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be non-convergent. The loss of convergency is the main proof, that the introduction of weighting for constraints is not 
only more exact, but sometimes even necessary! 
In following sections, standard deviations of the constraints are disscussed. In all cases, the constraints have been consid- 
ered as non-correlated and the formula 
2 
Pe =1/m: ; (5) 
was used for computing weights pc from standard deviations mo (here of a general constraint) in the adjustment. 
3.3 Formulation of implemented constraints and their standard deviations 
3.3.1 Constraints weighted as hard constraints Two constraints creating this group have very simple weighting due 
to their character. Namely the quaternion constraint on quaternions q, — q4, defining a rotation matrix 
Vd-ded-d-1-0 , (6) 
and a similar constraint on the length of the normal vector n = (n,,n,,n.) of an object plane 
Jritniinz-1=0 , (7) 
are the exceptions, because only these two constraints are defined as axioms. These constraints are given very small, but 
non-zero standard deviations. These constraints do not affect the geometry of the model at all. 
3.3.2 Co-planarity constraints The formulation of co-planarity of a point using parameters allowed by the model is 
(Xi; n;) ime In; =0 , (8) 
In; — distance between the plane and the origin of a chosen coordinate system. 
( , ) — notation for the inner (dot) product. 
For its weighting, riz; p is used, entered as a system parameter. It expresses two effects. One of them is the position 
uncertainty of the point with respect to the plane due to a face that is not perfectly planar. An error caused by the 
generalisation is the other effect, and probably a more significant one. 
Therefore, the weight of this constraint expresses and prevents an influence of a possible irregularity, which is either 
difficult to describe or even to estimate. If we want to use a polyhedral generalisation, then decreasing the weight of the 
co-planarity equations results in a violation against this principle. Therefore, weighting of this constraint may remain a 
subject of further investigation. 
3.3.3 Constraints for control points In contrast with the previous constraint, the precision of control points has a 
clear meaning. Generally, a triple of constrained point coordinates has a (3x3) full co-variance matrix. The constraint on 
one particular coordinate (here for a coordinate z) is realised as an equation of the type 
xz; = const . (9) 
A combination of at least seven control coordinates is required to avoid the singularity of the system (22). Theoretically, 
for a full co-variance matrix of the coordinates one obtains the full weight matrix, which is its inverse. We have im- 
plemented a common standard deviation mo p characterizing a complete file of control points. In practice, co-variance 
information about control points is hardly ever available. 
3.3.4 Parallelogram constraint This special type of constraint was discussed in (van den Heuvel, 1997), where it was 
applied in image space. Here, it is applied to the co-ordinates of the corner points of a parallelogram in object space. The 
assumption results to a set of three equations 
X: —21- 22 -23— 134 =0 (10) 
y,z: similarly : 
The set has the advantage of linearity and can also be useful for a simple computation of approximate values of coordi- 
nates. Assuming non-correlated x; — 14, for simplicity with the same variance m2 for each coordinate, then computing 
the propagation gives the variance of the constraint 
2 2 
mpi — 4 m, : (1 1) 
In practice, it is a difficult task to find a good estimation for my. We use the value of standard deviation of the co-planarity 
constraint moop for my. 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 383