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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
3. GEOMETRIC INVESTIGATION 
Although a CMOS array is mechanically quite stable and the 
pixels have a fixed geometric relationship, the use of these 
imaging systems for metrology purposes requires calibrating or 
checking of these camera systems. Online calibration of cameras 
during data collection process is possible for many types of 
photogrammetric work, but offline calibration and checking is 
more recommendable in the following cases: 
When information is desired about the attainable 
accuracy of the measurement system and thus about the 
measurement accuracy at the object; 
when online calibration of the measurement system is 
almost impossible for system immanent reasons so that 
some or all other system parameters have to be 
predetermined; 
when complete imaging systems or components are to 
be tested by the manufacturer for quality-control 
purposes; 
* when digital images free from the effects of the imaging 
system are to be generated in preparation of further 
processing steps (such as rectification). 
When setting up measurement systems it will be necessary to 
determine the position of cameras or other sensors in relation to 
a world coordinate system to allow 3D determination of objects 
within these systems. 
3.1 Mathematical Modelling for Calibration of a CMOS 
Camera 
There are several mathematical models for offline calibration of 
geometric structure of the present CMOS system (Ehlers, 1997; 
Madani, 1999). These formulations conceptually can be divided 
into two main groups: Rigorous Sensor Models (RSMs) and 
Generic Sensor Models (GSMs). 
RSMs reconstruct the spatial relations between sensor and 
object based on using conventional colinearity equations. The 
method is highly suited to frame type sensors and non-linear 
effects caused by lens distortion. Film deformations are 
modelled by additional parameters or by corrections after the 
linear transformation. Such image-specific parameters often 
include the approximate sensor position coordinates and sensor 
attitude angles. As RSMs basically are nonlinear models, the 
linearization and the requirement for suitable initial values of 
the unknowns are inevitable. Therefore there is a need for 
alternative approaches in case that we could not easily apply the 
RSMs. 
GSMs are presented as a sophisticated solution for overcoming 
the RSMs limitations. Although GSMs have been adopted a 
decade ago (Paderes et. al, 1989; Greve, 1992), the attempts to 
study both theoretical properties and empirical experimental 
results have started to appear only recently and are still rarely 
reported. 
In this study the tests are conducted based on evaluation of 
several different generic mathematical models, e.g. 2D 
Projective, 3D Affine, Rational function (Atkinson K.B., 1996) 
models for offline calibration of Canon EOS-1Ds CMOS 
camera. 
® Projective Transformation 
The projective transform describes the relationship 
between two planes. It is defined by eight parameters, 
which can be derived from four object points lying in a 
plane and their corresponding image coordinates. 
ZN dica) 
C X +e Y +l 
bX +b,Y +b2 
c,X+cgY +1 
Where x, y are coordinates of points in original image; 
X.Y are coordinates of points in object space; and ay, a», 
as, by, by, bs, cy, c; are projective parameters. 
e 3D Affine Transformation 
The model for 3D analysis of linear array imagery via a 
3D affine model is given by Hattori (2000). 
x=a, X +a,Y+azZ+a4 
y=b, X+b,Y+b2,Z+b4 
Where x, y are coordinates of points in original image; 
X,Y,Z are coordinates of points in object space; and a, to 
b, are affine parameters. 
e Rational Functions 
The RFM uses a ratio of two polynomial functions to 
compute the x coordinate in the image, and a similar ratio 
to compute the y coordinate in the image. 
m, m» ma my n» na 
  
S > > ap X'rizt TT sy X TE 
_ i=0 j=0k=0 | j-0j-0k-0 ' 
m M M RA Hp H3 gh Ga 
Sev y pa X Y!z* S.* Sax zh 
i joke * i=0 j=0k=0 * 
X, y are normalized pixel coordinates on the image; X, Y, 
Z are normalized 3D coordinates on the object, and Cts 
Dijk» Cijk» dix are polynomial coefficients. The polynomial 
coefficients are called rational function coefficients 
(RFCs). 
3.2 Experiments and results 
To assess the geometric stability of Canon EOS 1-Ds images, a 
images from a calibration environment have been taken (Figure 
11). With large amount of control and check points visible in 
the image (Figure 11) the selected area can be well used for the 
evaluation of the geometric parameters if interest.