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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
It is based on a constrained Delaunay triangulation using all
points of the DTM (mass points and structure elements) and the
polygons of the topographic objects of the 2D GIS data (section
3.1). The linear structure elements from the DTM and the object
borders are introduced as edges of the triangulation, the result is
an integrated DTM TIN.
Then, certain constraints are formulated and are taken care of in
an optimization process (section 3.2) In this way, the
topographic objects of the integrated data set are made to fulfill
predefined conditions related to their semantics. The constraints
are expressed in terms of mathematical equations and
inequations. The algorithm results in improved height values
and in a semantically correct integrated 2.5D topographic data
set.
A basic assumption of our approach is that the general terrain
morphology as reflected in the DTM is correct and has to be
preserved also in the neighbourhood of objects carrying implicit
height information. Therefore, any changes must be as small as
possible. A second assumption is that inconsistencies between
the DTM and the topographic objects stem from inaccurate
DTM heights and not from planimetric errors of the topographic
objects.
3.1 Non-semantic data integration
As mentioned in section 1.2 there are several approaches for the
integration of a DTM and 2D topographic GIS data based on a
TIN. Because Lenk’s approach has some advantages, we use a
variant of his algorithm. First, a DTM TIN is created using the
DTM mass points and the structure elements in a constrained
Delaunay triangulation. Second, the heights for the topographic
objects are derived using the height information of the TIN by
interpolating a height value for each object point. Next, the
points of the polygons representing the topographic objects are
introduced into the TIN by re-triangulating the neighbourhood
of the objects. Here, the Delaunay criterion is not re-
established. Then, the object lines are considered as constraints.
This is done in such a way, that the intersection points between
the object polygons and the edges of the DTM TIN (Steiner
points) are introduced as new points. The edges of the DTM
TIN and the lines of the object polygon are split.
a) b)
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Figure 2: Integration of a DTM and an object “lake”, a) original
DTM TIN and object “lake”, b) integrated data set
Figure 2 shows an example of the integration of a DTM and an
object “lake” of a 2D GIS data set. The original points of the
bounding polygon of the lake are shown in light blue. After the
integration, the intersection points between the DTM TIN and
the object polygon are new points of the integrated data set
(Figure 2b, coloured by black).
Another example is given in Figure 3. A road is an object
modelled by lines (Figure 3a, black line) which is buffered
using an attribute “road width". First, all intersection points
between the middle axis and the DTM TIN are estimated
UA
UA
(Figure 3a, light grey points of the centre axis). This is done
because every triangle has a ditferent inclination and the middle
axis should be best fitted to the terrain represented by the initial
DTM TIN. After buffering the left and right side of the road
contain as much points as the centre axis. Then, the object is
triangulated in such a way that the cross sections situated at the
points of the road centre axis border the triangles of the object
TIN. Thus, it is garanteed that for a profile a change in slope is
allowed. The bordering polygon is then introduced using the
variant of Lenk's algorithm (Figure 3b).
a) b)
\
Figure 3: Integration of a DTM and an object "road", a) original
DTM TIN and object “road“ after buffering, b)
integrated data set
3.2 Optimization process
As mentioned, there are topographic objects of the 2D GIS data
which contain implicit height information. Within the integrated
data set these objects have to fulfill certain constraints which
can be expressed in terms of mathematical equations and
inequations. To fulfill these constraints or to achieve semantic
correctness, the heights of the DTM are changed. Up to now the
horizontal coordinates of the polygons of the topographic
objects are introduced as error-free.
The heights of the topographic objects are estimated within an
optimization process which is based on a least squares
adjustment; these values are unknown parameters. The heights
of the corresponding part of the DTM are introduced as direct
observations for the unknown heights at the same planimetric
position. Equality constraints are introduced using pseudo
observations. Thus, a Gauss-Markov adjustment model is used
and the adherence to the constraints is controlled via weights
for the pseudo observations. Furthermore, inequality constraints
are introduced. The resulting inequality constraint least squares
adjustment is solved using the linear complementary problem
(LCP) (Lawson & Hanson 1995; Heipke 1986; Fritsch 1985;
Schaffrin 1981).
3.2.1 Basic observation equations: The heights which
correspond to the topographic objects of the 2D GIS data and
the heights of the neighbouring terrain are introduced as:
0, - Z, - Z, (1)
The height Z; refers to the original height, the value Z , denotes
the unknown height which has to be estimated.
In order to be able to preserve the slope of an edge connecting
two neighbouring points P; and P, of the DTM TIN (one of the
two points is part of the polygon describing the object, the other
one is a neighbouring point outside the object) and thus to
control the general shape of the integrated TIN additional
observation equations are formulated:
