A quantitative fuzzy-valued intersection matrix for obtaining fuzzy relationships between vague spatial objects

Topological relation between geographical objects is one of the key components of the Geographical Information System (GIS). Considering the objects to be exact with sharp boundaries, four intersection, and nine intersection matrices initiated the study of calculating the topological relations. Due to the extensive existence of vague spatial phenomenons, the study of uncertainty object modeling has grown rapidly. Parallelly, the relationship calculating tools have been upgraded to handle vague concepts. The fuzzy 9-intersection matrix and Egg-Yolk methods are popular tools. Although the geographical objects are uncertain, surprisingly, most of these tools generate certain relations between them. The qualitative nature of these tools overlooks the essence of their uncertainty. We propose a quantitative fuzzy-valued 9-intersection matrix to overcome these drawbacks to obtain a fuzzy relationship between uncertain geographical objects. Then two new similarity computation techniques are introduced to calculate the membership grade between the proposed matrix and the known crisp matrices. These similarity computations allow two spatial objects to have partial membership against the eight established topological relations. The quantitative calculations indicate the strength of the relationship. The superiority of the proposed model is exhibited through comparative studies. Further, certain linguistic variables are linked to the evaluated membership grades to generate an immediate association with the known crisp relations. Examples are provided to demonstrate the applicability and efficacy of this association.