Dictionary of neighbourhood configurations using fuzzy spatial and topological relations (work in progress).

Building on our previous work on Non-Gibbsian Markov Random Fields, we investigate the use of fuzzy representations for spatial and topological relationships. A fuzzy concept is one that holds to a variable degree which is determined by a membership function associated with that concept. The membership function typically maps a set of measurements to the unit interval. We introduce three membership functions that specify the degree to which a region is above, to the right, or surrounding another. Each region pair is thus associated with a three dimensional membership vector. The image shows a two-dimensional projection of the vectors corresponding to all region pairs of our training collection (vertical axis: containment, horizontal axis: aboveness). A neighbourhood configuration involving a focal region and N labelled neighbours is represented by a Nx3 label-augmented membership matrix. In order to obtain a small number of representative configurations, we introduce a distance between configurations and apply the k-medoid algorithm to all configurations observed in the training set. This gives us a set of what we call configuration prototypes. By noting the label at the focal region of each configuration associated with a prototype, we are now able to determine a conditional probability distribution over labels, and thus to define our fuzzy non-Gibbsian Markov random field.

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