Hrabacek, Jan 
  
2.3 Relations 
The possibility to compute coordinates of object points using images with known exterior orientation, is provided by the 
relationship, which is visualised at the figure 2. An object point (denoted O) belongs to an edge, which has a corresponding 
observation (image line) in the image. Therefore, the object point also belongs to the interpretation plane, that is generated 
by the image line and the projection center, denoted P. Let's denote r the position vector of the projection center P, x the 
position vector of object point O. Because both the point P and the point O belong to the interpretation plane, then the 
vector connecting P and O 
PÓsx-r ^ (3) 
also belongs to the interpretation plane. Perpendicularity of n;; to each vector of the interpretation plane gives the inner 
product 
n;; (x—r) = 0 , (n 
The equation (4) is the basic equation of the model, on which the least squares principle is applied. 
3 CONSTRAINTS IN THE MODEL 
3.1 Functionality of constraints 
Several groups of constraints can be distinguished in the model. We concentrated on the group of geometrical object 
constraints, but also other constraints were investigated. 
Due to the formulation of the mathematical model, the set of constraints is always non-empty. Each plane leads to a 
constraint on the length of its normal vector. The exterior orientation parameters cause the presence of a similar constraint 
on the four quaternion elements. This constraining is a price for the advantages of using redundant parameters. 
The control point coordinates are also introduced as weighted constraints. A minimum of seven coordinate constraints is 
required to define the coordinate system. The co-planarity constraint has a particular significance in the model. It can be 
classified as the special case of the geometrical object constraint on the distance between a point and a plane. But its main 
function is to ensure that the faces of the model are (almost) planar. The constraint is set for each point at least once. If 
the point is in more than one plane a constraint is set for each plane the point belongs to. 
All the constraints above are always present in the mathematical model, in constrast to the geometrical object constraints. 
They are added to the system after manual specification. The main purpose of these constraints can be described as 
“quality improvement”. There are several important factors, which influence the quality of the result. The first factor is 
the quality of the approximate exterior orientation parameters. Thanks to the integration of geometrical object constraints, 
convergence of the adjustment is obtained, even for approximate exterior orientations of low quality or bad intersection 
angles. Quality and quantity of the acquired data is the next factor. The number of images can be limited, due to either 
restrictions for image acquisition, or due to the need to economise processing. Obstacles and poorly visible features in 
images also have a quality decreasing effect. All these factors result in a lack of the desired properties of the model, with 
respect to our proclaimed goal. 
We know that architectural objects are constructed with regular, or even perpendicular features, some angles or distances 
may be known. When the photogrammetric network is too weak to avoid bad intersection angles, the non-measurement 
information is a welcome new source of information. This information is expressed in the form of geometric object 
constraints. 
3.2 Weighting of constraints 
The model integrates the image observations, with pseudo-observation equations for the constraints. In section 3.5 it is 
explained how this integration is done. The importance of a consistent weighting of different types of observations is well 
known in the adjustment theory. 
In principle, the weighting of the image observations is straight forward (part (19) of the model). A propagation derived 
from (4) is computed for the observations. Then, depending on the stochastic model applied, shorter observed lines are 
given lower weights, than longer lines. Those weights are absolut in the sense that they are derived from an assumption 
on the measurement precision in the image. 
In order to obtain a well balanced model, we have tried to use realistic standard deviations for computing weights of 
constraint equations. The constraints should be characterised by standard deviations too. In this application, the con- 
straints cannot be interpreted like geometrical axioms with infinite weights. For an example of the effects of omitting 
constraints, we point to section 4.1. Experiments showed that hard weighting of constraints can cause the adjustement to 
  
382 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 
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