ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision", Graz, 2002
Triangulation. However, if no new on-terrain point can be
found from the “on-terrain point stack” after the upward divide-
and-conquer triangulation is completed over the entire current
model space, our terrain reconstruction process is terminated
(see Figure 4).
4.3.1 Observation Model
Once the upward divide-and-conquer triangulation process is
triggered for certain areas reconstructed by a planar terrain
surface model ¢. , the remaining problem is to look for “the
most reliable” on-terrain point from LIDAR dataset S; located
over @, so that this local area is fragmented into more planar
terrain surface models. To this end, a tetrahedron model I is
adopted for terrain fragmentation of 9, , in which the base
triangle of T corresponds to 9, and the remaining three lateral
facets of T; are hypothesized as planar surface models where i is
the index of the tetrahedron model candidates generated over
@, and k is the index of facets which comprise a tetrahedron
model T; (see Figure 6(a)).
on-terrain point
LIDAR point
on-terrain candidate ——:
real terrain surface
current model
model candidate
Hoe
Figure. 6 Illustration of the generation of tetrahedron model
candidates. (a) Tetrahedron model 7 , where H is the height
I! . (b) The generation of two different tetrahedron model
candidates.
Since three vertices of 9, are labelled as on-terrain points at the
previous iteration step of our terrain reconstruction process, the
remaining vertex of the tetrahedron r is used to hypothesize
an on-terrain point out of S;. Thus, a set of tetrahedron model
candidates (77) is generated, sharing its base triangle with 9,
and using each point of S; as the remaining vertex of I (see
Figure 6(b)) However, during the generation of {T; s ri
satisfying the following ^ condition is rejected;
Vke(L2,3) 's* «0, where @* is one of three lateral facets
of T; and s! is vector of LIDAR dataset belonging to the
model candidate 6) :
For simplifying mathematical notations, let us consider one of
three lateral facets 9; as ¢. Suppose then LIDAR dataset S is
located over ¢ as seen in Figure 7(a). Since the underlying area
is hypothesized as a plane terrain surface by ¢ , the vertical
height of each point of S is recomputed relative to ¢ so that z
values of LIDAR points S, are projected into a flat horizontal
plane. Then, a set of labelling observations f for S is generated
by Eq. (6) when 6, is given (see Figure 7(b)).
(XY)
© (d)
Bl ons [] bus [J offs @ onterrainpoint © off-terrain point
Figure. 7 Illustration of observation model used for the
polarity measurement.
In order to make inter- and intra-relationships for on- and off-
terrain points, a TIN is constructed over f as seen in Figure 6(c).
Now, let us introduce a new observation variable y, for the
terrain polarity measurement using this TIN. Suppose that we
have a labelling function R which assigns a new labelling
observation y, to each triangle A, of TIN from a semantic label
set {ons, bufs, offs} (see Figure 7(d)); a “ons” is assigned to A,
when all the three vertices of A, are labelled as on-terrain
points by Eq. (6); similarly to this, *bufs" is assigned when A,
is comprised of the mixture of on- and off-terrain points;
otherwise, *offs" when all the vertices of A, are labelled as off-
terrain points. This labelling function R can be described as
follows:
y={y}; Vs,eS, 7, =R(A;); 7, € {ons,bufs, offs} (7)
ons if F({s;,8,,5,}|9,0,) = {on}
R(A,)=jbufs if F({s,,s,,5,}19,6,)={on,aff} (8)
off — if F((s.s,,5,3]6.0,) - (off)
Where in Eq. (8), (s,,5,,5,,) are three vertices of A, and F is the
labelling function for a single LIDAR point. Now, we can
measure the closeness between ¢ and y in terms of terrain
polarity measurement, in which A0 serves as a parameter to
determine a degree of the smoothness and discontinuity polarity
depending on a label assigned to A,. This AO is defined as
follows:
A6 -le, -e,| (9)
where AG is the angle difference between the slope of a
triangle A, and the one of a planar surface model ¢, namely
0, and 8, respectively.
A - 34]