  Current approaches to generic segmentation start by creating a hierarchy of
nested image partitions and then specifying a segmentation from it. Our first
contribution is to describe several ways, most of them new, for specifying
segmentations using the hierarchy elements. Then, we consider the best
hierarchy-induced segmentation specified by a limited number of hierarchy
elements. We focus on a common quality measure for binary segmentations, the
Jaccard index (also known as IoU). Optimizing the Jaccard index is highly
non-trivial, and yet we propose an efficient approach for doing exactly that.
This way we get algorithm-independent upper bounds on the quality of any
segmentation created from the hierarchy. We found that the obtainable
segmentation quality varies significantly depending on the way that the
segments are specified by the hierarchy elements, and that representing a
segmentation with only a few hierarchy elements is often possible. (Code is
available).
