bul 2004 
riae i 
Be 
image. 
CGD) is 
d target 
vith the 
). 
1 
2 
sponding 
ng x by 
in Figure 
t ellipse 
T. 
(4) 
(5) 
ch within 
of c, and 
the semi 
  
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part BS. Istanbul 2004 
Formula 5 describes, for the adjustment used, the best-fit 
equation where s defines a scale factor (€ can only provide 
values between 0 and 1) and f is background noise. By 
modelling the background, the thresholding process needed for 
the intensity-weighted centroiding can be avoided. 
Figures 5 and 6 afford a better understanding of the parameters 
of the target function T; they show graphs using different 
parameter sets for T. Especially noteworthy is the last 
parameter o (0.1 and 0.3), which defines the sharpness of the 
target signal. For completeness, it should be mentioned that 
horizontal sections of T are ellipses. 
  
Figure 6: Graph of T(200, 20, 0, 4, 4, 3, 0, 0.3). 
The function T turns out to be an ideal target function to 
perform a best-fit adjustment where the grey values of the 
pixels are taken as observations and the parameters of T as 
unknowns. As can be seen, this estimation process is non-linear 
and therefore partial derivatives of T with respect to each 
parameter are required. Since the derivations are straight 
forward, only the final equations are presented here. The partial 
derivative with respect to the ellipse parameters (Equations 6-8) 
contain only the Gaussian distribution G itself since the integral 
within the CGD Q and the derivation cancel each other out. 
  
2] 
oj ecpeeego ns 4 ysme 
&, a b: (6) 
e os 
a (7) 
  
  
  
e 
aT 15] ]^ 
€ =2s-c-o-G(-E)xy —-— 5) (8) 
06 5" 
OT 
do 
OT (9) 
LL = (=F) 
28 
op. 
zZ 
Q 
  
--E-s-G(- E) 
1 
2.2 Target Plane Adjustment by Observing the Implicit 
Ellipse Parameter 
Kager (1981) has shown that only two different circles in space 
with the same radius can project onto the same ellipse within 
the image. Hence, two images of the target have to exist to 
resolve this ambiguity. This is of no concern since resolution of 
such ambiguity is a fundamental requirement within 
photogrammetry. On the other hand, because of the small 
diameter of the ellipse, its elements (semi major and semi minor 
axes, and bearing) can only be determined with low accuracy. 
Consequently all ellipse images of one target have to be used to 
achieve the highest possible accuracy of the target plane. 
Therefore VM can offer ideal prerequisites since targets are 
generally imaged multiple times (74). 
The proposed target plane adjustment is performed for each 
target at the time of observing the implicit ellipse parameters of 
all images with known EO. To perform this, the mathematical 
connection between the circular target in object space and the 
ellipse parameters in image space has to be derived. 
— Image plane 
  
— Circle plane 
  
Figure 7: View cone which touches circular target. 
An oblique cone is defined (Figure 7) which touches the 
circular target and the apex of the cone is positioned at the 
projection centre of the image. Then the cone is intersected 
with the image plane. If the resulting section figure can be 
presented in the same mathematical form as an implicit ellipse 
equation the problem is solved. That way, the ellipse 
parameters are represented by functions, which depend on the 
circle parameters and an adjustment of indirect observations 
can be performed. 
In the object space, the cone can be described by 
X =C+2-r-(cosa-e, +sina-e,)+2-(M—C) (10)