function
digit | —————— — :digit
d. ional
function «cannons function
transformation
Figure 1: Function, Functional and Transformation
into complex numbers, i.e. if the space is a function
space every function corresponds to a digit and the
functional analysis is the analysis in these spaces (see
also figure 1).
The functional analysis includes a topological and an
algebraical part. The topology is realized by neigh-
bourhood definition and measures. On the other hand
with algebraical means we can define basic operators
like addition, scaling, and multiplication. This is very
important for the realization and also for modelling of
data bases and real-time mapping.
The standard measure in mathematics and physics are
normes || . || and scalar products (., .). The HILBERT-
space methods — using the scalar product as a special
norm — are powerful and constructive tools in algo-
rithmization. The transformations within HILBERT
spaces can be designed and analyzed by scalar prod-
ucts realized by integrals or series in the sense of L?-
and /?-HILBERT spaces. Than the image space trans-
formations (77, — integration, Tp, — projection) are
modeled:
Ta = Gy oa
Ter = (.,6)
9 is a weight-function corresponding to an area A and
6 is the DIRAC distribution depending on the projec-
tion direction (see also [9]).
Numerous examples in image processing demonstrate
this fact (duality between spatial and frequence spaces
especially discreteness and periodicity, using the or-
thogonality by decoupling of coordinate-directions in
integral transformations). Completed by algebraic
principles several approximative techniques were de-
veloped.
The discretization in spatial coordinates and in time is
the basis for computer realizations. This digital mod-
elling leads consequently to the digital scene analysis.
A great amount of processes in nature are described
by differential equations and their digital analogon —
the difference equations. Typical mathematical meth-
ods in real-time mapping processing are fast converg-
ing iterations of difference operator equations. In [2]
this is demonstrated by a comparison of variation with
least squares methods.
2.2 Scene Analysis Modelling
For a photogrammetric modelling in scene analysis it
is necessary to generalize the description in time di-
rection.
There are three time levels (point-series-continuity)
with a qualitative jump from the discreteness to the
continuity connected with new objects like image cube
and trajectories. This includes also a new mathemat-
ical description level characterized by variation meth-
ods and functional analysis. In figure 3 this fact is
shown in a symbolic kind.
It is typical that the search for a solution of prob-
lems in science and technology corresponds to finding
the extrema of functionals. Often such functionals
are measures of the difference of structures (e.g. least
square means and variation methods). In [6] such a
calculus in computer vision theory is demonstrated.
3 Problems of Real-Time Pro-
cessing
The task of scene correspondence is to define the ge-
ometric relations between two or more images of the
same domain. Thereby the time is running contin-
uously and typically the time points are discrete on
the time axis in the image space. Consequently these
images also correspond to discrete levels in the image
space (see also figure 3).
The correspondence problem is solved with the help
of image information (grey values, image frequencies,
textural features etc.) by using the variation calculus
to minimize corresponding functionals. The minimiz-
ing problem leads to the solution of EULER equations,
e.g. systems of partial differential equations (see also
[2).
Because a sequence of images is a set of discrete planes
in the image space one direction of generalization is
the change over to the time continuum and to get a
complete image space. In this model it is possible to
define termini as image flow or trajectories of objects.
The transformations act nearly exclusive in the image
space. As results of the discretization of the image
space we get sequences of points. The correspondence
is defined by the solution of attached difference equa-
tions being also an analogon in the digital sense to the
differential equations. By using an iterative scheme it
can be formulated as a complete digital photogram-
metric task (see [2]). Further image processes can be
designed by finding attached transformations within
the image space especially by separation and qualified
algorithmization in geometry and time. In dependence
on the complexity of algorithmization the use of prin-
ciples of parallelization is possible (see [5]).
Figure 2 shows the difficulties at working with image
flows and digital images in one calculus because from
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