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Proceedings of the Symposium on Progress in Data Processing and Analysis

Research Centr of Geodesy and Cartography
Karl-Rothe-Str. 10 - 14, Leipzig, DDR-7022, G. D. R.
1. Mathematical background of digital data and processes
In terms of structure images are two-dimensional manifolds and can be
described by functions of two space co-ordinates (perhaps vector-valued).
A discrete representation in form of number pairs can be assigned to a
function. Thereby a countable set {x^j is selected from the admissable
continuum of the arguments x. One speaks about a digital representation,
if also the range of values is limited to a finite set.
Such a discrete function concept led to generalized functions, the so-called
distributions. With the help of DIRACs ^-distribution discrete functions
can be written as sum of weighted displaced ¿'-distributions:
f ~~ L M k • i (x - x k ). (1)
Detailed explanation concerning this and the following is given in /1/.
The formula (1) can be interpreted in such a way, that the discrete point
x^ carries the mass p^. In generally the value x can be here a more-
dimensional place and the index k a multiindex. For the representation of
processes (also called transformations or operators) it is favourable to
use inherent characteristics of these processes. For this purpose appropriate
abstract mathematical spaces are defined. The most simple constructive
structure element is the scalar product, which leads to the so-called
HILBERT-space. With the help of the scalar product the orthogonality is
defined as central term. Using orthonormal bases the elements of this space
can be represented now by this basis. From the’point of view of processes to
be investigated function systems can be defined too. The eigenfunctions of
the appropriate operator are favourable - which lead to simple representations
of these processes (similar to the main axes transformation).