International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
where N is the number of GCPs and M is the number of terms in a
bivariate polynomial. The basis for the MQ and RMQ, may be
expressed as follows :
h(x, y) 2 J(d? t c^ )anah(x, y) - Vi c) ain
Where d is the distance between a point (x.y) and a neighbouring
point, and ¢ is the MQ parameter. The effect from a given point to
all equidistant points is the same, being expressed as a translation
of the radial function A. A polynomial is first used to model the
general geometric image to object transformation and an
interpolating function is then used to separately fit the resulting
vectors of residuals in X and Y at each control point. Weights can
be assigned to each control point and the effect of local distortions
measured GCPs are calculated using an interpolating matrix
developed from the distance between GCPs. Matrix formulations
of the MQ, which is particularly suited to the rectification of
remote sensing images of large scale and locally varying
geometric distortions, are given in Ehlers.
2.2.2 Thin Plate Splines (TPS)
Thin plate splines are analogous to the bending of an infinitely
thin plate when subjected to point loads X and Y at locations (x, y).
The formulation of TPS basis function may be given as:
h(x, y) 2 d? logd orA(x, y) » d^ logd? (12)
Derived functions in TPSs have minimum curvature between
control points, and become almost linear at large distances from
the GCPs. The influence of individual GCPs is localized, and
diminishes rapidly away from their locations. In order to represent
a warping transformation accurately, TPSs should be constrained
at all extreme points of the warping function.
2.3 Orbital Parameter Model
An orbital parameter model can be applied to the pushbroom
images in order to determine their exterior orientation parameters.
An orbital resection method has been developed to model
continous changing of position and attitude of the sensors by
finding the orbital parameters of the satellite during the period of
its exposure of the image. A bundle adjustment has been
developed to determine these parameters using GCPs. This
program has been already tested for SPOT Level 1 A and IB stereo
pairs (Valadan and Petrie, 1998) as well as, MOMS-O2 stereo
images (Valadan, 1997) and IRS-1C stereo pairs (Valadan and
Foomani, 1999).
The well known collinearity equation relates the points in the CT
object coordinate system to the corresponding points in the image
coordinate system. The relationship between these two coordinate
systems is based on three rotations using combinations of the
Keplerian elements mentioned above but computed with respect to
the CT system using the transformation parameters between the
CT and CI systems, plus three rotations, ®, ¢, k, for the additional
undefined rotations of the satellite at the time of imaging. The off-
nadir viewing angles of the linear array sensor must also be
included as angle a and B. The following equations will then
result:
Xi = X X 7A
Y,~ Ya {=2SRY In (13)
eC Li -Z
where
R-R()RGRGOR(GORGOR( a2) 72 2R (7 2~DR(
and
S is the scale factor;
o,B :is the cross-track and along-track viewing angles;
X;, V; ‘are the image coordinates of object point 7;
Xp» V, :are the image coordinates of. principal point;
Xs y Z, : are the object coordinates of image point /;
X y, Z, : are the coordinates of the position of the sensor's
perspective centre in the CT system;
c is the principal distance = focal length of the linear array
imaging system, and
R, :defines the rotation around the / axis, where j=1, 2 or 3.
Because of the dynamic geometry of linear array systems, the
positional and attitude parameters of a linear array sensors are
treated as being time dependent. The only available measures of
time are the satellite’s along-track coordinates. Thus the major
components of the dynamic motion, the movement of the satellite
in orbit and the Earth rotation are modelled as linear angular
changes of / and © with respect to time, defined as f and 0 ;
Thus:
1, = J * Ji.
0=20,+0x (14)
where,
f and çà : are the true anomaly and the right ascension of the
ascending node of each line i respectively;
f and dm : are the true anomaly and the right ascension of the
ascending node with respect to a reference line, for example the
centre line of the scene; and
f and e
and Q,.
During the orientation of a pushbroom image, nine parameters of
the orientation (f ll Da Pag) find the
position in space of the satellite and its sensor system and its crude
attitude. Considering the attitude of a scan line as a reference, the
attitude parameters @,(P, and Æ of the other lines can therefore
are the first values for the rates of change of y
be modelled by a simple polynomial based on the along-track (x)
image coordinates as follows:
2
(OO xTOX
2
P = P, + PX + P2X
K=Ky + KX + KX
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