GES
o-referenced
IKONOS as
tributed. For
pensive sub-
dy Standard
ng level 1B-
available as
oved at least
1e 3D-affine
described by
n depending
Chis requires
re. The view
own satellite
the smallest
o-referenced
bed method,
| with one
metry. The
direction is
in relation to
M
SS
e
against the
'ction
ed with fast
1g a fast and
S satellite is
n (figure 2)
the mirror
he effect of
‘the covered
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV. Part Bl. Istanbul 2004
area. This yaw correction is generating scenes with sides
perpendicular to each other, but still rotated against the national
net coordinate system. IKONOS and QuickBird even can
generate images with sides parallel to the coordinate axis of the
national coordinate system. Of course the rigorous geometric
relation has to be respected by the mathematical model based
on the correct imaging geometry.
2.2 Rational polynomial functions based on sensor
orientation
Based on the sensor geometry and orientation, the relation
between the image and the ground coordinates (geographic or
national net coordinates) can be determined by a three
dimensional interpolation in the object space with polynomials
(Grodecki 2001).
Pr.
!
P.CX.Y.7),
ERI GA m e
B(X. V 7) ma E qoYcoag X AAA
+a -Y-Z+a,-X-Z+ag-Y' +a, -X +a, 2+
dy Xl Ltd, Lihat AU. ailes
a Y Xa A rea Z a. Y ka, X sz
3
d s.
Formula I. rational polynomial functions
Usually the image coordinates are expressed with a third order
polynomial of the national net coordinates X, Y, Z (formula 1),
so 80 coefficients are required. Of course this is an
approximation, but with the high number of coefficients the loss
of accuracy against a rigorous model is negligible. The RPCs
just based on the direct sensor orientation (sensor depending
RPCs), have to be improved by a shift to at least one control
point. As image coordinates also national net coordinates are
used for geo-referenced scenes. The sensor depending RPCs
optimally do use the available sensor information.
2.3 Rational polynomial functions based on control points
Not in any case the sensor based RPCs are available. With
control points, the basic idea of a three dimensional
interpolation can be used. Of course it is not possible to adjust
80 coefficients - this would require at least 40 three dimensional
well distributed control points. So the number of unknowns has
to be reduced. This also called terrain dependent solution is
only able to determine the lowest order terms shown in formula
I. The number of unknowns is depending upon the number and
distribution of control points.
This method is not using the available, but in some cases
restricted sensor orientation information, by this reason more
and well distributed control points are required. If all control
points are located in the same height level, the position of
points in a different elevation cannot be determined. So the
control points are required three-dimensionally distributed
around the area of mapping. In general an extrapolation out of
the area of the control points has to be avoided. Even the low
order polynomials can lead to extreme errors and random errors
at the control points are enlarged outside the controlled area.
2.4 Three dimensional affinity transformation
The high resolution space sensors do have a small view angle,
allowing the replacement of the perspective geometry in the
CCD-line by a three dimensional affinity transformation
(formula 2). This model can be improved by some corrections
for a sufficient use of the perspective geometry in the CCD-line
direction.
X = A+ AX+A;Y + A4, Formula 2: 3D-affinity
y = Ast AX + AY + AgZ transformation
Like the RPCs based on control points, the orientation
information of the sensor is not used and the control points must
be located three dimensional around the mapping area. This is
often causing problems in mountainous arcas because the
control points are usually located in the valleys or lower areas
and not on top the mountains, leading to geometric problems of
the more elevated areas.
2.5 Reconstruction of the imaging geometry
m
pe ERIS
Figure 3. geometric situation of IKONOS Geo, QuickBird
OrthoReady and other level 1B-type images
With the exception of the QuickBird Standard Imagery, the
level 1B-type images like IKONOS (CARTERRA) Geo and
QuickBird OrthoReady are projections to a plane with constant
height. In all cases the azimuth and the incidence angle from the
scene centre to the satellite are directly or indirectly available,
allowing a reconstruction of the imaging geometry using also
the published information about the sensor orbit. This has to
respect also the change of the view direction in relation to the
orbit during imaging with the extreme case of the imaging
against the sensor movement (figure 2). The rectified images
are usually available with the georeference. This may be given
directly for any pixel like for IKONOS and QuickBird or the
corner positions of the scenes are known and a transformation
between ground and pixel positions is required for the
georeference of any pixel. Depending upon the data set a
similarity, an affinity or a perspective transformation is
necessary.
Based on the reconstructed sensor geometry, the view direction
for any scene position can be calculated and based on the height
