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.