99 
3. Some geometric Considerations 
The ZEISS Planicomp System has been designed for processing conven- 
tional photogrammetric imagery, which follow the geometric laws of 
central projection, expressed by the collinearity equations 
x? = ’ 
X; Xo 
ras (1) 
-C § iva 3 
Ziel, 
where ( x’, y',- ch is an image point, A; a scale factor, A a 
rotation matrix, ( ke re 22)! the perspective center and 
(X: Y, D a point on the ground. For opto-mechanical scanners, the 
image geometry is 
0 X. inox 
1 i oj 
in 6 |= — A. Prim » 
C Si : i j Yi Y oj (2) 
-C cos © i Z; = Z oj 
(see KONECNY 1972). This rigorous expression incorporates the scan angle 
9 and the time - dependency of the parameters indexed by j. The x? - 
coordinate is equivalent to the time. Anyway, the formulas (1) and (2) 
look somewhat similar. They can for practical use indeed be treated as 
identical, when introducing the following restrictions: 
a) time invariant parameters 
b) small scan angle 
C) scan frequency, instanteneous field of view, platform speed 
and altitude in correspondance. 
a) and b) can be fully accepted for LANDSAT MSS imagery, item c) only 
aproximatively. Even small variations of one of the parameters in c) 
cause, in first order, a scale difference in one direction, i.e. an 
affinity. The affinity is indeed the most sensitive factor contributing 
to geometric quality of LANDSAT MSS imagery (BAHR 1978). 
Going back to using an analytical plotter for LANDSAT MSS imagery, we 
may simply correct any affinity during the absolute orientation, and 
beside this possibility when plotting at the table, which compensates 
for (affine) paper contraction. As formula (1) and (2) are practically 
identical for LANDSAT MSS imagery, we are allowed to process this type 
of data using an analytical plotter. 
When processing LANDSAT data digitally, a second-order polynomial 
approach is sufficient for most cases: 
9 ’ 31,3 2 "o 
a, + ax’ + ay’ + agx’y’ + x * àgy r (3) 
by + b,x + boy + bx y' * b,x + Doy 
"X 
The 3 first terms correspond to the affine transformation. The second- 
oder polynomial approach assumes still more restrictions: