rors 
‚al. 
are 
scale 
ta 
€n 
La 
for 
he 
j more 
linear 
te 
rrain. 
(13) 
Cf. 
ween 
ious- 
corre- 
| de- 
Sm resp., 
~7 
P r 
rivatives of the function g'x) with respect to shift and scale are z, . and DU, 
is leads to the con- 
where cz, is the origin of the coorcinate system (usually z, - 0 1S chosen), th 
V : : id 
dition Zc? . (z,-,) s 0 , from which eq. (20) is derived. 
“ 
m 3 7 f) 
we, v e V 
Fig. 6 S 
: A 
On the optimal choice of the 
i 
transferred point when a scale 9.4 ted 
difference is present. C and C, Ld i ete] 
2 
eRe) 
ei 
—9 
———. 
——— 
a 
bee 
e 
  
  
  
  
  
; i Cov(s.g,) 
= weighted centres of gravity. 1 3 4 5 6 7 9 9 10 Mi | 1 
M = middle of template g. Trans-  g,(x,) M 2 e 
ferred points C' and M': error | ? ! f 
i Ct, =0.2 0 4 mm per re pe 
M'-M,=3,error C'-C,=0. e NE ° 
| | mek cl ft A i et i a 
"Xu v S NE AN Noi iiM. T 
aix 8) 6 4 pw J o0 1 2 3 4-5 
imp | X=2 
remet | : 
gw * | s $ x? ! 
hl Bh Th fe bl ble 
po ue Lig 0.3 498 = 
¢ M 
An example is given in fig. 6. The object g(æ,)and the signal 2, (=; differ by a scale difference. 
The covariance function leads to a shift of z = 2. If the centre M of the template is transferred, 
i. e. shifted by 2, the bias M' -M,-3 is very large. But if one transfers the centre C of gravity 
one obtains the point C' which is very close to the true centre C. of gravity. The bias of 0.2 
is due to discretization errors. 
The centre of gravity is not optimum in two-dimensional patches, but clearly reduces the effect 
of unmodelled geometric distortions. 
3.5 Convergence and pull-in-range 
The convergence and the pull-in-range are essential for the economy of the least squares procedure. 
Especially the requirements for the approximate values are detemined by the pull-in-range. We will 
analyse the pull-in-range for the case where no noise is present. Again we restrict the discussion 
to one-dimensional signals. 
The shift is estimated from the maximum of the cross correlation function or which for the 
noiseless case reduces to the autocovariance function E (x). Suppose the signal is ideal band limi- 
ted resulting from sampling with pixelsize Az and assuming the signal to be white. Then R (x) = 
R, si mwz/Ac. The least squares algorithm eq. (11b) is equivalent to a Newton-Raphson approach to 
search for the maximum, adapting the gradient to the actual approximate values. The gradient tech- 
nique, where the gradient is kept constant, and the Newton-Raphson iteration scheme for this appli- 
cation have been excellently discussed by Burkhardt and Moll (1978). 
They showed that the convergence rate is cubic. This is due to the symmetry of the autocovari- 
ance function at the maximum. In general the Newton-Raphson iteration scheme only reaches quadra- 
tic convergence. 
Moreover they proved that the pull-in-range is rather small compared to the gradient technique. 
Thus the approximate values must fullfill the condition | «1. 52/1* hz = 0.48 Ax, thus must be 
better than half a pixel. However, though the actual position of the object is not known the gra- 
dient of By 0 i.e. the curvature of R (0) at the maximum is known. Burkhardt and Moll showed 
that it can be used to increase the pull-in-range of the Newton-Raphson approach by nearly a fac- 
tor 3. The approximate values then only have to be accurate up to 1.5 pixels.This result can be 
used also in multiparameter estimation with the least squares approach, as usually good approxi-