The mathematical model corresponding to this figure is 
A r"(t) 2 r"(t) * sR"s(t)p", (1) 
where 
2p 
ete peese iuste inane 
     
      
  
   
p Uncorrected 
\ Image vector 
Corrected| \ 
image \ 
vector b 
SgP 
or 
Y 
\ 
"uer \ Py 
Georeferenced Uncorected 
Yom positon position 
   
Ellipsoid 
Figure 1: Georeferencing of Airborne Sensing Data 
Ar" is the position vector of an image object in the 
chosen mapping frame; 
r" is the coordinate vector from the origin of the 
mapping frame to the centre of the position 
sensor on the airplane, given in the m-frame 
R™,  isthe three-dimensional transformation matrix 
which rotates the aircraft body frame (b-frame) 
into the mapping frame; 
S is a scale factor which depends on the distance 
between camera and object; 
p° is the vector of image coordinates given in the 
b-frame. 
Equation (1) is only a first approximation of the actual 
situation. Since the three sensors for positioning, attitude 
determination, and imaging are at different locations in 
the aircraft, another set of transformations is necessary 
to relate all sensors to the origin and the axes of the 
chosen body frame in the aircraft. The parameters of 
these transformations can be considered fixed for one 
mission and are obtained through a pre-flight calibration, 
for details see Schwarz et al. (1994). 
The resulting modelling equations are 
A r"(t) = r"(t) + R"h(t) { s dR% (p° + dr’) + dr®} (2) 
The additional vectors and matrices in equation (2) are 
as follows: 
dR is the transformation matrix which rotates the 
camera frame (c-frame) into the body frame; 
p is the imaging vector in the c-frame; 
dr? is the translation vector between the centre of 
the attitude sensor and the perspective centre of 
the camera; and 
dr“ is the translation vector between the centre of 
the positioning sensor and the perspective 
centre of the camera. 
68 
The camera frame is defined as having its origin in the 
perspective centre of the camera, its z-axis along the 
vector between the perspective centre and the principal 
point of the photograph, and its (x,y)-axes in the plane of 
the photograph with origin at the principal point. The 
corresponding image vector is therefore of the form 
graben (3) 
Ef 
In case of pushbroom scanners and CCD frame imagers, 
the second vector component is replaced by 
y, 7 (y - yo) / ky 
where ky accounts for the non-squareness of the CCD 
pixels. 
It should be noted that the origins of the position and 
attitude sensors are not identical. Furthermore, the 
vectors r" and A r", as well as the rotation matrix R™p are 
time dependent quantities while the vectors p^ and dr^, as 
well as the matrix dR" are not.. This implies that the 
aircraft is considered as a rigid body whose rotational 
and translational dynamics is adequately described by 
changes in Ar” and R™p. This means that the 
translational and rotational dynamics at the three sensor 
locations is uniform, in other words, differential rotations 
and translations between the three locations have not 
been modelled. It also means that the origin and 
orientation of the three sensor systems can be 
considered fixed for the duration of the flight. These are 
valid assumptions in most cases but may not always be 
true. 
The quantities Ar", R"y and p^ in equation (2) are 
determined by measurement, the first two in real time, 
the third in post mission. The quantities dR’; and dr” are 
determined by calibration, either before or during the 
mission. To determine dR" by calibration, a minimum of 
three well determined ground control points are required, 
while dr? can be obtained by direct measurement on the 
ground. The scale factor s changes with flying altitude of 
the aircraft above ground. It can, therefore, either be 
approximated by assuming a constant flying altitude, 
calibrated by introducing a digital terrain model, or 
determined by measurement, using either stereo 
techniques or an auxiliary device such as a laser 
scanner. For precise georeferencing, the latter technique 
is the most interesting to be investigated because it 
provides all necessary measurements form the same 
airborne platform. Thus, datum problems can be 
avoided. For a more detailed discussion of calibration 
issues, see Schwarz et al. (1993). 
3. DETERMINATION OF GEOREFERENCING 
PARAMETERS BY GPS AND INS 
For the georeferencing process, the parameters A r" and 
R", are obviously of prime importance. They are usually 
determined by combining the output of an inertial 
measuring unit (IMU) with that of one or several receivers 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B6. Vienna 1996 
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