Defuzzification of spatial fuzzy sets by feature distance minimization

We present a novel defuzzification method, i.e., a mapping from the set of fuzzy sets to the set of crisp sets, and we suggest its application to image processing. Spatial fuzzy sets are, e.g., useful as information preserving representations of objects in images. Defuzzification of such a spatial fuzzy set can be seen as a crisp segmentation procedure. With the aim to provide preservation of selected quantitative features of the fuzzy set, we define the defuzzification of a fuzzy set to be a crisp set which is as close as possible to the fuzzy set, where the distance measure on the set of fuzzy sets, that we propose for defuzzification, incorporates selected local and global features of the fuzzy sets. The distance measure is based on the Minkowski distance between feature representations of the sets. The distance minimization, performed in the suggested defuzzification method, provides preservation of the selected quantitative features of the fuzzy set. The method utilizes the information contained in the fuzzy representation for defining a mapping from the set of fuzzy sets to the set of crisp sets. If the fuzzy set is a representation of an unknown crisp original set, such that the selected features of the original set are preserved in the fuzzy representation, then the defuzzified set may be seen as an approximate reconstruction of the crisp original. We present four optimization algorithms, exhibiting different properties, for finding the crisp set closest to a given discrete fuzzy set. A number of examples, using both synthetic and real images, illustrate the main properties of the proposed method. An evaluation of both theoretical aspects of the method, and its results, is given.