Similarity Between Random Sets Consisting of Many Components

Abstract

Random sets play an essential role in modelling several phenomena in biology, medicine and material science. However, sometimes it is hard to describe them using a specific model. Therefore it can also be difficult to classify them or to compare their realisations. This contribution proposes a similarity measure between two random sets whose realisations consist of many components based on just one realisation of each of them. The similarity measure is obtained in a non-parametric way taking into account the shapes and the positions of the components. The procedure is justified by a simulation study and consequently applied to real biomedical data of histological images of mammary tissue.

Two-Step Method for Assessing Similarity of Random Sets

“…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.…”

“…Realisations of both these models are simulated as realisations of the random disc Quermassinteraction process (Moeller and Helisová, 2008) with suitable chosen parameters. More details about the parameters can be found in Gotovac et al (2016) and Gotovac (2019). 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).…”

“…More details about the parameters can be found in Gotovac et al (2016) and Gotovac (2019). 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.…”

“…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.…”

“…However, it is closely tied to the placement of components in realisations, which is not always desirable. Further approach can be found in Gotovac (2019), where the author also focuses on similarity of shapes and positions of components in realisations, but the positions can be omitted in special cases. The considered components are either the connected components or some more specific set components obtained by further decomposing of the observed set (e.g.…”