Qualified topological between objects with possibly vague shapes

Qualified topological between objects with possibly vague shapes

Previous works

Spatial vagueness According to (Erwig and Schneider 1997, Hazarika and Cohn 2001, Pfoser et al. 2005), spatial vagueness can characterize the position and/or shape of the spatial extent of a given object. From this perspective, the shape vagueness refers to the difficulty of distinguishing an object shape from its neighborhood. Shape vagueness is an intrinsic property of an object that 91 certainly has an extent in a known position but cannot or does not have a well-defined shape (Erwig and Schneider 1997).

For example, a region has a vague shape when it is surrounded by a broad boundary instead a sharp one. One could normally use the term « fuzziness » to speak about «shape vagueness» since it would correspond to the unclearness of an object shape as it is defined in a general ontology (i.e., to the definitions found in the Oxford and the Cambridge dictionaries). Nevertheless, in order to avoid confusion with the mathematical definition found in the specialized ontology of Fuzzy Set Theory (Zadeh 1965) which is used in several GIS-related papers (e.g., Altman 1987, Burrough 1989, Brown 1998, Schneider 2001), we have decided to use the expression “shape vagueness”.

Accordingly, one must not confuse “fuzziness” as defined in Fuzzy Set Theory with the concept of “shape vagueness” as defined in the present paper. Spatial vagueness can also characterize well-defined (or crisp) objects when there is uncertainty about objects’ positions despite their sharp shapes; we refer to this scenario as positional vagueness.

Positional vagueness is a measurement imperfection related to the accuracy and precision of the instruments and processes used in the measurements (Mowrer 1999). Figure 3.3 shows this categorization of spatial vagueness into « shape vagueness » and « positional vagueness ».

In this paper, we only deal with the formal representation of spatial objects with vague shapes and the topological relations between them. Spatial vagueness Shape vagueness Positional vagueness Figure 3.3 Categorization of spatial vagueness In general, we distinguish between at least two categories of models used to represent spatial vagueness. In the first category, crisp spatial concepts are transferred and extended to formally express spatial vagueness; we speak about exact models (Cohn and Gotts 1996, Clementini and Di Felice 1997, Erwig and Schneider 1997) as explained in the next section.

In the second category, three principal mathematical theories are generally used: (1) models based on the Fuzzy Logic (Zadeh 1965) (e.g., Altman 1987, Burrough 1989, Brown 1998, Schneider 2001, Tang 2004, Hwang and Thill 2005, Dilo 2006), which can be used to represent continuous phenomena such as temperature, (2) models based on rough sets (e.g., Ahlqvist et al. 1998, Worboys 1998),

which represent the objects with vague shapes as a pair 92 of approximations (upper and lower approximations), and (3) models based on probability theory (e.g., Burrough and Frank 1996, Pfoser et al. 2005), which is principally used to model errors of positions and attributes.

Formal definitions of objects with vague shapes In the original version of paper, this section reviews previous works that formally define objects with vague shapes. In the present manuscript, this review literature has been transferred in Chapter 2 (cf. Section 2.3) in order to reduce redundancies and improve the readability of the thesis.