Susumu Hattori 
  
time-variant functions. Instead, a single coefficient value relates to all scan lines in the orientation process, 
which implies that the following conditions are met: Firstly, the satellite moves in space linearly and at uniform 
velocity with constant orientation parameters. This supports use of the 3D-GK as the object space coordinate 
system, as in Kruck (1988), whose model is very similar to Eqs. 4 except that the parameters Bi there constitute 
elements of a rotation matrix. This first condition is reasonable for near-nadir pointing sensors where variation 
in terrain height is modest, but it is not so suitable in high-latitude areas. The second condition is that the 
collinearity equation holds between image coordinates and object space coordinates, which is fulfilled when 
utilising the 3D-GK through application of an Earth curvature correction. 
The 2D affine model (Eqs.5) strictly only holds if the field view is infinitesimally small, which is clearly not 
the case in practise. The difficulty of applying this model to satellite line scanners with fields of view of a few 
degrees or more (eg 4° for SPOT) is overcome by an initial transformation of the 2D perspective image to an 
affine projection, a process that will be further discussed in the following section. 
Eqs.4 and 5 form a solid model in space just as with conventional stereo frame imagery. In order to absolutely 
orient the model in object space, the number of GCPs required for the parallel perspective and 2D affine 
models are five and four, respectively. To illustrate this, we let the space be reconstructed from a pair of stereo 
photographs in which image and object space coordinates are related by general projective equations: 
B,X + BY + BaZ + B4 
Bo X ad BıoY zz B11Z c 1 
  
B5 X + BgY + B7Z + Bg 
BgX + B1gY + B11Z +1 
  
The model space coordinates X,,, Y,,, Z,, and object space coordinates X,Y, Z are related by linear transfor- 
mation of a homogenous coordinate system of fourth degree: 
Ci Cr Cs Cs 
C5 1,01. C8 
T0, Co Cu Cr 
Cis Cia Cis 1 
(7) 
N 
BEP 
$ 
Mp N X 
Since the coefficients C; are determined by five GCPs, the rank of the observation equations formed by the 
coplanarity condition is at best 7 (22-15). The case of Eqs.4 is the same and accordingly the equivalence to 
Eq.7 for the 2D affine model is 
Xm Dy. Ds Ds X Dio 
Ym = D,° Ds De Y + Dii (8) 
Zm D; Ds Dg Z Di» 
More than four GCPs are thus necessary to determine the 12 unknowns, and the rank of the observation 
equations via the coplanarity condition is 16-12 — 4. 
4 TRANSFORMATION: CENTRAL PERSPECTIVE TO AFFINE PROJECTION 
Considering that the field of view of SPOT is four degrees, the projection of a SPOT image effectively stands 
between central perspective and affine. Errors due to discrepancies in projection when applying the affine 
model can be eliminated by a transformation of images from central perspective to affine projection (Okamoto 
& Akamatsu, 1992a;b ). This procedure, however, requires a knowledge of terrain height, which is invariably 
unavailable for the area covered by the imagery. The contradiction is overcome by a straightforward iterative 
procedure of stereo measurement. For the discussion of the image transformation, the following symbols are 
used: v is the pixel coordinate in the central projection image, va is the corresponding coordinate in the affine 
projection image, c is the focal length, p, is the principal point and 2a is the field view angle. 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 361