ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
Due to these two disadvantages, the use of non-metric 
cameras in photogrammetric applications is considered under 
very strict constraints. However if non-metric cameras are 
calibrated properly they could be used in more applications. 
In section 3.1 a brief summary of the systematic errors 
introduced by the non-metric cameras is presented. Section 
3.2 describes the mathematical models employed to 
transform image space coordinates to the object space 
coordinates taking into account the effect of the systematic 
errors. 
3.1 Systematic Errors Introduced By  Non-Metric 
Cameras 
"Systematic errors in non-metric cameras are the deviation of 
the physical imaging event from the projection mathematical 
model", Karara (1989). There are two types of systematic 
errors; the external systematic errors, and the internal 
systematic errors. The sources of the external systematic 
errors are: the atmospheric refraction and the comparator 
errors. However, according to (Marzan and Karara, 1976) the 
atmospheric refraction errors are neglected for object 
distances less than 300 meters. The causes of the internal 
systematic errors are: the lens distortion, the film 
deformation, and film unflatness. 
Lens distortion is considered as the major source of errors for 
the non-metric cameras. Lens distortion is introduced by two 
components: symmetric lens distortion and asymmetric lens 
distortion. The adopted mathematical model for symmetric 
lens distortion according to (Brown, 1971) is given by 
Equation (5). 
Ar = kyr + kor +r” +... (5) 
ere k „‚k „‚k , 7 constants, 
Wieie t 25. 
r ^ radial distance from the point of symmetry, and 
Ar = symmetric lens correction in the radial direction 
The asymmetrical lens distortion is due to the lenses 
decentering. We employed the correction model presented by 
(Faig and Moniwa, 1972), Equations (6). 
Ax = by (r? +2x2)+2byxy 
(6) 
Ay = by (r? +2y°)+2b,xy 
Where b D b, — constants, 
r — radial distance from the point of symmetry, 
x, y 7 point coordinates in image space, and 
Ax, Ay = symmetric lens corrections 
The film deformation inside the camera is caused by bad 
flatness of the film. The film deformation outside the camera 
is due to lack of control over temperature, and the humidity 
during processing and storage of non-metric films. In 
(Robson, 1992) the 6-paramer transformation model is used 
to model the total film deformation. 
3.2 Transformation Between Image Coordinates and 
Ground Coordinates 
We employed four models to transform the image 
coordinates and the object coordinates. The first model is the 
collinearity representation with no systematic error 
correction parameters. Nine transformation parameters are 
included in this model. The second model is a modified 
version of the collinearity representation, 6 more parameters 
are added to handle the systematic errors, Equation (7). The 
third model is the DLT representation, (Abdel-Aziz and 
Karara, 1971). The fourth model is an adapted form of the 
DLT representation, Equation (8). The DLT is a linear 
treatment of a non-linear problem so its results are 
approximate. 
a, *axa,yt kar? - 
my =X) mo(Y -Y.)* ma -Z.) 
m3yX —X,)* myyY -Y,) t ma33Z —-Z.) 
b, bx by ko yr! - 
aX - X) myyY -Y.)* mayZ 7Z«) 
my (X — X) *myyY - Y.) + M33(Z = Ze) 
  
(7) 
  
Where f — camera interior parameters, 
x. Y, Zz, = exposure station coordinates, 
x, y = image space coordinates, 
X, Y, Z = object space coordinates, 
m through m. rotation matrix elements, 
r = radial distance from the point of symmetry, and 
k,k,b,b,b ,a ,a,a =systematic error 
12 ro iy ela 
correction parameters 
xlX LY +I3Z la loxX * lgxY el XZ + xr? 0 
V+ Is X +16Y +17Z + Ig +1gxX + [i xY ++I xZ ++kpyr? = 0 
(8) 
Where x, y ^ image space coordinates, 
X, Y, Z ^ object space coordinates, 
I through un = DLT parameters, 
r = radial distance from the point of symmetry, and 
k, k, = systematic error correction parameters 
4. DIGITAL APPROACH FOR OBJECT SPACE 
RECONS TRUCTION 
Two object space reconstruction systems are presented in this 
paper. The first system employs a cheap 35mm camera, Zenit 
E, and an inexpensive flatbed scanner, while the second 
system uses only a low-resolution digital camera, Casio QV- 
A10. First 50 target points are fixed as shown in Figure 3. 
The points are fixed on wood plates with different thickness 
varying from 5 mm to 10cm. A theodolite and a total station 
are employed to find the reference ground coordinates of the 
target points.