Different averages of a fuzzy set with an application to vessel segmentation

Image segmentation is a major problem in image processing, particularly in medical image analysis. A great number of segmentation procedures produce intermediate gray-scale images that can be understood as fuzzy sets. Additionally, some segmentation procedures tend to leave free tuning parameters (very influential in the final binary image) for the user. These different binary images can be easily aggregated (into a fuzzy set) by making use of fuzzy set theory. In any case, a single binary image is required so our interest is to associate a crisp set to a given fuzzy set in an intelligent and unsupervised manner. The main idea of this paper is to define the averages of a given fuzzy set by using different definitions of the mean of a random compact set. In particular, the average distance of Baddeley-Molchanov and the mean of Vorob´ev have been used. A theoretical study of some new definitions of fuzzy set averages has been performed. In particular, these averages have been obtained for L-R fuzzy numbers. Finally, we present a medical image application, that of retinal vessel detection. Some recent segmentation procedures have been revisited and modified using these new averages. The experimental results are very promising.