Fuzzy Models of Spatial Relations, Application to Spatial Reasoning

Spatial relations are an important component of image content, that proved to be useful for recognition of individual objects and for image understanding. Indeed, spatial relations provide structural information about the scene, which is often more stable that individual object characteristics, can help disambiguating objects of similar appearance, and is often available as prior knowledge. A typical example is anatomy, where relations between anatomical structures are described in anatomical textbooks or dedicated web sites, and can be used to drive the interpretation of medical images. This will be illustrated on magnetic resonance images (MRI) of the brain, for segmenting and recognizing internal brain structures. This is a typical example where shape and appearance information may not be sufficient for recognition, in particular in pathological cases, while using structural knowledge is relevant and helps solving the problem. Similar examples can be found in understanding aerial and satellite images. One important characteristic of spatial relations is that they often have a clear intuitive meaning in natural language, but crisp mathematical models are often to restrictive, not robust enough, and do no model the intrinsic imprecision attached to the linguistic descriptions of the relations. Fuzzy models are better suited, and allow accounting for imprecision both in the relations and in the objects. This was already mentioned in [21]. My work on fuzzy models of spatial relations was initiated while I was visiting Lotfi Zadeh’s lab in Berkeley, where I spent a few months in 1995 and 1997, enjoying the stimulating environment and fruitful discussions, with researchers from different fields of fuzzy sets theory. The main approach I proposed to model fuzzy spatial relations relies on mathematical morphology [32], because of its strong algebraic framework, which allows developing consistent models in different settings (from purely quantitative ones on sets, to purely qualitative ones in various logics), the fuzzy sets setting being a midway [8]. Another feature is that different types of representations can be proposed, expressing relations as numbers, fuzzy numbers, intervals, distributions, or fuzzy regions of space.