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All data to solve this three dimensional vector equa- 
tion can be taken from the reports describing the opti- 
cal design and the camera calibration, respectively. 
The equation is identical for the transformation of the 
parameters of additional refracting surfaces, if these 
are defined in a vector oriented form. 
3.1.4. Computation of Target Image Models and 
Definition of the Photogrammetric Image Points 
Two conditions must be fulfilled by the mathematical 
models of the target images: they must allow the 
identification of the intersection points of the corre- 
sponding rays and they have to be realistic models of 
the density distribution in the real target image. 
Using ray tracing for the computation of the target 
image models fulfills the first condition: Starting at 
an object target a representative number of rays is 
traced numerically through the optical system and in- 
tersected with the image plane (see in detail Kotowski 
1988 pp.324ff and Pegis 1961, pp.8f). The type of each 
ray is defined in advance. This means, each of the 
traced ray is either a principle, a meridional, a sagit- 
tal or a skew ray, it starts with a certain incident 
angle from the object target and has one specific 
wavelength. One of these rays, the principle ray of 
one colour, is defined as the corresponding ray be- 
tween object point and image point. The intersection 
of this ray as well as of all other rays results in a two 
dimensional distribution of intersection points, the 
first state of the target image model. The intersection 
point of the corresponding ray, well defined in this 
model, is defined as the photogrammetric image 
point. 
If the density distribution of all intersection points in 
a target image model is equivalent to the density dis- 
tribution of the real target image, the second condi- 
tion is fulfilled. To achieve this representative 
distribution, the correct selection of rays to be traced 
is essential. Their number and their spatial distribu- 
tion have to approximate the radiation that is sent 
back from the object target and darks the sensor sur- 
face sufficiently. 
The abstraction from the radiation to be modelled to 
the distribution of discrete rays is processed in three 
steps. In the first step only that cone of radiation is 
modelled, that is assigned to only one wavelength 
and starts only from the center of the object target. 
The resulting intersection points in the image plane 
are equivalent to the point spread function, proposed 
in this shape by Herzberger (Herzberger 1947, 1954, 
1957), called a spotdiagram. To achieve a realistic 
distribution of the rays one of the pupils is divided 
into surface elements of equal size and each ray is as- 
signed to one of these pupil elements (Herzberger 
1958, p.106 und Cox 1964 p.378). 
In the second step the object target itself is divided 
into small surface elements, which are of equal size, 
if a constant radiation can be assumed over the whole 
object target. Starting at the center of each of these 
surface elements, one spotdiagram is computed, lead- 
ing to a group of overlapping spotdiagrams in the 
image plane. This distribution of intersection points, 
accumulated for one wavelength, shall be called the 
monochromatic target image model. 
In the third step the whole spectrum of wavelengthes 
passing the optical system is divided into constant in- 
tervals. For each of the resulting wavelengthes the 
monochromatic target image model is computed as 
described in the previous paragraph. The spectral 
characteristics of the illumination, the target radi- 
ation, the transparency of the optical system and the 
sensor surface have to be considered by introducing 
an adequate weight for the rays of each of the several 
wavelengthes. 
So, the required ray tracing for the computation of a 
target image model is complete. The three accumu- 
lated varieties of intersection points are overlayed by 
a raster with a raster field size equivalent to the pixel 
size of the real digitized target image. The intersec- 
tion points are counted in each field separate and the 
number of points in one field is interpreted as a den- 
sity value, resulting in a digital image with a certain 
density characteristic. 
To represent the density characteristic within the tar- 
get image realistically, the number of traced rays 
must be large enough. To prove this the difference be- 
tween the density values derived from two intensifi- 
cations of numbers of traced rays is computed: If this 
difference is still significant, a further intensification 
has to be performed, i.e. more rays have to be traced, 
as long as the density characteristic does not change 
any more. 
As mentioned above, for each target the principle ray 
of a constant wavelength, starting from the object tar- 
get center, is defined as the corresponding ray and its 
intersection point with the image plane is the photo- 
grammetric image point. Its position within the digi- 
tal target image model is known; to identify it in the 
real target image, it has to be transferred there. 
3.2.Identification of Comparator Coordinates 
for the Defined Image Points 
To be able to identify the photogrammetric image 
points in the real target image, this and the digital 
target image model have to be matched, e.g. by least