DESCRIBING DIGITAL PHOTOGRAMMETRY 
BY FUNCTIONAL ANALYTICAL METHODS 
DR. ERHARD PROSS 
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) 
k 
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).