A similarity measure between fuzzy regions to obtain a hierarchy of fuzzy image segmentations

In image segmentation it is well known that a given image can be analyzed with different detail levels, this is why some hierarchical approaches have been proposed to give a different segmentation for each detail level. Most of these proposals are specially designed for precise and well defined regions. However regions usually have blurred contours, soft color shades, and brightness that give rise to the problem of the imprecision in the regions. In this paper we face both problems considering the imprecision of the regions at the definition of the criteria to obtain a hierarchy detail levels. Concretely, we propose to calculate a similarity relation between fuzzy regions, based on two measures that take into account the imprecision in the transition between the regions, as well as the likeness of their characteristics. Then we use this fuzzy similarity relation to obtain a nested hierarchy of fuzzy segmentations by means of its alpha-cuts. In this way we obtain a tool to easily change the detail level and obtain a new fuzzy segmentation of the image, just changing the value of alpha.