## Multiseeded Fuzzy Segmentation

The idea behind multiseeded fuzzy segmentation is to generalize the approach described in the previous section to multiple objects: each of the objects in the image has its own definition of strength for the links and its own set of seed spels. Each of the objects is then defined as the collection of those spels that are connected entirely within the object to one of its own seed spels in a stronger way than to any of the other seed spels. This intuitive notion will be made precise. An essential feature of our approach is that it does not simply calculate, for every spel, the grade of membership to each of the individual objects of that spel and then assigns the spel to the object for which its grade of membership is maximal (such an algorithm is discussed in [27]). The reason for this is that if a spel is separated from the seed points of Object 1 by spels belonging to Object 2, then it should not be assigned to Object 1. The gestalt that we are trying to capture here is a segmentation in which the chains that determine "belonging to an object" must lie entirely in that object.

A potentially time-consuming step in finding such objects is the calculation of the multiple fuzzy connectedness of all the spels to the seed spels. We devised a greedy and efficient algorithm that provides the desired segmentation. We demonstrate its performance on various mathematically-defined and physically obtained (real) images. The output of the process is a segmentation into fuzzy sets in the classical sense ([25], p. 39) that, for each spel, we also produce a "grade of membership" in the object(s) to which it belongs.

Similarly to the method presented in last section, we rely on the user of our method to identify seed spels that definitely belong to the various objects into which we desire to segment the images, and we suggest (as other advocates of segmentation based on fuzzy connectedness have done before us) that the user-selected seed spels can be used for automatic calculation of the definitions of the strengths of links in each one of the objects. Since our choice implies that the output of our algorithm is user-dependent, we report on experiments (in which five users segmented five images, each five times) that validate the accuracy and robustness of our approach.

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