Fuzzy segmentation is a technique that assigns to each element in an image (which may have been corrupted by noise and/or shading) a grade of membership in an object (which is believed to be contained in the image). In an earlier work, the first two authors extended this concept by presenting and illustrating an algorithm which simultaneously assigns to each element in an image a grade of membership in each one of a number of objects (which are believed to be contained in the image). In this paper, we prove the existence of such a fuzzy segmentation that is uniquely specified by a desirable mathematical property, show further examples of its use in medical imaging, and illustrate that on several biomedical examples a new implementation of the algorithm that produces the segmentation is approximately seven times faster than the previously used implementation. We also compare our method with two recently published related methods.
