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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