Full text: Proceedings of the Symposium on Progress in Data Processing and Analysis

These eigenfunction systems are the key to better understanding the mathematical 
correlations. 
3d). 
An especially interesting and important function system are the functions 
f t (x) = e ixt . 
Because these functions are the eigenfunctions of numerous operators 
(regular discretization, displacement, difference and differential operators) 
they play an outstanding role. With their help a natural access to discrete 
and digital representations as well as to digital filters is created and the 
fundamental role of the FOURIER-transformation is justified. 
3 
able 
2. Process of discretization 
Dn, 
The discretization process can be understood as assignement of numbers to a 
continuous fact. The rule, that assigns numbers to a function, is called a 
-called 
functional. The correspondino mathematical discipline is the functional 
tions 
analysis. Numbers can be assigned not only to functions, but also to general 
transformations T in form of its eigenvalues. 
If 
(1) 
—i 
X 
II 
> 
X 
/. 
where X is a number, then these numbers A are the eigenvalues and the 
corresponding functions x are the eigenfunctions of the transformation T. 
Dint 
So one can assign a (analogue or continuous) state or event as well as 
Dn Of 
their transformations to numbers. Only in this way one is able to represent 
special processes and data - in the exact sense - onto computers. 
to 
[■opriate 
The discretization of functions is carried out logically in two directions 
3 
namely digitization in the place range and in the range of values. 
For a digitization in the place range the importance of the uniform decompo- 
is 
sition had already been underlined. Depending on the dimensionality of the 
space 
sses to 
place range, uniform decompositions according to Fig. 1 arise. 
Geoscientific facts, which are processed by photogrammetric means, are two- 
is of 
dimensional, because of the existence of the photographic layer on a plain 
sntations 
base. Whereas a uniform decomposition of the one- and three-dimensional 
spaces is only possible by intervals and cubes, the decomposition of the 
plain is possible in various ways. From the practical point of view often 
only the square is chosen, because many technical devices (scanner, display) 
work in this way and the mathematical tools are clear (e. g., matrix calcula 
tion). 
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