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Hrabacek, Jan 
  
A possibility of simple testing of constraints, offered by the adjustment, has been implemented. However, it is difficult to 
find a realistic precision for each constraint. A possibly weak geometry and the generalisation applied while extracting 
image lines may affect the truthfullness of constraints more than inaccuracies caused by the construction of the building. 
In our experiments an “educated guess” for the weights was made with which a good result was obtained in the sense of 
the goals proclaimed above. 
In section 2 of the paper, basic principles of the mathematical model are introduced. Section 3 presents all implemented 
features regarding the integration of constraints into the model. Reasons for using constraints, and why to use them 
with weights, are discussed. At the end of the section, the way in which the mathematical model joins the image line 
observations and the non-measured information, is presented. The example of processing a test data set fills section 4. 
2 PRINCIPLES OF LINE-PHOTOGRAMMETRY, USED MATHEMATICAL MODEL 
2.1 Object parametrisation 
As the basic type of parameters, 3D coordinates of object points are used to de- 
scribe the model. The assumption, that the model is polyhedral, is an important 
part of the concept. It has several consequences. The model must involve a cer- 
tain level of generalisation. Due to the polyhedrality, each set of points creating 
a face belongs to a common plane. In our approach the parameters of the planes 
are used as the second type of parameters. Figure 1 shows possible parameters 
for a model. In principle, an object could be defined only by planes, whose 
object points are intersections, and a topology description. For the presented 
models both point and plane parameters are used. It means a redundancy of 
parameters and it automatically leads to the need to formulate constraints that 
Figure 1: Object parametrisation force the points to corresponding planes. Although the over-parametrisation 
by planes and points solved by constraining could seem unwise, it has advantages. A relationship 
between an observation and the object point is easy to formulate. But the use of 
points as the only parameters would cause difficulties in handling plane entities. The use of object plane parameters has 
the advantage of a simple formulation of relationships and constraints on planes. 
  
A third (optional) group of parameters are the parameters of exterior orientation. Four quaternion elements parametrise 
a rotation matrix R;, see the equation (2). Replacement of three unknown rotation angles by four quaternions is another 
example of over-parametrisation. The advantage is the absence of singularities. 
2.2 Observations 
The observations are characterised by image coordinates of beginning and end 
point of each extracted line. Using the camera system, camera coordinates of 
such a point are expressed by the spatial direction vector 
dzí(r,y,—f) ’ (1) 
x, y — image coordinates of the image point. 
f — focal length. 
Deformation of the image and a lens distorsion are supposed to be already cor- 
rected. 
For later formulation of needed relationships, so called interpretation plane is 
introduced. The plane is generated by the projection center of the camera and 
by the extracted line. Figure 2 explains the genesis of an interpretation plane. 
  
Figure 2: Interpretation plane 
The formula 
n;; = R; (d; X d3) > (2) 
d;, d» - vector (1) of begin and end point of the line. 
R; — rotation matrix of camera exterior orientation. 
( x ) — cross product of vectors. 
expresses the interpretation plane, parametrised by its normal vector and rotated to the object system, where nj; denotes 
the normal vector of the interpretation plane, in the object system. 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 381