process must be 
In this study, we 
f the mechanical 
)bject space. The 
ect under control. 
> configuration of 
t the condition of 
k is to achieve a 
ible to direct this 
etween the image 
nmetric treatment 
ic analysis of the 
matching by least- 
we applied for the 
f the block defined 
itended to provide 
with several fields 
ape and dimension 
actors. 
e the physical field linked to the recording of information 
based on meta-data (geometric and semantic). 
e the optical field which allows treatments by comparison. 
Hence, photogrammetry is an indirect measurement technique 
of objects recorded in the form of photographic perspectives. 
Thus the image of the object tends to replace the object itself as 
a data carrier during the actual measurement. 
2 Models in Digital Photogrammetry 
Digital photogrammetry has existed since 1980, through an 
adapted instrumentation and various models of restitutors were 
presented at the congress of Ottawa [3]. The suggestion of 
interesting solutions in terms of cost / effectiveness, amongst 
which the use of a PC type platform, appears to be appropriate 
for most applications [4]. The aim of this work is to determine 
by the calculation the external elements of two perspective 
beams, using points known on the object, and of points of 
connections common to both beams. 
Analytical restitution leads to specific determinations by 
exploiting the measurements made directly in the plan of the 
plate. The object point / image point correspondence is done 
analytically applying mathematical relations. Depending on 
whether the photogrammetric unit is the beam or the model, one 
of the two following conditions is applied: 
e colinearity 
e coplanarity 
the characteristics of a restitutor on PC are close to those 
required of a video game, and so it is worth looking closely into 
the programming techniques, since different applications [5] 
pose different problems regarding the acquisition of three 
dimensional co-ordinates and their stereoscopic visualization. 
2-1- Equation of Colinearity 
The problem arising from this condition is to find a rotation 
which makes the image vector Om and the object vector OM 
collinear, figure (1). If K indicates the scale factor relating to 
point M, the following relation will occur: 
  
x X-X, 
yl-kR | Y-—Y, (1) 
-C Z-20 
O 
X 
-C 
< 
M 
Figure 1 : Co linearity Condition 
2-2 Equation of Coplanarity 
The problem to be treated is no other than that of the formation 
of the image - analytical model, i.e. the realization of the 
relative orientation and it results in: 
l. Finding the rotation matrices R, and R;. 
2. Or finding the rotation matrix R; and the translations 
according to axes which make the three vectors coplanar, as 
defined in figure (2) below: 
The base b=0,0, 
The homologous rays: O;m; and O,m, 
Zi Yi 
  
  
  
  
  
  
  
  
  
M 
Figure 2: Coplanarity condition 
If the left perspective center O, is taken as origin, the vectors 
co-ordinates will then be: 
O;O,, Om,, Om, 
b. X. x +b | 
0,0,=|by| Om =|y |;0m =|y.b 
bz Z Zz +b 
1 
In mathematical terms, the condition of coplanarity is expressed 
in the equation (2): 
b b. b 
X, Y, z 1=0 (2) 
x tb y +b z +b 
In order to restore the object, either the beam unit or the model 
unit may be chosen, but it is necessary to make the correction of 
the plate coordinates which must be used to realise the different 
orientations. 
2-3- DLT Approach 
The direct linear transformation method (D.L.T.) of the co- 
ordinates comparator with the ground co-ordinates, figure (3) 
was developed in [1]. It is based on the following pair of 
equations: 
-]11-