Résumé : |
(Auteur) Uncertainty is an inherent property of observations. Abstracting the real world to conceptional objects is a step of generalization, and the measurement, taking place on abstracted objects, propagates this uncertainty, due to additional systematic, gross or random errors. Each spatial analysis is infected by these sources of uncertainty. It is necessary to introduce propagation of uncertainty in spatial reasoning to allow an assessment of the results. The scope of the thesis is to combine the process of observation with a mathematical model of qualitative spatial relations, modelling the randornness of the observations. A methodology is presented for probability-based decisions about spatial relations.
When determining spatial relations from positional uncertain objects, one has to distinguish between quantita-tive relations, which become imprecise, and qualitative relations, which become uncertain. Topological relations, being of qualitative nature, may or may not be true in presence of positional uncertainty. Assuming the overlay of two independent objects indicates a very small overlap, the question arises whether the two objects could be neighbored in reality. Comparing the degree of overlap to the size of uncertainty will allow to make a decision, and to assess this decision.
Because of the essential importance of considering uncertainty in spatial analysis, a theoretically well-based model of uncertainty is preferred against limitations in validity or meaning. Therefore, in this thesis, positio-nal uncertainty will be described stochastically, and the inference from this description to the uncertainty of derived spatial relations is treated with a statistical classification approach. Probabilities of single relations are determined, and the relation with maximum probability, given the evidence from observation, is chosen.
At the beginning the separation of abstraction and measurement within the observation process is discussed. The complexity of regions is limited by the smoothness of the boundary when related to its uncertainty.
In the following chapter the space of possible relations is reduced to two sub-graphs of the conceptual-neigh-borhood-graph of binary relations between regions. Depending on the sub-graph that the relation between two regions belongs to, different sets of intersection sets between the two objects become uncertain. To describe the uncertainty a morphological distance function is defined, based on the skeleton.
It is shown that it is sufficient to use the minimum and maximum distance, and to classify the relation, depending on the signs of these two values. For the first time the imprecision of measurement and the uncertainty of abstraction are described using probability densities. They are used to determine a vector of probabilities the sign of the distance values have, which is the basis for a Bayesian classification. From these distance classes the decision about the topological relation of the two objects is derived.
In the last chapter examples show the versatility of the proposed methodology.
Other approaches for handling uncertainty are discrete, using error bands, or fuzzy, with the problem of weaker results. With the strong connection to an observation process we hope to give a more valuable decision method, with probabilities as interpretable results, which should be useful! for the assessment and propagation in spatial reasoning processes. |