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it was shown, how this one-dimensional transformation principle can be
extended to two-dimensional transformations. Thereby, using two seccessively
performable transformations in row and column direction, in connection with
a decoupling of the transforming function, exactly the above mentioned
algorithmic dimension reduction is realized.
Orthogonality is the decisive mathematical aid for decoupling the co-ordinate
axes. With those orthogonalitiy relations two-dimensional procsses can be
divided into two independent one-dimensional processes. A classic example for
that are the integral transformations, because the corresponding characteristic
functions fulfill exactly this orthogonality. The example of the two-dimensional
FOURIER transformation is sufficiently known, which can be realized as twofold
one-dimensional FOURIER transformation in each co-ordinate.
4. Prospects
Each problem to be solved has got corresponding data structures, which have a
natural dimension. Many problems in photogrammetry and neighbouring disciplines
in geoinformatics have got a two-dimensional position reference, completed by a
semantic feature. With the help of the stereoscopic principle the terrain
altitude can be determined as special "feature", which can also be related to
a two-dimensionally structured situation.
Besides the dimensionality of data the basic structuring - raster or vector
data - is of decisive importance. Vector-oriented data, which are produced
within the photogrammetric plotting process especially during the derivation
of cartography-oriented data bases, are naturally unproblematic in regard of
the dimension number. Raster-oriented data - e. g. digital image structures -
with a sequential order basically contradict former photogrammetric approaches.
The term (p, x) means, that the feature p exists in the place x.
The transformation T thereby means, that this is also decomposed in regard of
feature and space:
T - T(M,G).
There is M = M(p) und G = G(x).
