The original version of the article was published in Mathematical Morphology - Theory and Applications 2 (2017) 55–75. Unfortunately, the original version contains a mistake: in the definition of Dif (C1, C2) in Section 3.6, max should be replaced by min. In this erratum we correct the formula defining Dif (C1, C2).

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This article is a rst attempt towards a general theory for hierarchizing non-hierarchical image segmentation method depending on a region-dissimilarity parameter which controls the desired level of simpli cation: each level of the hierarchy is "as close as possible" to the result that one would obtain with the non-hierarchical method using the corresponding scale as simpli cation parameter. The introduction of this hierarchization problem in the form of an optimization problem, as well as the proposed tools to tackle it, is an important contribution of the present article. Indeed, with the hierarchized version of a segmentation method, the user can just select the level in the hierarchy, controlling the desired number of regions or can leverage on any of the tools introduced in hierarchical analysis. The main example investigated in this study is the criterion proposed by Felzenszwalb and Huttenlocher for which we show that the results of the hierarchized version of the segmentation method are better than those of the original one with the added property that it satis es the strong causality and location principles from scale-sets image analysis. An interesting perspective of this work, considering the current trend in computer vision, is obviously, on a speci c application, to use learning techniques and train a criterion to choose the correct region.

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