reconstructed terrain surface is a priori 
assumed as being "flat" and relative vertical heights of 
underlying LIDAR points are recomputed from $,. Then, the 
constant criterion J, straightforwardly assigns corresponding 
labels to underlying LIDAR points S, located within ó,; ifa 
relative height of a point measured from 9, is less than ó,, an 
on-terrain label is assigned to this point; otherwise, an off- 
terrain one (see Figure 2). This labelling process using Ó, is 
universally applied to the entire LIDAR data regardless of 
terrain surface variances and thus, §, is independent of 9, . Eq. 
(2) can be rewritten as follows: 
JU Vs,e S, f x E(s19,5,): f; € ton, off] (6) 
where f. is a set of labelling observations generated when $; is 
given. Once the model 9, reconstructs a local terrain surface, it 
is required to determine whether this terrain reconstruction 
process would continue over the underlying area. If a condition 
to trigger the process is satisfied, the underlying terrain is 
fragmented even further in order to be made more flattened, and 
this process continues until its termination condition is satisfied. 
The ó, selected is used to provide a triggering and terminating 
condition for terrain fragmentation. As seen in Figure 2, assume 
that we have a set of LIDAR points S; and ÿ, is used as a local 
plane terrain surface model. According to Eq. (3), on-terrain 
points populated by using J, must be satisfied with following 
condition; ps” —0 where s^" is vector of on-terrain points 
JJ j 
belonging to S;. If there exists any on-terrain point with which 
this conditioning property is not satisfied; $,s" #0, it indicates 
terrain surfaces having different slopes coexist within the 
underlying area and 9; is not enough to make it flattened. Thus 
a terrain fragmentation process is triggered to seek more planar 
surface models to reconstruct underlying terrain surface as 
being plane. 
We shall define a “buffer space” as one located between the 
terrain surface model ÿ, and ó,, which needs to be empty of 
any LIDAR point to make the terrain fragmentation process 
terminate (Figure 2). The emptiness of “buffer space” 
characterizes a plane terrain surface in a sense that when a local 
area is properly flattened by ÿ, , there must be a discontinuity 
within the “buffer space”, in which any LIDAR point cannot be 
located. Hence, both emptiness of “buffer space” and prior 
assumption of plane terrain surface for underlying area control 
trigger and terminate overall terrain surface reconstruction 
process in our research. 
Off-terrain 
space 
   
  
Buffer space 
  
  
On-terrain 
space frequency 
@  on-terrain point O 
off-terrain point 
== real terrain surface —:— plane terrain model 9, 
Figure 2. Illustration of a “buffer space” used to provide a 
triggering and terminating condition for the terrain 
fragmentation process. 
3.2 Terrain Polarity Measurement 
If the terrain fragmentation process is triggered, the remaining 
problem is to seek the “best fit” planar surface model to 
reconstruct underlying real terrain surface as being plane. To 
this end, it is necessary to discuss a criterion for this model 
selection problem. Let us define an intra-relationship of on- 
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