Part-Whole Relations as Products of Metric Spaces

Humans recognize objects by combining the processes of composition/decomposition and similarity matching. When we see an object, we try to match its whole, global aspect to some internal similar pattern that describes it. If this is not enough to recognize the object, it is decomposed into a structure of parts, where each part is considered as a whole to be matched. In this paper, we propose a formal concept representation theory that supports this recognition strategy. It is a specialization of Gardenfors' theory of conceptual spaces and Aisbett and Gibbon's formulation of the same theory based on pointed metric spaces. In our work, a whole and its parts are represented as regions in metric spaces and their relationship is given by their product. The resulting framework is intended as a concept representation model for computer vision programs.