(c 7 50mm) from a distance H4, — 1.2m so as to obtain photos 
of a scale approximately k — 1:25. The photo base chosen 
ranged between 20 — 30 cm, in order to have a satisfying 
overlap in all cases. Additionally, during the photographing, 
colour slides Tungsten (ASA 200) were used with artificial 
lighting provided by two Soft-boxes. 
The scanning of the obtained photos has been done with a 
resolution of 600 dpi. Finally, for the determination of the photo 
orientations, the collection of the DEM data and the digital 
process of the final products (orthophotos, line-drawings etc), 
various commercial and non-commercial programs were used. 
SURFACE FITTING 
The first stage of the process is the fitting of a mathematically 
defined surface on the 3D point cloud. For this process it was 
decided to work only with the geodetically determined control 
points, considering them as more accurate than those of the 
DEM. Taking into account that map projections are going to be 
used, the desired model surface is that of a sphere or an 
ellipsoid. In this application, the most appropriate mathematical 
surface proved to be a sphere. 
Generally, the surface of a sphere is defined as a set of points, 
which satisfy the following equation (1): 
(1) 
F(x,z) = (x-X0)" + (Y-Yo)” HZ-20)Ÿ + R° =0 
where  R = the radius of the sphere 
Xo, Yo, Zo = the position of the centre of the sphere 
X, y, Z — object coordinates in ground coordinate 
system 
F(x, z) = the algebraic distance between the position 
of a point x and the surface z; 
This very model is used for the least squares adjustment. In 
particular, the least squares method was applied. However, in 
order for this method to be successful, the vector of the 
approximate values of the unknowns has to be determined 
(Faber, 2000). The method presented in that article is based on 
the general equation of the second order surface (2). The 
specific method mainly involves the calculation of the 
parameters of a triaxial ellipsoid; but by adjusting the method 
properly, the results may as well be used in this case leading to 
a very satisfying outcome. 
2) 
^) 2 2 
F(Z,x) = 01x" + apy” + 0332" + 205Xy + 2043X2 + 2023yZ + 
Q44X + Oy + 034Z + 0g = 0 
where  [a;, ay, as3,..., 244] = the equation coefficients 
X, y, Z — object coordinates in ground coordinate 
system 
Following the proposed process and solving the generalized 
eigenvalue problem, the values of the coefficients are calculated 
using all of the available control points. 
After the values of the coefficients have been calculated, the 
geometrically interpretable parameters of the desired surface 
may be derived. In addition to the author's suggestions for the 
determination of the position of the centre of the surface, 
another step is suggested in this contribution in order for the 
other parameters to be derived. 
------ Formation of the matrix a ------ 
a — [a11 a12/2 a13/2; a12/2 a22 a23/2 ; a13/2 a23/2 a33] 
------ Solution of the eigensystem a (rotation matrix Q and tle 
eigenvalues b) 
[Q,b] = eig(a) 
meee Calculation of the angles f, w, k ---------- 
9 — asin(Q(1,3)) 
cose — cos(qQ); 
coso — Q(3,3)/coso; 
® = acos(cosm) 
cosk = Q(1,1)/coso; 
K — acos(cosk) 
------ Calculation of the axes of the ellipsoid --- 
MI = [X-xm Y-ym Z-zm]; 
M = Q'*M1'; 
M = M; 
D = zeros(L,1); 
fori=1:L 
D(1) = M(1,:)*b*M(1,:)"; 
end 
d = mean(D) 
al ^ sqrt(d/b(1,1)) 
bl = sqrt(d/b(2,2)) 
cl = sqrt(d/b(3,3)) 
In case a rotational ellipsoid is going to be used, the derived 
values, for the rotations and the axes, can be used after some 
proper adjustment e.g. after averaging of some values (This 
process has successfully been applied on simulation data of 
rotational ellipsoids). 
For the case of the sphere surface the rotations are considered 
indifferent at this stage, the coordinates of the centre are used as 
approximate values and the approximate value of the radius is 
calculated by averaging the values al, bl and cl. 
Having calculated the approximate values of the unknowns, à 
least squares adjustment can be applied. Considering the 
position of the centre of the sphere (x,, y, z,) and the radius (R) 
as unknown and treating the 3D space coordinates of the control 
points (x, y, z) as observations, a least squares adjustment has 
been applied. The adjustment process, apart from the values of 
the unknown parameters, gives the corresponding variations and 
the overall standard deviation of the adjustment, which may be 
considered as an index for the quality of the adjustment. 
REFERENCE SYSTEM DEFINITION 
Usually the object coordinates refer to a geodetic system, which 
has a random origin and orientation within the 3D space. Such à 
system is not always proper for projection purposes and this is 
the main reason for the definition of a new coordinate system. 
—464— 
  
Error in the x position (cm) 
e 
  
  
Figur: 
  
Error in the y position (cm) 
ce 
-90 
  
Figure