Formalizing fuzzy objects from uncertain classification results

Concepts of fuzzy objects have been put forward by various authors (Burrough and Frank 1996) to represent objects with indeterminate boundaries. In most of these proposals the uncertainties in thematic aspects and the geometric aspects are treated separately. Furthermore little attention is paid to methods for object identiŽ cation, whereas it is generally in this stage that the uncertainty aspects of objects become manifest. When objects are to be extracted from image data then the uncertainty of image classes will directly eŒect the uncertainty of the determination of the spatial extent of objects. Therefore a complete and formalized description of fuzzy objects is needed to integrate these two aspects and analyse their mutual eŒects. The syntax for fuzzy objects (Molenaar 1998) , was developed as a generalization of the formal syntax model for conventional crisp objects by incorporating uncertainties. This provides the basic framework for the approach presented in this paper. However, the model still needs further development in order to represent objects for diŒerent application contexts. Moreover, the model needs to be tested in practice. This paper proposes three fuzzy object models to represent objects with fuzzy spatial extents for diŒerent situations. The Fuzzy-Fuzzy object (FF-object) model represents objects that have an uncertain thematic description and an uncertain spatial extent, these objects may spatially overlap each other. The Fuzzy-Crisp object (FC-object) model represents objects with an uncertain spatial extent but a determined thematic content and the Crisp-Fuzzy object (CF-object) model represents objects with a crisp boundary but uncertain content. The latter two models are suitable for representing fuzzy objects that are spatially disjoint. The procedure and criteria for identifying the conditional spatial extent and boundaries based upon fuzzy classiŽ cation result are discussed and are formalized based upon the syntactic representation. The identiŽ cation of objects by these models is illustrated by two cases: one from coastal geomorphology of Ameland, The Netherlands and one from land cover classiŽ cation of Hong Kong.