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MATHEMATICAL PROBLEMS OF REAL-TIME MAPPING 
AND DATA BASE MODELLING 
Erhard Pross 
Institute of Applied Geodesy 
Leipzig, Germany 
Commission II, Working Group 1 
Abstract 
In real-time processing the time is the main processing direction controlled by discrete operators. Functional 
analytical methods integrating discrete and continuous description and methods lead to constructive tools in 
data base modelling and real-time mapping. The realization of image operators as difference operators depends 
on the the fast convergence and on the techniques' stability and algorithmic ideas of switching between spatial 
and time iterations and processing directions. 
KEYWORDS: database modelling, theory, scene analysis, optical flow 
1 Introduction 
With the triumphal progress in computer techniques 
in several fields also a digital photogrammetry and 
real-time mapping was formed. Numerous ideas and 
principles in physics, electronic engineering, and new 
disciplines like robot and machine vision influence the 
digital photogrammetry in a very great amount. So 
we must recognize not only an integration of geodesy, 
photogrammetry, and cartography in geoinformatics 
but also new concepts and models based on modern 
mathematics. The separate developments in mod- 
elling must be integrated in one model (see [1]). The 
integration of different sensor types in one real-time 
mapping model from the data capture to the the stor- 
age and the post-processing of the data is investigated 
in [4]. 
The idea of this paper is to view modelling in real- 
time mapping from an abstract mathematical point 
of view and give some directions to the algorithmic 
realization of techniques. 
2 Mathematical Background of 
Data Base Modelling 
2.1 Abstract Space Definition 
For the definition of an abstract mathematical space 
corresponding to tasks in geoscience it is necessary to 
define measurements of nature signals as elements of 
such a space and technological processes and technical 
course as corresponding transformations between suit- 
able spaces. Spaces are characterized by the contents 
(images, features, ...), by the discreteness (analog, 
digital) or by the structure (vector, raster). 
The art of abstract mathematical modelling leads to 
such models, the concrete realization of which leads 
to special views on projections into and integrations 
within function spaces. The functional analysis is a 
mathematical building with powerful tools to realize 
such a modelling. In geoinformatics we know image 
spaces — including images, image sequences, raster 
maps, ...— and feature spaces (see [3], [10]). 
A mathematical model demonstrating a functional 
analytical description in geoscience especially remote 
sensing, photogrammetry, and cartography in one cal- 
culus was developed (see [7], [8]). 
From the semantic level point of view image spaces 
have a low level and the feature space a middle or 
high level. 
Coordinates are the spatial z-, y- and z-coordinate 
— in topography especially in smallscale topography 
the third coordinate (height) plays an important role 
by 2.5—, 2+1- or 3-D-GIS-modelling —, the features 
or attributes m and last but not least the time t. 
The significance of the time leads to qualitative differ- 
ent models in regard of time 
e low influence — time as date or up-dating num- 
ber 
e common coordinate 
e dominant processing parameter for instance in 
real-time mapping 
These several levels differ with different methods and 
techniques. For that reason the common methods in 
photogrammetry, remote sensing, and GIS — corre- 
sponding to the first and second level — cannot be 
transferred to real-time mapping processing. 
À functional is a transformation of an abstract space 
455