nbul 2004 
f the road 
se points. 
2 
Ps 
/ 
#4 
P, 
an object 
ope values 
2 object is 
(7) 
sive points 
direction 
values of 
(8) 
the centre 
side of the 
'oduced for 
from the 
centre axis 
points are 
ed to form 
ts the road 
pe (Figure 
horizontal. 
ections and 
(9) 
p3 and p, as 
of equation 
e unknown 
yrdinates of 
stimated. A 
'OSS section 
Itre axis. AS 
b. Equation 
p, and the 
), and again 
f profile p; 
process the 
es can be 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004 
calculated. This straight line represents the non-horizontal 
profile p». 
3.2.3 Inequality constrained least squares adjustment: 
The basic observation equations (section 3.2.1) and the equation 
and inequation constraints (section 3.2.2) have to be introduced 
in the optimization process which is based on an inequality 
constrained least squares adjustment. The stochastic model of 
the observations (basic observations and equation constraints) 
consists of the covariance matrix which can be transformed into 
the weight matrix. Assuming that the observations are 
stochastically independent, the diagonal of the weight matrix 
contains the reciprocal accuracies of the observations. To fulfill 
the equation constraints the corresponding pseudo observation 
has to have a very high accuracy and the corresponding 
diagonal element of the weight matrix has to be large. The 
possibility to solve the optimization process, i.e. the semantic 
correctness of the resulting integrated data set depends on the 
choice of the individual weights. The algorithm is formulated as 
the linear complementary problem (LCP) which is solved using 
the Lemke algorithm (Lemke, 1968). For more details see Koch 
(2003), the LCP is explained in detail in Lawson & Hanson; 
1995; Heipke, 1986; Fritsch, 1985 and Schaffrin, 1981. 
4. RESULTS 
The results presented here were determined using simulated and 
real data sets. Two different objects were used - a lake which 
can be represented by a horizontal plane and a road which can 
be composed of several tilted planes. The simulated data consist 
of a DTM with about 100 height values containing one 
topographic object. The heights are approximately distributed in 
a regular grid with a grid size of about 25 meters. 
The real data consist of the DTM ATKIS DGMS, a hybrid data 
set containing regularly distributed points with a grid size of 
12,5 m and additional structure elements. The 2D topographic 
vector data are objects of the German ATKIS Basis-DLM. 
Three different lakes were used bordered by polygons. The 
objects are shown on the left side of Figure 1. 
4.1 Simulated data 
In case of a lake, the basic observation equations (1) and (2), 
the equation constraints (3), (4) and the inequation constraint 
(5) are used. The unknown lake height is identical to the mean 
value of the heights inside the lake. This is true if the 
neighbouring heights outside the lake are higher than the mean 
height value, i.e. if the inequation constraints (5) are fulfilled 
before the optimization begins. It is also true if neighbouring 
heights outside the lake are somewhat lower than the mean 
height value and equation (3) has a very high weight. Here, 
equation (3) was given a weight of 10° times higher than all 
other observations. Equation (4) had a rather low weight 
because the heights are not original heights of the DTM. 
After the optimization process the equation and inequation 
constraints are fulfilled, and thus the neighbouring heights 
outside the lake are higher than the estimated lake height. All 
heights inside the lake and at the waterline have the same height 
level; the integrated data set is consistent with the human view 
of a lake. 
If some heights outside the lake are too low and the heights of 
the bank have a high weight, the lake height is pushed down. 
Then, the heights outside are nearly unchanged, consistency is 
again achieved. 
327 
The second simulated data set represents a road with five initial 
polyline points. The maximum height difference is 6 m, the 
road length is 160 m and the width is 4 m. 
The investigations were carried out by using different weights 
for the basic observation equations (1), (2) and the equality 
constraints (8), (9). Furthermore, the inequation constraints (6) 
and (7) were used. Equation (1) was considered for all points of 
the bordering polygon, the points of the centre axis and the 
points outside the object which are connected to the polygon 
points. Equation (2) represents the connections to the 
neighbouring terrain. Using the same weight for all 
observations results in a road with non-horizontal cross sections 
and differences to the tilted planes. After the optimization 
process the inequation constraints are fulfilled and the 
maximum differences between the initial DTM heights and the 
heights of the integrated data set are in an order of half a meter. 
Using higher weights (10° times higher than other weights) for 
the basic equation (2) and the equation constraints (8) and (9) 
leads to horizontal cross sections and nearly no differences to 
the tilted planes. The maximum differences between the initial 
DTM heights and the heights of the integrated data set are 
somewhat bigger than the differences before. 
If just the equation constraints (8) and (9) have a high weight, 
the equation and inequation constraints are fulfilled exactly. 
However, compared to thé results before, the terrain 
morphology has changed considerably. 
The results show, that a compromise has to be found between 
fulfilling the equation constraints and changing the terrain 
morphology. Using a higher weight of 10° leads to fixed 
observations, i.e. the equation constraints are fulfilled exactly. 
But, the terrain morphology is not the same as before. 
4.2 Real data 
The real data sets representing lakes consists of three ATKIS 
Basis-DLM objects with 294 planimetric polygon points. The 
DTM contains 1.961 grid points with additional 1.047 points 
representing structure elements (break lines). The semantically 
correct integration was carried out by using the same equations 
as in the simulations and high weights for the equation 
constraints (3) and for the basic observation equation (1) (10^ 
times higher than other weights). 
The number of basic observations and equation constraints is 
2.754; 533 parameters had to be estimated and the number of 
inequation constraints is 530. The results show, that all 
constraints were fulfilled after applying the optimization. The 
differences between the estimated lake heights and the initial 
mean height values are very small. The first mean height value 
is reduced by 2 mm and the second one by 4 mm. The third lake 
is 3,7 cm lower than the original mean height value which is 
caused by a higher number of heights at the bank which did not 
fulfill the inequation constraint (5). 
Js 
     
    
   
Y* 4 y 
z^ Md 
A ALIE 
fur 
     
WP 
  
Figure 6: Residuals after the optimization process, blue: terrain 
is pushed down, red: terrain is pushed up 
Figure 6 shows the residuals after the optimization process. The 
blue vectors correspond to adjusted height values which are