ute 
ite 
74) 
the 
ree 
ww 
to 
task 
in 
where T (gip,) and TP stand for arbitrary radiometric or geometric transformations which 
depend on the parameter vectors p, and p. We thus reach the problem of multiparameter estimation. 
Tr and T_ might e.g. be linear transformations. T, then compensates for contrast and brightness and 
T for differential perspective, i.e. affine distortions, neglecting the local curvature of the 
terrain (cf. Pertl 1984): 
rg ea, g++ 
- 
r 2 
Ty) = = = [b b, | [5 (6) 
f 7 7 3 3 
LCL le, bio Xs) 
Other parameters could describe nonlinear transformations or e. g. the width of the point spread 
function R(x) (cf. Thurgood and Mikhail 1982). 
0f course the additional parameters have to be predicted using a priori information or estima- 
ted from the available data. The necessity to compensate for scale and rotation, possibly affini- 
ty and the unability of the classical matched filter to provide an estimate lead to more or less 
sophisticated prediction schemes (cf. eg. Hobrough (1971), Kreiling (1973) or Panton (1978)). 
Direct solutions are given by Emmert and McGillem (1973) for affinity parameters and by Casasent 
and Psaltis (1976) for scale and rotation. They used invariance properties of the amplitude spec- 
trum. In general estimating two or more parameters requires a refined searching strategy to 
reduce the numerical effort. Hill-climbing or gradient methods for estimating additional parame- 
ters are widely used and discussed in section 3.2 (cf. e.g. Schalkoff and McVey 1979, Wild 1979, 
Huang and Tsai 1980, Meyers and Frank 1980, Fórstner 1982, Thurgood and Mikhail 1982). 
3. The idea of deriving parameters from the amplitude spectrum, which is invariant to shifts, 
suggests to look for algorithms which do not extract the information from the gray levels directly 
but derive the shifts or other parameters from functions of the gray levels. 
The most promising approach is to use invariants of the images. Thus instead of comparing g and . 
, one compares I(g) and I(g;)s where I is a function invariant to transformations T. € Cr out of 
a class c_of expected gray level transformations: ILg(æ)]= I(T [g(z)l). Similarily one could use 
functions which are invariant to an expected class of geometric transformations. 
Using the gradient or edge image is the most commonly method being invariant to a large class 
of temporal gray level changes of the image (cf. Hobrough 1959, Anuta 1970, Wong 1978, Makarovic 
1980, Wiesel 1981). As taking the derivative of an image is high pass filtering this yields a 
sharper peak in the correlation function (cf. Pratt 1974). It can therefore be a surrogate for one 
of the filters mo, T. OP m, for obtaining a ó-like correlation function. The precision of the 
a 
x 
2 
estimated shift however is not increased by this means (cf. Anuta 1970 and sect, 3.3.1). 
Other functions which are used for correlation are invariant moments (cf. Wong and Hall 1978) 
and Hadamardcoefficients (v. Roessel 1972). The concept of Fourier desriptors commonly used in 
the field of pattern recognition seems to be very powerful for object location as the identifica- 
tion and the location process are performed simultaneously within the same mathematical framework 
besides being invariant to scale and rotation of the image in concern (cf. Wallace and Mitchell 
1980, Mikhail et al. 1983) The proposal of Masry (1981) to correlate entities, i.e. features ex- 
tracted from the image, seems to be a quite general concept as not only objects with closed boun- 
dary can be handled (cf. Lugnani, 1982). 
Quite differently motivied image functions are the complex exponentiation (Gopfert 1977) and 
the binary or polarity correlation (cf. Makarovic 1980, Marckwardt 1982). The complex exponentia- 
tion T.(g(x)) = exp(-jpeg(z)), with p depending on the variance of g, aims at whitening the signal. 
Though this is not valid in general, e. g. for a rectangular function, the correlation function 
T. (g)«T, (g. (7«)) for normal imagery yields a much sharper peak than the cross correlation. This