Mathematical Modeling of the Relationship "between" Based On Morphological Operators

The spatial relationship “between” is a notion which is intrinsically both fuzzy and contextual, and depends in particular on the shape of the objects. The few existing definitions do not take into account these aspects. We propose here definitions which are based on morphological operators and a fuzzy notion of visibility in order to model the main intuitive acceptions of the relation. We distinguish between cases where objects have similar spatial extensions and cases where one object is much more extended than the other. These definitions are illustrated on real data from brain images.

[1]  Isabelle Bloch,et al.  Mathematical morphology and spatial relationships: quantitative, semi-quantitative and symbolic settings , 2002 .

[2]  Yann Mathet.  Etude de l'expression en langue de l'espace et du deplacement : analyse linguistique, modelisation cognitive et leur experimentation informatique , 2000 .

[3]  James M. Keller,et al.  Quantitative analysis of properties and spatial relations of fuzzy image regions , 1993, IEEE Trans. Fuzzy Syst..

[4]  G. Matheron.  Random Sets and Integral Geometry , 1976 .

[5]  B. Landau,et al.  “What” and “where” in spatial language and spatial cognition , 1993 .

[6]  Roberto Marcondes Cesar Junior,et al.  Approximate reflectional symmetries of fuzzy objects with an application in model-based object recognition , 2004, Fuzzy Sets Syst..

[7]  Isabelle Bloch,et al.  Fuzzy Relative Position Between Objects in Image Processing: A Morphological Approach , 1999, IEEE Trans. Pattern Anal. Mach. Intell..

[8]  Isabelle Bloch,et al.  Fuzzy mathematical morphologies: A comparative study , 1995, Pattern Recognit..

[9]  Azriel Rosenfeld,et al.  Degree of adjacency or surroundedness , 1984, Pattern Recognit..

[10]  Ulrich Bodenhofer,et al.  Fuzzy “Between” Operators in the Framework of Fuzzy Orderings , 2003 .

[11]  Yannick Larvor.  Notions de méréogéométrie : description qualitative de propriétés géométriques du mouvement et de la forme d'objets tridimensionnels , 2004 .

[12]  Isabelle Bloch,et al.  Description of brain internal structures by means of spatial relations for MR image segmentation , 2004, SPIE Medical Imaging.

[13]  Johan van Benthem,et al.  A ModalWalk Through Space , 2002, J. Appl. Non Class. Logics.

[14]  Anca L. Ralescu,et al.  Spatial organization in 2D segmented images: representation and recognition of primitive spatial relations , 1994, CVPR 1994.