transformation are depicted in Figure 1, Schwarz et-al 
(1993). 
The mathematical model corresponding to this figure is 
Ar"() - TG) Fam (D pb G) 
where 
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; 
m 
Ry is the three-dimensional transformation matrix which 
rotates the aircraft body frame into the mapping 
frame (roll, pitch, and yaw are measured by the INS); 
S is a scale factor derived from the height of the sensor 
above ground; 
  
  
     
  
   
  
  
  
  
  
  
p^ is the vector of image coordinates given in the b- 
frame. 
Zb 
Yb 
P) V m... 
Flight Path uet 
p b Uncorrected 
image vector 
\ 
\sppP 
\ 
\ 
ud Sr \ Ellipsoid 
eq d Uncorrected * 
position position 
m 
Figure 1:  Georeferencing of Airborne Sensing 
Data 
Equation (3) is, however, only a first approximation of the 
actual situation. The three sensors for positioning, attitude 
determination, and imaging are physically separated, and it 
can therefore not be assumed that they function in the same 
measurement frame. The actual situation is shown in Figure 2 
which enlarges the area around point P(t) in Figure 1. It has 
been assumed that the remote sensor, for example a 
photogrammetric camera, is mounted in the stable area of the 
airplane, that the positioning sensor, a GPS antenna is 
mounted on top of the airplane, and that the attitude sensor, 
an inertial measuring unit is mounted in the interior of the 
aircraft, somewhere close to the remote sensor. In this case, 
aircraft position is defined by the antenna centre of the GPS 
receiver (m-frame) and aircraft attitude is given by the 
internal axes of the inertial measuring unit (b-frame). 
They do in general not correspond to the position and 
attitude of the remote sensing device which is given by the 
position and orientation of the camera frame (c-frame). This 
frame has its origin in the perspective centre of the camera, 
its z-axis is defined by the vector of length f between the 
perspective centre and the principal point of the 
photograph, and its (x,y)-axes are defined in the plane of the 
photograph and are measured with respect to the principal 
point. The corresponding image vector is therefore of the 
form 
X Xp 
c— _ 
pP =/Y Yp (4) 
—f 
      
C. X 
Camera 
  
yc 
Figure 2: Coordinate Transformations Between 
Airborne Sensors 
In case of pushbroom scanners and CCD fram imagers, the 
second vector component is replaced by 
y* 2 (y - yp)/ ky 
where ky accounts for the non-squareness of the CCD pixels. 
The resulting modelling equations are 
A rI (t) » ,m (0 R7 (0 ( sdR? pC - drP } (5) 
where the subscripts and superscripts correspond to the 
frames defined above. The additional notations in Equation 
(1) are as follows: 
bir : : : 
dR, 1s the transformation matrix which rotates the camera 
frame into the body frame; 
is the imaging vector in the c-frame as given by 
Equation (4) 
dr? is the translation vector between the GPS antenna 
centre and the centre of the INS, and 
dr“ is the translation vector between the GPS antenna 
centre and the perspective centre of the camera. 
This equation, in a somewhat simplified form, has been 
discussed in detail in Schwarz et al (1993). A few remarks 
will therefore suffice here. It should be noted that the origins 
of the position and attitude sensors are not identical. 
Furthermore, the vectors r™ and Ar™, as well as the rotation 
a2 m ‘ 2% : 
matrix R, are time dependent quantities while the vectors p^ 
b 
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 RE 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 as functions of time 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™ , Ry and p^ in Equation (1) are 
determined by measurement, the first two in real time, the 
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