ded, 
ors, 
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discussed in this paper. First we wish to review the ways in which the various approaches differ : 
a) Some of the methods involve partial calibration while others lead to complete calibration of 
the camera. Partial calibration considers only a limited number of elements of interior orientation 
(e.g. lens distortions-radial and in some cases ælso decentering). In complete calibration, all 
elements of interior orientation are considered, 
b) Some methods are designed for a specific type of test object, while others do not require any 
test field. Among the methods not requiring test objects are Brown's self calibration method 
briefly outlined in Section 3.1 and Jacobi's approach sketched in Section 3.2. Test objects used 
in some of the calibration methods discussed in this paper may be grouped as follows : 
A - One dimensional test object, as used by Wôälpert and briefly outlined in Section 3. 3. 
B - Two-dimensional test objects. In most cases planar target ranges are used and the target 
coordinates are determined to a high degree of accuracy by geodetic methods. For example, 
Hallert uses circular grids (targets arranged in concentric circles), as briefly sketched in 
Section 3.4. On the other hand, Brown uses a number of planar plumb lines as his test object, as 
briefly described in Section 3.6. 
C - Three-dimensional test objects. In most cases target ranges are used, in which the spatial 
coordinates of the targets are determined to a high degree of accuracy by geodetic measurements. 
This approach is used by Torlegàrd (Section 3. 7) and Harley (Section 3. 8). Dóhler uses a rotatable 
staff with targets (whose coordinates are determined to a high degree of accuracy) arranged in 
seven parallel planes, as briefly outlined in Section 3. 5..A geometric approach which incorporates 
a three-dimensional test object consisting of planar sets of parallel straight lines and a number 
of rods extending beyond the plane, is used by Abdel-Aziz as briefly outlined in Section 3. 9. 
3.1. Brown's Analytical Self- Calibration Approach 
  
Without requiring any absolute object space control, this powerful calibration method 
uses multiple exposure stations for the calibration of radial and decentering lens distortions. It 
incorporates Brown's [ 9] extension of Magill's formula into the process of multiple station 
analytical stereotriangulation developed by Brown [6] . This method "requires that lens distortion 
coefficients be precalibrated for one object plane and regards as unknown the distortion coefficients 
for the particular object plane on which the camera is focussed. It also regards coefficients of 
decentering distortion as being unknown". 
While no absolute control is required in this approach, it can be incorporated in the 
solution if it is available. 
"If highly convergent geometry from three or more exposure stations is exercised, the 
program can also accomplish an accurate calibration of the principal point Xp» Yp: If." in 
addition, at least one distance between targets in object space is known, the program can also 
recover the principal distance c of the camera (otherwise a pre-established value of c must be 
enforced) '. 
Among its many applications, this approach was used recently for complete calibration 
of non-metric cameras [10], and the results were highly satisfactory. 
3.2. Jacobi's Method of Calibration of Non- Metric Cameras 
  
Jacobi's method [20] is designed for partial camera calibration in connection with object 
photography. By referring the radial lens distortion to the optical axis of the lens (and not as usual 
to the principal point of the photograph), the radial distortion can be determined beforehand (e. g. 
using a three-dimensional test object) and thereby be separated from the other interior orientation 
elements. This way, the number of object-space control points necessary for the calibration of 
every single photograph is reduced. 
The position of the optical axis is defined mathematically through the introduction of two 
angles (a and in Fig. 1) as additional inner orientation unknowns, 
Except for the lens distortion, all parameters of interior orientation are determined 
analytically together with the elements of exterior orientation for each photograph. 
95.