Fusion of ground truths and hierarchies of segmentations

This paper introduces a new fusion paradigm that reorders the contours of an input hierarchy of segmentations utilizing any external function. Ground truth partitions, markers, numerical functions, and other additional sources are used to obtain a new reordered hierarchy. This transformation aids in combining sets, functions, and partitions in different ways, resulting in a hierarchy of partitions. This result is shown to be a lattice whose infimum operation is the Jordan net opening. Applications are demonstrated over hierarchies and ground truths from the Berkeley dataset. Experiments include hierarchy transformation by proximity to ground truth, fusion of two hierarchies, measuring structural changes produced by said transformations. We also demonstrate how intrinsic properties of the Jordan curves, like curvature, can be used to produce directional openings on contours of the partitions and reorder the hierarchy, in particular the L-opening over Quadtree partitions.