Hierarchical image segmentation provides region-oriented scale-spaces: sets of image segmentations at different detail levels in which the segmentations at finer levels are nested with respect to those at coarser levels. Guimarães et al. proposed a hierarchical graph-based image segmentation (HGB) method based on the Felzenszwalb-Huttenlocher dissimilarity. It computes, for each edge of a graph, the minimum scale in a hierarchy at which two regions linked by this edge should be merged according to the dissimilarity. We provide an explicit definition of the (edge-) observation attribute and Boolean criterion which are at the basis of this method and show that they are not increasing. Then, we propose an algorithm to compute all the scales for which the criterion holds true. Finally, we propose new methods to regularize the observation attribute and criterion and to set up the observation scale value of each edge of a graph, following the current trend in mathematical morphology to study criteria which are not increasing on a hierarchy. Assessments on Pascal VOC 2010 and 2012 show that these strategies lead to better segmentation results than the ones obtained with the original HGB method.