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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part BS. Istanbul 2004 
  
Figure 3. Photos with homologous points and polylines 
From now on, the homologous points have photo-coordinates, 
named (x', y"). Their object-coordinates (X, Y, Z) can be 
determined through Bundle Adjustment. 
The unknowns of this photogrammetric method are the external 
orientation elements (Xo, Yo, Zo, &, '/, &:) of each shot and the 
object-coordinates of the control points. The camera parameters 
(X'o, y'o, c) must be known or estimated. 
For each point of each shot, the linearized observation equations 
of a Least Square estimation can be written. Each point gives 
two equations coming from the equations of the central 
projection (calculation of the point's photo-coordinates from 
approximated values of the unknowns). 
In a matrix form, this can be written as below (Kraus, 1997) : 
ysB*X-L (1) 
where v is a matrix containing the residual vector of the 
photo-coordinates 
B  isamatrix containing the partial derivatives 
(regarding the unknowns) of the calculated 
photo-coordinates (these derivatives of the 
collinear equations are calculated from 
approximated values of the unknowns) 
X  isamatrix containing the unknowns to calculate 
L isa matrix containing the subtraction of the 
measured photo-coordinates from the calculated 
The Least Square estimation permits to write that in the 
following form (named normal equation) : 
B'*BAX=B EL (2) 
For what follows, we adopt the notation : 
N=B *B nzB'*L (3) 
In order to be able to resolve the normal equation for several 
shots, the previous matrix must be written as below : 
NIl N12 IXI 
[| — T = n=| 
(4) 
TINi? N22 
i 
nl 
n2 
  
  
The unknowns are then calculated by Least Square estimation : 
X1=(NI11-NI12*N22"'*N12" y' * (nl- N12*N22'#n2 
X2=N22'*(n2-N12'*X1 ) (5) 
where X1 contains the external orientation elements 
(Xo, Yo, Zo, &, v, K) of each shot 
X2 contains the object-coordinates of all object 
points . 
Thanks to the 3D coordinates of the object points, we are now 
able to draw a vectorial model of the column. This model will 
represent the theoretical column's surface, calculated from the 
3D control points. 
2.3 Layout of a theoretical cone from the control points 
It seems that the theoretical surface of Egyptian columns is a 
cone with a very acute angle at the summit. To obtain a 
vectorial view of the column, a 3D polyline has been drawn, 
based on points describing this cone geometrically. 
Parameters of the cone have been calculated by a mathematical 
algorithm, which carries out the surface matching of a cone. 
(Shakarji, 1998) 
These parameters are : 
- the director numbers of the axis 
- one axis point 
- the summit angle 
- the distance between the axis point and a generatrix of 
the cone 
A solution is found through an iterative modification of the 
cone’s parameters. The calculation ends when the average 
distance between the cone’s points and the calculated surface 
stops decreasing. 
The 3D polyline describing the cone is constructed and recorded 
by seven points : 
the summit [1] 
the point of intersection between the axis and the 
perpendicular plan determined from the higher control 
point (represented by the high circle [HC]) [2] 
the point of intersection between the axis and the 
perpendicular plan determined from the lower control 
point (represented by the low circle [LC]) [3] 
the point of intersection between the *(OX) axis" and the 
circle [LC] [4] 
the point of intersection between the *(OX) axis" and the 
circle [HC] [5] 
the point of intersection between the *(OY) axis" and the 
circle [HC] [6] 
the point of intersection between the *(OY) axis" and the 
circle [LC] [7] 
It is worth noting that to determine the best cone approaching 
the points can lead to a cone, whose axis is not necessarily 
vertical. Indeed, the column might have suffered a slight 
subsidence or inclination in time.