Av 
Hrabacek, Jan 
  
It has to be noted, that for weighting of equations (16), m?, has to be multiplied by the length of nz, ny, respectively. 
This is caused by the non-unit length of decomposition vectors (17). 
3.5 Integration of constraints into the model 
The mathematical model integrates constraints with the observation equations. The basic equation for observations is (4). 
With (2) we see that it contains both parameters and observations. In more general point of view, (4) can be rewritten into 
a formal equation, here in a form with adjusted values 
e3-0 | (19) 
where q( , ) denotes a set of functions — conditions. Arguments of the functions are the vector of adjusted parameters 
x and the vector of adjusted observations 1. So-called least squares adjustment of condition equations with observations 
and parameters is the result of applying the least squares principle on the equation (19). Similarly, the formal equation 
for the set of constraints is expressed by 
p(X) =0 , (20) 
with the same notation as for (19). This set can be interpreted as pseudo-observation equations. With this formulation 
constraints can be given weights by means of a submatrix of the weight matrix of the observations. The final step of the 
task is to find a least squares solution respecting (19), (20). Linearisation of the equations (19), (20) gives 
B-v + A-dx + 1, = 0 
CU en 
v — vector of corrections to observations. 
dx — vector of corrections to parameters. 
A, B — matrices of partial derivatives of (19). 
C? — matrix of partial derivatives of (20). 
',, le; — vectors of misclosures. 
The system (21) is solved in two steps. In the first step, the vector dx is computed from the system 
A ! 
[n] ED : 
on condition z Q:!z — min, where Q. is the co-factor matrix of the system. Then the left part 
Biv iiz2A ; (23) 
is solved with v as unknown, with a similar condition v7 Q;'v = min for the co-factor matrix Q, of observations. The 
computation has to be done iteratively due to non-linearities of the model. For more details, see (van den Heuvel, 1999) 
or (Mikhail, 1976). 
4 EXAMPLE 
Two reconstructions have been selected from a number of experiments. The results of other experiments are briefly 
discussed in the conclusions. Presented examples illustrate the constrast between the result obtained almost without 
constraints and the constraint-based reconstruction of the same model. Both examples use a stochastic model, which 
gives a shorter line relatively lower weight. The precision is derived from the standard deviation of the coordinates of the 
end points of the lines, estimated as the size of one pixel. For the used images, this size is approximatelly 0.0065 x 0.0065 
mm. The stochastic model is described in (van den Heuvel, 1998). The CIPA test data set of Zürich City Hall was used, 
particulary four images taken with an OLYMPUS C1400L camera (8.2 milimeters focal length) were selected. Figure 4 
shows the used images and gives an impression of the data. For more details regarding the data set, consult CIPA web site 
(CIPA, 1999). 
# of image # of parallelogram st. dev. of result st. dev. of cons. common 
lines/params. consts/others coords. mean/max. angle+parall. settings 
Model #1 112/154 0/0 0.17 m/0.47 m moop -— 0.05 m 
Model #2 393/426 99(33)/41(27) 0.14 m/0.25 m 25'/50'/80'/150! mcp =0.10m 
Table 1: Statistics of the models 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part BS. Amsterdam 2000. 385