Assessing Similarity of Random sets via Skeletons

Two-Step Method for Assessing Similarity of Random Sets

“…The power is larger than the method in Gotovac et al (2016) and one of the methods in Gotovac (2019), which also distinguishes realisations based on the shape of the components, and is comparable to another method in Gotovac (2019) and to the method in Debayle et al (2021), which, however, takes into account the placement of components. Another advantage with respect to the method in Gotovac et al (2016) is that it is not heuristic, and moreover, it does not require a lot of input parameters as needed in Gotovac et al (2016) and Debayle et al (2021). A small complication occurs when components of compared realisations are too close to each other, because the method assumes independence of the components, which is not satisfied when the components are too densely placed.…”

“…In the simulation study, we first focus on four models which illustrate the usage of the procedure. The models and approach to simulation can be found in Debayle et al (2021), Gotovac (2019), Gotovac Ðogaš and Helisová (2021) and Gotovac et al (2016). For all considered models, we simulate 200 realisations and compare 100 vs 100 realisations of the same models as well as 100 vs 100 realisations of different models.…”

“…Note that the same repulsive model is employed for simulation studies in all above mentioned papers, similarly as the Boolean model, while the same cluster model is used only in Gotovac (2019). In Debayle et al (2021), Gotovac Ðogaš and Helisová (2021) and Gotovac et al (2016), another cluster model is considered, which is not suitable for our current study since its realisations consist of too few components. The fourth model is the Boolean model with grains to be ellipses (called the ellipse model in the following paragraphs).…”

“…New methods taking this into account have been developed in the last five years (Debayle et al, 2021;Gotovac, 2019;Gotovac Ðogaš and Helisová, 2021;Gotovac et al, 2016). In Debayle et al (2021), morphological skeletons (Serra, 1982) of compared realisations are constructed and then, similarity of two realisations is defined through a function describing mass growth around selected points of the skeletons. This method shows the highest power in simulation study, compared to the methods in the remaining three papers.…”

“…New methods taking this into account have been developed in the last five years, see [3], [4], [5] and [6]. In [3], morphological skeletons [17] of compared realisations are constructed and then, similarity of two realisations is defined through a function describing mass growth around selected points of the skeletons.…”