ine sce AZ 
I. m Xo, Yo,Zo 
€ e 9 9 e i X , 
me o e e em / 
/ 
1050mm / 
. / 
e e e e e again. 
EN E = a E25 ? 
abl M 
(a) (b) 
Fig. 3 Configuration of the simulated data 
(a) projected and control points; 
(b) separation between the reference planes; 
As explained earlier a constraint must be introduced. In 
this case, the A, (plane 1050mm apart from the camera) 
was fixed as: 
A, = 1050 mm 
The same procedure was used to generate a second set 
of simulated data with a smaller AZ (50mm). The 
projection planes were supposed to be at: 
Z, =-1050mm; Z,=-1000mm and Z,=-950mm 
6.1.2 Calibration results 
In order to analyse the experiments the estimated values 
ofthe parameters are compared with the true ones, 
which are known from the simulated data. The true error 
(€,) can be defined as the difference between the 
estimated and the true parameter values. The standard 
deviation is defined as the square root of the estimated 
variance. As several groups of parameters were 
estimated (direction cosines for several projected lines) 
the mean-square values of the true errors (€,…) Were also 
computed encompassing all the projected points. 
With the simulated data available, space resection 
parameters of the camera were obtained and used to 
compute object space coordinates of the projected points 
lying at the p" reference plane. This process is repeated 
for all reference planes. The discrepancies between the 
true and the computed values, at the first plane 
(1050mm), for the XY and Z coordinates are within 
O.3mm and 1mm respectively. 
The coordinates of the projected points for the three 
reference planes were used as observations in order to 
estimate the coordinates of the projector perspective 
center and the direction vectors of the projected straight 
lines. 
Table 1 Results of the two case studies with different 
separation for the reference planes 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
AZ = 100mm AZ = 50mm 
€, 0, €, 0, 
I 0.0076 0.0005 0.0069 0.002 
m, | 0.0090 0.0005 0.0091 0.0018 
n, 0.0372 0.0016 0.0362 0.0079 
Xo | 0.383 0.420 0.492 2.069 
Y; | 0.251 0.370 1:079 1:825 
Zo. \ 0.293 1.691 1.498 8.333 
  
  
Results obtained from the two sets of simulated data 
(AZ=100mm and AZ=50mm) are presented in table 1. 
The mean-square values of the true errors of the 
estimated direction parameters and the true error for the 
projection center are presented in the first and third 
columns. The estimated standard deviations are 
presented in the second and fourth columns. The 
estimated standard deviations are similar for all the 
projected straight lines. The coordinates are given in 
millimeters and the direction cosines are undimensional. 
No significant improvement in the accuracy of the 
direction cosines could be verified as an effect of a 
greater AZ. However, the coordinates of the projection 
center were better estimated with a larger AZ (100mm). 
The estimated standard deviations of the projection 
center and the direction cosines are higher in the second 
case (smaller AZ, 50mm). With these results we can 
conclude that a stronger geometry is obtained in the first 
case, in which a greater separation (AZ) between 
reference planes was used. 
Table 2 Residuals of the XYZ coordinates after 
estimation of the projector parameters 
  
AZ = 100mm AZ = 50mm 
MSV MSV 
0.302 mm 
0.244 mm 
0.317 mm 
0.577 mm 
0.276 mm 
0.232 mm 
  
Table 2 presents the mean-square value of the residuals 
in the X, Y and Z coordinates after the estimation of the 
projector parameters, for both cases. It can be verified 
that these residuals were similar for both cases, except 
in the Z coordinates for the second case, which were 
higher. Notice that in all cases these residuals were 
smaller than 1mm. 
Using the estimated projected parameters and the 
intersection procedure presented in section 3, the 
coordinates of the projected fifteen points were 
computed using the first reference plane and the 
372 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B2. Vienna 1996 
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