scanners carried on satellite and aircraft (Poli, 2003). The model 
is based on the photogrammetric collinearity equations, because 
each image line is the result of a nearly parallel projection in the 
flight direction and a perspective projection in the CCD line 
direction. 
The model can be applied to single- and multi-lens sensors. In 
case of multi-lens sensors, like SPOT-S/HRS, additional 
parameters describing the relative orientation (displacements 
and rotations) of each lens with respect to a suitable central 
point are introduced. During the georeferencing of images from 
linear CCD array scanners, particular attention must be paid to 
their external orientation, because each image line is acquired 
with a different external orientation, that cannot be estimated 
with a classical bundle adjustment, due to the large number of 
unknowns (6 for each image line). The sensor position and 
attitude are modeled with piecewise 2" order polynomial 
functions depending on time. The platform trajectory is divided 
into segments according to the number and distribution of 
available Ground Control Points (GCPs) and Tie Points (TPs) 
and for each segment the sensor position and attitude are 
modeled by 2™ order polynomials. At the points of conjunction 
between adjacent segments constraints on the zero, first and 
second order continuity are imposed on the trajectory functions. 
Additional pseudo-observations can fix some or all parameters 
to suitable values. For example, if the 2™ order parameters are 
fixed to zero, the polynomial degree is reduced to 1 (linear 
functions). This option allows the modeling of the sensor 
position and attitude in each segment with 2" or ]* order 
polynomials, according to the characteristics of the trajectory of 
the current case study. In case of sensors carried on aircraft, 
additional GPS and INS observations can be included in the 
model (Poli, 2002). 
The sensor model includes aalso a self-calibration, which is 
required for the correction of the systematic errors due to: 
principal point displacement (d,, d,), focal length variation (d.), 
radial symmetric (k;, &;) and decentering lens distortion (p;, p»), 
scale variation in CCD line direction (s,) and the CCD line 
rotation in the focal plane (0). 
Finally the functions modeling the external and the internal 
orientation are integrated into the collinearity equations, 
resulting in an indirect georeferencing model. Due to their non- 
linearity, the complete equations are linearized according to the 
first-order Taylor decomposition with respect to the unknown 
parameters. For this operation initial approximations for the 
unknown parameters are needed. The resulting system is solved 
with a least squares method. An overview of this sensor model 
is given in Figure 4. 
The sensor model was applied in order to orient the stereopair 
and estimate the ground coordinates of the Check Points (CPs). 
The available ephemeris (sensor position and velocity) were 
used to generate the approximate values for the parameters 
modeling the sensor external orientation (position and attitude) 
in fixed Earth-centred geocentric Cartesian system. 
The GCPs coordinates were transformed into the same system. 
From the available 41 object points, a group of them was used 
as GCPs and the remaining as CPs. Different tests have been 
carried out in order to choose the best input configuration. 
   
  
  
  
   
  
    
  
   
   
  
  
   
  
   
  
  
   
  
  
  
  
  
  
  
  
  
   
   
   
  
  
  
  
  
   
  
  
  
   
  
   
   
   
   
   
  
   
  
   
  
  
  
   
  
  
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part Bl. Istanbul 2004 
  
| COLUNEARITY EQUATIONS FOR ONE-LENS SENSORS 2 
  
  
  
EXTENSION TO 
MULTI-LENS SENSORS 
  
  
pr 
| 
EXTENSION TO GPSANS | 
| OBSERVATIONS X a 
  
| EXTERNAL ORIENTATION MODELLING | 
      
[ ES 
i 
«C GPSINS? >> 
NO 
SELF-CALIBRATION | 
Y 
  
f 
| INTEGRATION IN COLLINEARITY EQ. | 
  
| LEAST SQUARES ADJUSTMENT | 
Y 
  
Figure 4. Flowchart with main components of the rigorous 
sensor model. 
The tests were set as follows: 
e external orientation modeling with quadratic functions, 
varying the number of segments and GCPs configurations, 
no self-calibration; 
e external orientation modeling with linear and quadratic 
functions, best GCPs configuration and best trajectory 
segments, no self-calibration; 
e self-calibration with best external orientation modeling 
configuration. 
The choice of the unknown self-calibration parameters to 
include in the modeling is based on the analysis of the cross- 
correlation between the self-calibration parameters, the external 
orientation parameters and the ground coordinates of the TPs. 
The best results in the CPs were obtained by modeling the, 
external orientation with 2 segments and 2™ order functions and 
with self-calibration. The self-calibration parameters that mostly 
influenced the model were &;, k», p» and s, for both lenses and 8 
for both lines. The other self-calibration parameters were not 
used because they correlated highly (79596) with the external 
orientation parameters. By changing the number of GCPs and 
TPs, the RMSE were always less than 1 pixel (Table 3). Figure 
5 shows the residuals in planimetry (top) and in height (bottom) 
using all the object points as GCPs. 
Table 3. RMSE for all points using rigorous orientation model. 
  
  
  
  
  
  
  
  
Number of RMSE in RMSE in RMSE in 
GCPs + CPS East (m) North (m) Height (m) 
8 +3] 3.68 6.52 4.75 
16 +25 3.46 6.22 3.75 
41 * 0 3.24 5.52 3.68 
  
   
Inter 
North 
83} 
5.29! 
Figur 
4.2 
The 
Polyı 
came 
ratior 
rema: 
FN Ir eee 0 ASN wa IN ei