||This Monday I will present a novel defuzzification method, i.e., a mapping from the set of fuzzy sets to the set of crisp sets. Fuzzy sets have many applications in image processing and defuzzification of a fuzzy spatial set can be seen as a crisp segmentation procedure.
We define an optimal defuzzification of a particular fuzzy shape as a crisp shape which is as close as possible to the fuzzy shape. The distance measure we propose for defuzzification incorporates selected features of the two sets at different scales and is based on the Minkowski distance between feature representations of the two sets. Our method allows to combine two defuzzification approaches; it utilizes the information contained in the fuzzy representation both for defining a mapping from the set of fuzzy sets to the set of crisp sets, and for the approximate reconstruction of an unknown crisp original.
Fully utilizing the information available in a fuzzy (discrete) representation of a shape enables the defuzzification to be performed so that the crisp discrete representation is generated at an increased spatial resolution.
The proposed defuzzification method can thus be used as a method to perform segmentation at sub-pixel resolution. Different search algorithms for finding the crisp set closest to a given discrete fuzzy set will be discussed and I will present a number of examples to illustrate the main properties of the proposed method, for both 2D and 3D spatial fuzzy sets.