International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part Bl. Istanbul 2004 
  
y = | 1 = AZ ule p 0 (3) 
ad COS (0 
where y = central-perspective coordinate 
Ya Ya = affine coordinate without/with height correction 
f= focal length 
0 — roll angle of the sensor 
AZ = height from the average height of terrain 
   
| Reference plane 
t 
ya: 7 o { 
Figure 1. Conversion from central-perspective to affine imagery 
Equations 2 and 3 are required for the central-perspective to 
affine image conversion. It can be seen that the correction 
becomes more important when either the roll angle or field of 
view of the sensor are large (Equation 2) or the terrain is 
mountainous (Equation 3). In other words, for a high-resolution 
sensor with a near-nadir view direction over low relief terrain, 
the perspective-to-affine image correction becomes negligible 
and conversion is not warranted. Of importance in an evaluation 
of the impact of the conversion is the invariance of the 
coefficients in Equations 2 and 3 (Fraser & Yamakawa, 2004). 
If the image is georectified to a reference plane, which is the 
case for IKONOS Geo and QuickBird Standard imagery, the 
georectified point Paco” is observed instead of p. Note that there 
is a linear relationship between the y-coordinate (sample 
coordinate) of Bo and P (Ya 5 YgeoCOS@). This means that the 
correction of Equation 2 is implicit in the generation of 
georectified imagery. An image conversion for georectified 
imagery is therefore not required for moderately flat terrain. 
This is consistent with findings of practical applications 
employing IKONOS imagery (Fraser et al., 2002). However, 
for mountainous terrain, the image conversion may well have 
an impact upon geopositioning accuracy, as will be shown later. 
A 'rigorous' theory for satellite sensor orientation based on 
affine projection has also been proposed by Zhang & Zhang 
(2002). As it happens, the conversion coefficient in their model 
is of the same form as that in Equations 2 and 3 (Fraser & 
Yamakawa, 2004). 
3.3 Sensor dynamics 
Agile HRSI satellites can dynamically rotate and swing so that 
the sensor is tilted to 20-40 degrees off nadir. This has 
advantages of shortening the revisit period and offering flexible 
imaging configurations, including along-track stereo recording. 
The ability to view obliquely is quite common for earth 
observation satellites and is, needless to say, required for 
across-track stereo coverage. A concern arising here in the 
context of the affine model is the possible introduction of non- 
linear perturbations as a result of dynamically re-orienting the 
satellite during image recording. IKONOS offers two imaging 
modes: *Forward mode' and ‘Reverse mode’. In the Forward 
mode, the sensor pointing direction is moving backwards, 
opposite to the satellite motion and changing at around one 
degree per second. In *Reverse mode' the sensor is close to 
steady, maintaining a near constant view direction. 
In terms of geometrical characteristics, dynamic variation in 
pitch angle requires special attention because it could cause 
non-uniform resampling. Although imagery products are 
georectified by utilising a very rigorous sensor geometry model, 
high frequency accelerations or perturbations of the sensor 
might not always be perfectly recovered and completely 
corrected for in the imagery. This concern is more pronounced 
with QuickBird, simply because of its continuous re-orientation 
during image capture. 
The standard formulation of the affine model treats the 
orientation parameters as time-invariant. However, as an 
approach to accounting for the presence of non-linear image 
perturbations, time-variant coefficients can be considered: 
x=A OX +A, +A4,(DZ + A, (t) (4) 
y 7 A,QU)YX  AGQ)Y 9 A, (0Z 7 A, (0) 
where 4(1)=a; +a;t=a; +b;x (5) 
For geometrical interpretation, Equation 4 can be rearranged to: 
(X ray +a, 2 + a, ~x)+x(b'X +b°Y +b Z+h')=0 (6) 
(aX Fay ka 7 tay -y)tx(b X +h Y+b Z+bh)=0 
where IT, = ay X + ag Y+ ag z+ a,’ = 0 relates to the image 
plane in line direction at x = 0 
Il) = aj X^ af Y ag Z* dap y = O relates to the image 
plane(s) in sample direction at x ^ 0 
As illustrated in Figure 2, in the line (across-track) direction the 
imaging plane IH (ag! X ay Y ag Z4 dg X0), which is 
parallel to IT;, rotates around an axis defined as an intersection 
of planes IT,’ and IT, (P) X b) Y by Z* bj!—0), as time goes 
on. Similarly, in the sample direction the imaging plane IT, 
rotates around an axis defined as an intersection of planes IT, 
and IL (bj X bj Y b,/Z+ b,5=0). 
  
Figure 2. Geometrical interpretation of time-variant affine 
parameters. 
The affine model with time-variant parameters requires eight 
GCPs instead of the usual four. The significant rise in the 
number of unknown parameters is not too desirable in terms of 
the numerical stability of the computation. Equation 4, therefore,