Susumu Hattori
of satellite line scanner imagery can be carried out on standard digital photogrammetric workstations using
traditional models with minor modifications (eg Konecny et al., 1987; Kruck & Lohmann, 1986; Kruck, 1988).
In some respects, the affine model to be discussed can be regarded as an expansion of that proposed by Kruck
(1988), which utilised the Gauss Krueger projection plane and ellipsoidal heights as a reference coordinate
system. This system will be referred to as 3D-GK in the following discussion. Orientation angles forming
the collinearity equation can then be fixed as constant by assuming the satellite travels linearly in the object
space, at constant velocity. This assumption enables parameter suppression to avoid instability in the orienta-
tion /triangulation. For compensation of non-linear fluctuations of parameters, which cannot be accounted for,
eight additional parameters are added to the collinearity model. These comprise the distortion correction for
Earth rotation, along with perturbation terms for interior and exterior orientation parameters.
3 MATHMATICAL MODELS OF AFFINE PROJECTION
A line-scanner image constitutes a 2D central perspective projection. The conventional central projection
equation (collinearity equation) relating image coordinates u,v with object space coordinates X,Y, Z therefore
needs to be modified for push-broom scanner imagery to:
0 = au(X- Xo)+a12(Y — Yo) + a13(Z — Zo)
Jj a3 (X — Xo) + a22(Y — Yo) + a23(Z — Zo) (1)
a31(X — Xo) + a32(Y — Yo) + a33(Z — Zo)
where Xo,Yo, Zo are the coordinates of the projection center; u and v are in the flight direction and sensor
direction, respectively; c is the focal length; and aj; are the elements of the rotation matrix A. It is noteworthy
that the v coordinate is always zero. The exterior orientation parameters, which are unique for each scan line,
are modelled as continuous functions of time or line number, usually by low-order polynomials.
Okamoto (1988) modified Eqs.1 to the form of general projective equations, Eqs.2, in order to effect corrections
for linear distortions within the parameters:
0 = X+AY + AZ + A;
y - AY * As Ao (2)
E AY + AgZ +1
Eqs. 2, in which the coefficients A; are again modelled as functions of time or line number, will be referred to
here as the 1D Perspective Model. In order to avoid over-parameterisation in instances where the sensor view
angle is very narrow (eg less than 1 degree for the new generation of 1m satellites such as Ikonos), an affine
projection model can substitute for the expression for the in-line sensor coordinate v:
0
LU
AY + AsZ + Ag
Il
The model represented by Eqs.3 is here termed a 1D affine model, in which the coefficients are again described
by functions of time. In Eqs.1 through Eqs.3, the object space coordinate system is assumed as 3D Cartesian. As
will be suggested in the following, however, the latter two models are also applicable for the 3D-GK coordinate
system, so long as a height correction for Earth curvature is applied.
With regard to the time-dependent modelling of the coefficients A;, these parameters are expected to be constant
or at least piecewise linear within the relatively small extent of a single satellite image scene. For the reported
experiments, a linear variation model (Hoffmann, 1986) has been applied, which means that every parameter
introduces two unknowns into the resulting least-squares model for orientation/triangulation.
Okamoto et al. (1996, 1999) further extended Eqs.2 and 3 to the following two projection models:
4 = B1X + B,Y + B37Z + By
BsX + BeY + B7Z + Be (4)
B9X + B10Y + B11Z +1
3 = B,X + B,Y + B3Z + Ba (5)
v = BsX + BgY + B7Z + Bg
The first of these, Eqs. 4, is termed the Parallel Perspective Model , whereas the second, Eqs. 5, will be referred
toas the 2D Affine Model. It is noteworthy that in neither of these models are the parameters modelled as
360 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.
