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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.