be matched well because of the geometric distortions. Fig. 5 shows the influence of scale errors
onto the output signal to noise ratio. These theoretical results are adapted from Svedlow et.al.
(1976). They are based on the AR-image model with white noise perturbation. The results here are
given in dependency on the parameter c, with a - ó r (cf. sect. 3.1). Clearly the effect of scale
SNE
Fig. 5 v le |
Output signal to noise ratio for different lei E
values of scale distortion e versus the a x
length d of a square area (adapted from
Svedlow et.al. 1976). with autocovariance
mia
function 7 = R, e mim and ez 5» +
lel 20.25
Lad
ici = 0.5
a
1 i i
1 T a
2 4 6 d[rm]
distortion will increase with increasing scale difference and decrease with increasing correla-
tion length », thus increasing z, as the image is smoother for large z.The optimum patch size
2 d with Z = 2.5 n° /
pt op
Q t or
|c| proves to be
i
i
et
A
4 - /
c - V
opt
>: l2.
iei
5 (19)
For good imagery and a scale difference of 0.1 the optimum size is 2 mm^. Fig. 5 shows that for
smaller patches the increase of the signal to noise ratio is approximately proportional to the
patch side d. Svedlov et. al. also analysed the effect of unmodelled rotation differences and more
general deformations occuring in LANDSAT images. As for high precision correlation at least linear
geometric deformations should be compensated by the algorithm it would be worth to investigate
the influence of unmodelled nonlinear distortions representing the local curvature of the terrain.
As the effective bandwidth of the image is not much influenced by geometric distortions eq. (13)
also holds for the minimum variance c?.
Actually very small patch sizes of only 5x5 pels are used in TV image sequence analysis (cf.
Dinse et.al. 1981, Bergmann 1982). In photogrammetric applications the patch size varies between
11x11 pels (cf. v. Roessel 1972, Gambino and Crombie 1974) and 32x32 pels (e.g. Markarian et.al.
1973). Hence most systems do not seem to take full advantage of the accuracy potential. Obvious-
ly severe local nonlinearities, especially occuring in large scales, require to use smaller corre-
lation windows, accepting the decrease in accuracy.
2. Quite a different approach to compensate for unknown scale differences or other geometric dis-
tortions is to look for a point C(z y) within the patch whose transformed point CT Ress
is invariant to scale differences between the images. Thus the bias c'-C, is eliminated. It can be
shown for one-dimensional signals that this point is the whighted centre of gravity of the patch,
where the weights are the squares of the gradient
j 9
) € d
da CO
PEE
e
m. forte tt (20)
5
t
T2?
Mu
=
€
3
The proof uses the fact that additional parameters in a least squares problem do not influence
the result if they are orthotogonal to the other unknowns (cf. Fürstner 1982). As the partial de-
a.
Mot) "T]
— ia. — qu
