Measuring Hierarchiness of Image Segmentations

Numerous segmentation methods are able to produce several partitions of the same image by tuning a scale parameter. In such a series of multilevel segmentations, if every region at a given level is included in a single region of the segmentation at the next level, then the series is called a hierarchy. Hierarchies are often desired for multiscale image representation and analysis due to their mathematical properties, leading to accurate and efficient solutions. Although certain effective strategies may not produce a hierarchy, it is uncertain whether their multiscale output is close to be one. This work explores several cases when analyzing two consecutive segmentations, as full inflation and full merge, for instance. From those, we provide three measures for evaluating the hierarchiness between two subsequent partitions: (i) nestedness; (ii) refinement error; and (iii) inflation ration. Using our proposals in a in-sequence pairwise comparison, as shown by the experimental results, it is possible to verify whether a multiscale segmentation is a hierarchy and, if not, to analyze the nature and extent of the hierarchical errors that prevent it from becoming hierarchical.