The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
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these direct observations. Details about the generic pushbroom 
sensor model can be found in Weser et al. (2007). 
In the case of CARTOSAT 1, the metadata delivered with the 
images do not contain the information required to determine the 
parameters of the generic pushbroom scanner model, so that it 
cannot be used for direct georeferencing. However, Barista 
allows the determination of the pushbroom sensor model if at 
least some metadata plus enough GCPs are available. If the 
approximate latitude and longitude of the four comers of the 
image, the flying height of the satellite (H), the focal length, 
and the approximate look-angles of the camera are known, the 
pushbroom sensor model can be initialized, and approximate 
values for its parameters can be determined. In the case of 
CARTOSAT 1, the values for coordinates of the comers of the 
image, the flying height and the along-track look-angle are 
specified in the metadata file distributed with the CARTOSAT 
1 imagery. The values for the focal length (1945mm) and the 
sensor pixel size (7pm) are given in CARTOSAT lb (2006). 
Alternatively the focal length can also be coarsely estimated 
from the flying height and the swath width. 
Once approximate values have been determined, the precise 
values of the parameters of the pushbroom sensor model, i.e., 
the values of the parameters of the spline functions describing 
the time-dependant orbit path S(t) and attitudes R P (t), can be 
determined from GCPs. The parameters of the interior 
orientation cannot be improved, but this would hardly be 
possible anyway given the sensor geometry (very small opening 
angles). Thus, errors in the focal length will be compensated by 
shifts in the orbit path. The fact that no direct observations for 
the orbit path and attitudes are available has to be compensated 
by a larger number of GCPs than would otherwise be required 
and by modifications to the sensor model. In the following 
sections, the initialisation of the sensor model parameters and 
the modifications of the sensor model will be described. 
3.3.1 Initializing the approximate parameters of the 
generic pushbroom sensor model: 
To initialize the sensor model, the parameters c F , R M , C M , R P (7), 
S(t), and Ro in Equation 3 have to be approximately determined. 
Using the nominal values for the focal length and the pixel size 
and assuming the principal point to be in the centre of the image 
line results in c F = (N/2, 0,J) T , where N is the number of pixels 
in an image line and /is the focal length in pixels. The position 
C M of the camera in the satellite is assumed to be C M =(0,0,0) T . 
These values will be kept constant in the adjustment. All the 
spline parameters used to model the time dependant angles 
roll(t), pitch(t), yaw(t) parameterising R P (7) are also initialised 
with zero, which yields R P (7)=I. This means that the satellite 
orbit and satellite platform systems (Weser et al., 2007) are 
initially identical. R M , the rotation matrix from the camera 
system to the satellite platform system, can be computed from 
the along-track viewing angle a and the across-track viewing 
angle J3: 
Rm - [X M , Y m , Z m ] (4) 
with Z M — Zq / || Z 0 || 
Z 0 = [tan(a), -tan(fj), -1] T 
X M = [0, cos(a), -sin(/3)] T 
Ym = x X M 
The remaining parameters S(t) and Ro cannot be determined 
separately. Ro is computed from the satellite position at the 
scene centre (Weser et al, 2007) whereas S(t) can only be 
computed when Ro is known. This leads to an iterative process 
in order to determine both parameters. 
The centres of the first (M/r) and last (M^) image rows are 
determined from the four comer points. Extending the position 
vectors M P and in geocentric coordinates by the factor 
(/ + H/R), where R denotes the earth radius, yields two 
approximate orbit points S° F and S° L . The orbit path is 
approximated by a circle of radius R s = (R + H) connecting S° F 
and S° L and passing through the earth centre. The first 
approximation for R 0 , namely R/, can be determined from this 
path, as described in Weser et al. (2007). Since this 
approximation does not yet consider the viewing angles a and /?, 
it has to be improved. 
The third column vector of R M according to Equation 4, Z M , 
describes the viewing direction of the satellite camera in the 
platform system. Denoting the approximation for R 0 after 
iteration step i by R 0 ', the vector g' describing the viewing 
direction in the geocentric object coordinate system is given by 
g' = R r / • Z M . The improved positions of the orbit end points in 
iteration i + I are situated on straight lines parallel to g', thus 
S 1+1 j = M y + Xj ■ g' with j e {F, L}. The intersections of these 
straight lines with a sphere of radius R s yield the improved 
positions S'* 1 F and S' ’ 1 L , from which improved orbit path 
parameters and an improved rotation matrix R 0 ' w are derived. 
The iterations cease when the positions of S/r and S L change by 
less than a pre-defined distance threshold between two 
successive iterations. Back-projecting the GCPs to the stereo 
pair using the approximate values determined as described 
above results in offsets of up to 500 pixels in image space, 
which is close enough for the bundle adjustment to converge. 
3.3.2 Modification of the generic pushbroom sensor 
model: 
The parameters to be determined via bundle adjustment using 
GCPs are the coefficients of the spline functions used to model 
the time-dependant orbit path, S(t), and attitudes, R ? (t). In 
Weser et al. (2007), the components of S(t) and the angles 
roll(t), pitch(t), yaw(t) parameterising R P (t) were modelled by 
cubic splines. In order to reduce the number of parameters to be 
determined, the degree of the spline functions is reduced to 2 if 
no direct observations for the orbit path and attitudes are 
available. Furthermore, additional observations are used in the 
adjustment to act as ‘soft constraints’ to achieve a more stable 
solution for the spline coefficients. First, a fictitious observation 
of a point to be situated in a plane passing through the earth 
centre is added for several points along the orbit path, thus 
forcing S(t) into such a plane with a certain a priori standard 
deviation. Second, by direct observations of the position vectors 
S(t) and the tangential vectors dS(t) / dt being perpendicular at 
several discrete times t, the orbit path S(t) is forced to be almost 
circular. The additional observations should keep the number of 
GCPs required to determine the parameters in the model 
described by Equation 3 within reasonable limits, without 
compromising the accuracy of the model.