Random Tessellations Research Articles

Geometric modelling of corrosion inhibitor pigments in active protective coatings based on SR-nano-CT images

This paper reports on an essential step towards accelerating the development of new, environmentally friendly active protective coatings by structure optimization. The complex microstructure of the pigment particles within a coating have been observed non-destructively in 3D by nano-computed tomography using synchrotron radiation. For the first time, a stochastic geometry model is fitted based on spatial geometric features of the particles observed in the 3D images. The typical cell of a random Gibbs-Laguerre tessellation is used to model the particles' polyhedral shapes as well as the observed joint size and aspect ratio distributions.

Converting Tessellations into Graphs: From Voronoi Tessellations to Complete Graphs

A mathematical procedure enabling the transformation of an arbitrary tessellation of a surface into a bi-colored, complete graph is introduced. Polygons constituting the tessellation are represented by vertices of the graphs. Vertices of the graphs are connected by two kinds of links/edges, namely, by a green link, when polygons have the same number of sides, and by a red link, when the polygons have a different number of sides. This procedure gives rise to a semi-transitive, complete, bi-colored Ramsey graph. The Ramsey semi-transitive number was established as Rtrans(3,3)=5 Shannon entropies of the tessellation and graphs are introduced. Ramsey graphs emerging from random Voronoi and Poisson Line tessellations were investigated. The limits ζ=limN→∞NgNr, where N is the total number of green and red seeds, Ng and Nr, were found ζ= 0.272 ± 0.001 (Voronoi) and ζ= 0.47 ± 0.02 (Poisson Line). The Shannon Entropy for the random Voronoi tessellation was calculated as S= 1.690 ± 0.001 and for the Poisson line tessellation as S = 1.265 ± 0.015. The main contribution of the paper is the calculation of the Shannon entropy of the random point process and the establishment of the new bi-colored Ramsey graph on top of the tessellations.

Milkweed and floral resource availability for monarch butterflies (Danaus plexippus) in the United States

The global decline of pollinators, particularly insects, underscores the importance of enhanced monitoring of their populations and habitats. However, monitoring some pollinator habitat is challenging due to widespread species distributions and shifts in habitat requirements through seasons and life stages. The monarch butterfly (Danaus plexippus), a migratory insect pollinator that breeds widely throughout North America, presents a unique case study for testing a sampling framework to overcome these challenges. Monarchs exhibit discrete resource needs across life stages (e.g., larval requirement for milkweed, adult requirement for floral nectar), utilizing many land use types across their extensive geographic range during breeding and migration seasons. The Integrated Monarch Monitoring Program (IMMP) uses a standardized protocol with a generalized random tessellation stratified (GRTS) sampling design to gather spatially balanced and ecologically representative information on monarch habitats within the United States. The IMMP is applicable to various land use types and habitats used by breeding monarchs and may be extended to sites outside of the GRTS design to collect data on non-random sites of interest, such as legacy or conservation sites. Additionally, the IMMP’s modular design and publicly available training allows for broad participation, including involvement from community scientists. Here, we summarize habitat metrics (milkweed and floral resources) across 1,233 sites covering much of the monarch’s breeding range. We examine variation in milkweed density and floral resource availability on probabilistic (random) and non-probabilistic (convenience) samples and among land use types (site types). Additionally, we assess resource availability within core geographies for monarch breeding and migration, specifically within the U.S. Fish and Wildlife Service’s Monarch Conservation Units (western, northern, and southern United States). Milkweed density, floral frequency, and floral richness were higher on non-random sites and in the North region. Among site types, milkweed density was highest on Rights-of-Way and Unclassified Grassland, while floral frequency was lowest on Rights-of-Way. The IMMP represents the first field-based habitat monitoring program of this scale for monarchs, yielding a robust dataset on monarchs and their habitats across their breeding range and offering a framework for surveying the habitat of insect species with diverse habitat requirements or widespread distributions.

Tessellation-valued processes that are generated by cell division

AbstractProcesses of random tessellations of the Euclidean space $\mathbb{R}^d$ , $d\geq 1$ , are considered that are generated by subsequent division of their cells. Such processes are characterized by the laws of the life times of the cells until their division and by the laws for the random hyperplanes that divide the cells at the end of their life times. The STIT (STable with respect to ITerations) tessellation processes are a reference model. In the present paper a generalization concerning the life time distributions is introduced, a sufficient condition for the existence of such cell division tessellation processes is provided, and a construction is described. In particular, for the case that the random dividing hyperplanes have a Mondrian distribution—which means that all cells of the tessellations are cuboids—it is shown that the intrinsic volumes, except the Euler characteristic, can be used as the parameter for the exponential life time distribution of the cells.

Performance evaluation of spatially balanced sampling designs in fishery-independent surveys

Cost-effective fishery-independent surveys are critical for obtaining high-quality data to support stock assessment and fisheries management. Classical probability-based sampling designs may exhibit limitations in spatial coverage and flexibility. Spatially balanced sampling (SBS) designs offer the ability to efficiently distribute sampling stations across a survey area. This study evaluates the performance of SBS designs in multispecies fishery-independent surveys through simulation studies, involving two widely used classical sampling designs (Simple Random Sampling and Stratified Random Sampling) and two SBS designs (Generalized Random Tessellation Stratified Model and Balanced Acceptance Sampling). The aim of the simulation study was to assess the performance of different sampling designs in estimating the abundance index for four target fish species across different seasons. The results showed that SBS designs had relatively high accuracy and precision in terms of relative estimation error, relative bias, and coefficient of variation. Among the four sampling designs, the Generalized Random Tessellation Stratified Model performed optimally at the same sample size. In conclusion, this study suggests that SBS designs can enhance the precision of abundance index estimation for different fish species. Given its ability to effectively distribute sampling stations and its flexibility, a better understanding and application of SBS designs in fishery-independent surveys are warranted.

Evolution of polygonal crack patterns in mud when subjected to repeated wetting–drying cycles

The present paper demonstrates how a natural crack mosaic resembling a random tessellation evolves with repeated ‘wetting followed by drying’ cycles. The natural system here is a crack network in a drying colloidal material, for example, a layer of mud. A spring network model is used to simulate consecutive wetting and drying cycles in mud layers until the crack mosaic matures. The simulated results compare favourably with reported experimental findings. The evolution of these crack mosaics has been mapped as a trajectory of a 4-vector tuple in a geometry-topology domain. A phenomenological relation between energy and crack geometry as functions of time cycles is proposed based on principles of crack mechanics. We follow the crack pattern evolution to find that the pattern veers towards a Voronoi mosaic in order to minimize the system energy. Some examples of static crack mosaics in nature have also been explored to verify if nature prefers Voronoi patterns. In this context, the authors define new geometric measures of Voronoi-ness of crack mosaics to quantify how close a tessellation is to a Voronoi tessellation, or even, to a Centroidal Voronoi tessellation.

Second-Order Properties for Planar Mondrian Tessellations

In this paper planar STIT tessellations with weighted axis-parallel cutting directions are considered. They are known also as weighted planar Mondrian tessellations in the machine learning literature, where they are used in random forest learning and kernel methods. Various second-order properties of such random tessellations are derived, in particular, explicit formulas are obtained for suitably adapted versions of the pair- and cross-correlation functions of the length measure on the edge skeleton and the vertex point process. Also, explicit formulas and the asymptotic behaviour of variances are discussed in detail.

The β-Delaunay tessellation IV: Mixing properties and central limit theorems

Various mixing properties of [Formula: see text]-, [Formula: see text]- and Gaussian-Delaunay tessellations in [Formula: see text] are studied. It is shown that these tessellation models are absolutely regular, or [Formula: see text]-mixing. In the [Formula: see text]- and the Gaussian case exponential bounds for the absolute regularity coefficients are found. In the [Formula: see text]-case these coefficients show a polynomial decay only. In the background are new and strong concentration bounds on the radius of stabilization of the underlying construction. Using a general device for absolutely regular stationary random tessellations, central limit theorems for a number of geometric parameters of [Formula: see text]- and Gaussian-Delaunay tessellations are established. This includes the number of [Formula: see text]-dimensional faces and the [Formula: see text]-volume of the [Formula: see text]-skeleton for [Formula: see text].

Habitat Provision and Erosion Are Influenced by Seagrass Meadow Complexity: A Seascape Perspective

Habitat complexity plays a critical role in shaping biotic assemblages and ecosystem processes. While the impacts of large differences in habitat complexity are often well understood, we know less about how subtle differences in structure affect key ecosystem functions or properties such as biodiversity and biomass. The late-successional seagrass Posidonia australis creates vital habitat for diverse fauna in temperate Australia. Long-term human impacts have led to the decline of P. australis in some estuaries of eastern Australia, where it is now classified as an endangered ecological community. We examined the influence of P. australis structural complexity at small (seagrass density) and large (meadow fragmentation) spatial scales on fish and epifauna communities, predation and sediment erosion. Fine-scale spatially balanced sampling was evenly distributed across a suite of environmental covariates within six estuaries in eastern Australia using the Generalised Random Tessellation Structures approach. We found reduced erosion in areas with higher P. australis density, greater abundance of fish in more fragmented areas and higher fish richness in vegetated areas further from patch edges. The abundance of epifauna and fish, and fish species richness were higher in areas with lower seagrass density (seagrass density did not correlate with distance to patch edge). These findings can inform seagrass restoration efforts by identifying meadow characteristics that influence ecological functions and processes.

Shape modeling with spline partitions

Shape modelling (with methods that output shapes) is a new and important task in Bayesian nonparametrics and bioinformatics. In this work, we focus on Bayesian nonparametric methods for capturing shapes by partitioning a space using curves. In related work, the classical Mondrian process is used to partition spaces recursively with axis-aligned cuts, and is widely lied in multi-dimensional and relational data. The Mondrian process outputs hyper-rectangles. Recently, the random tessellation process was introduced as a generalization of the Mondrian process, partitioning a domain with non-axis aligned cuts in an arbitrary dimensional space, and outputting polytopes. Motivated by these processes, in this work, we propose a novel parallelized Bayesian nonparametric approach to partition a domain with curves, enabling complex data-shapes to be acquired. We apply our method to HIV-1-infected human macrophage image dataset, and also simulated datasets sets to illustrate our approach. We compare to support vector machines, random forests and state-of-the-art computer vision methods such as simple linear iterative clustering super pixel image segmentation. We develop an R package that is available at https://github.com/ShufeiGe/Shape-Modeling-with-Spline-Partitions.

Nonparametric bagging clustering methods to identify latent structures from a sequence of dependent categorical data

Nonparametric bagging clustering methods are studied and compared to identify latent structures from a sequence of dependent categorical data observed along a one-dimensional (discrete) time domain. The frequency of the observed categories is assumed to be generated by a (slowly varying) latent signal, according to latent state-specific probability distributions. The bagging clustering methods use random tessellations (partitions) of the time domain and clustering of the category frequencies of the observed data in the tessellation cells to recover the latent signal, within a bagging framework. New and existing ways of generating the tessellations and clustering are discussed and combined into different bagging clustering methods. Edge tessellations and adaptive tessellations are the new proposed ways of forming partitions. Composite methods are also introduced, that are using (automated) decision rules based on entropy measures to choose among the proposed bagging clustering methods. The performance of all the methods is compared in a simulation study. From the simulation study it can be concluded that local and global entropy measures are powerful tools in improving the recovery of the latent signal, both via the adaptive tessellation strategies (local entropy) and in designing composite methods (global entropy). The composite methods are robust and overall improve performance, in particular the composite method using adaptive (edge) tessellations.

The -Delaunay tessellation: Description of the model and geometry of typical cells

AbstractIn this paper we introduce two new classes of stationary random simplicial tessellations, the so-called $\beta$ - and $\beta^{\prime}$ -Delaunay tessellations. Their construction is based on a space–time paraboloid hull process and generalizes that of the classical Poisson–Delaunay tessellation. We explicitly identify the distribution of volume-power-weighted typical cells, establishing thereby a remarkable connection to the classes of $\beta$ - and $\beta^{\prime}$ -polytopes. These representations are used to determine the principal characteristics of such cells, including volume moments, expected angle sums, and cell intensities.

A comparison of design-based and model-based approaches for finite population spatial sampling and inference.

The design-based and model-based approaches to frequentist statistical inference rest on fundamentally different foundations. In the design-based approach, inference relies on random sampling. In the model-based approach, inference relies on distributional assumptions. We compare the approaches in a finite population spatial context.We provide relevant background for the design-based and model-based approaches and then study their performance using simulated data and real data. The real data is from the United States Environmental Protection Agency's 2012 National Lakes Assessment. A variety of sample sizes, location layouts, dependence structures, and response types are considered. The population mean is the parameter of interest, and performance is measured using statistics like bias, squared error, and interval coverage.When studying the simulated and real data, we found that regardless of the strength of spatial dependence in the data, the generalized random tessellation stratified (GRTS) algorithm, which explicitly incorporates spatial locations into sampling, tends to outperform the simple random sampling (SRS) algorithm, which does not explicitly incorporate spatial locations into sampling. We also found that model-based inference tends to outperform design-based inference, even for skewed data where the model-based distributional assumptions are violated. The performance gap between design-based inference and model-based inference is small when GRTS samples are used but large when SRS samples are used, suggesting that the sampling choice (whether to use GRTS or SRS) is most important when performing design-based inference.There are many benefits and drawbacks to the design-based and model-based approaches for finite population spatial sampling and inference that practitioners must consider when choosing between them. We provide relevant background contextualizing each approach and study their properties in a variety of scenarios, making recommendations for use based on the practitioner's goals.

Elastic properties of Laguerre approximations of random foams

Stochastic models are valuable tools to study the mechanical behaviour of foams whose characteristic features include random cellular morphology and cells of different size (polydispersity). Random Laguerre tessellations, additively weighted generalisations of Voronoi tessellations, possess these features which makes them natural models for foams. This work studies how well a random foam can be represented by a Laguerre tessellation. For this means the Surface Evolver is used to simulate random, polydisperse soap froths which are subsequently reconstructed by Laguerre tessellations. By design, the resulting Laguerre approximations resemble the cellular morphology of the underlying froth. The Young’s modulus of low-density solid foams with open cells is then calculated for random, polydisperse foams based on Laguerre approximations and simulated soap froths. For the special case of monodisperse foams, we compare Laguerre approximated foams with the well known Kelvin and Weaire–Phelan foams.

Polydisperse solid foams: Multiscale modeling and simulations of elasto-acoustic properties including thin membrane effects

This work is concerned with the prediction of the elasto-acoustic properties of polydisperse solid foam structures. A highly polydisperse foam sample is first characterized using microtomography and scanning electron microscopy. Relevant geometrical properties are then determined by image processing and utilized to model the partially closed cell system with random Laguerre tessellations. The macroscopic visco-thermal transport properties of the solid foams are next calculated by numerical techniques, using either finite element computations or pore-network simulations. The permeability and sound absorption coefficient at normal incidence are also measured and a good agreement is obtained with the calculations when the elasto-acoustic coupling is modeled from the Biot’s equations (including characterized visco-elastic parameters). Our results demonstrate that stochastic geometry provides a robust framework to understand the structure–property relationships of polydisperse foam.

Random Tessellations and Gibbsian Solutions of Hamilton–Jacobi Equations

We pursue two goals in this article. As our first goal, we construct a family $\mathcal{M}_G$ of Gibbs like measures on the set of piecewise linear convex functions $g:\mathbb{R}^2\to\mathbb{R}$. It turns out that there is a one-to-one correspondence between the gradient of such convex functions and $\textit{Laguerre tessellations}$. Each cell in a Laguerre tessellation is a convex polygon that is marked by a vector $\rho\in\mathbb{R}^2$. Each measure $\nu^f\in\mathcal{M}_G$ in our family is uniquely characterized by a kernel $f(x,\rho^-,\rho^+)$, which represents the rate at which a line separating two cells associated with marks $\rho^-$ and $\rho^+$ passes through $x$. To construct our measures, we give a precise recipe for the law of the restriction of our tessellation to a box. This recipe involves a boundary condition, and a dynamical description of our random tessellation inside the box. As we enlarge the box, the consistency of these random tessellations requires that the kernel satisfies a suitable kinetic like PDE. As our second goal, we study the invariance of the set $\mathcal{M}_G$ with respect to the dynamics of such Hamilton-Jacobi PDEs. In particular we $\textit{conjecture}$ the invariance of a suitable subfamily $\widehat{\mathcal{M}_G}$ of $\mathcal{M}_G$. More precisely, we expect that if the initial slope $u_x(\cdot,0)$ is selected according to a measure $\nu^{f}\in \widehat{\mathcal{M}_G}$, then at a later time the law of $u_x(\cdot, t)$ is given by a measure $\nu^{\Theta_t(f)}\in\widehat{\mathcal{M}_G}$, for a suitable kernel $\Theta_t(f)$. As we vary $t$, the kernel $\Theta_t(f)$ must satisfy a suitable kinetic equation. We remark that the function $u$ is also piecewise linear convex function in $(x,t)$, and its law is an example of a Gibbs-like measure on the set of Laguerre tessellations of certain convex subsets of $\mathbb{R}^3$.

A three-dimensional geometrical model for the microstructure of additively manufactured metals

In the last few decades, selective laser melting has been adopted in a broad range of applications, including automobile, aerospace, medical, and robotic industries, mainly because of the unique versatility of this method in fabricating complex parts. Despite all the advantages of the method, the quality and the mechanical properties of the manufactured parts are drastically affected by the microstructure and processing parameters. Since the fabrication and characterization of test samples are time-consuming and expensive, developing reliable and accurate models to investigate the correlation between the microstructure and the mechanical properties at the bulk scale is a necessity. In this paper, a three-dimensional geometrical model, which explicitly considers the grains, melt-pools, and melt-pool boundaries, is developed. In this model, first, the grain microstructure is constructed by random Voronoi tessellation of a cubic domain to coincide with reality. Then, considering the melt-pool boundaries’ position and orientation, the redundant portions of the grains are removed. Finally, the complete model is constructed by assembling all the remaining grains. In addition, a method for selecting the randomly distributed grain boundaries and melt-pool boundaries is introduced. The accuracy of the proposed model in considering the texture is validated by comparing the pole figures against those obtained from electron backscatter diffraction tests, and a good agreement is shown. The results show that the pole figures of the model converge to those of the experiment when the number of grains is increased. In addition, some geometrical models are generated by adjusting different process parameters, that is, hatch space, layer thickness, and scan strategy, to see how capable the model is in generating accurate misconstrues. The observations prove that the proposed geometrical model can resemble all the details with good accuracy and can be used for numerical simulation purposes.

Simulation of Ultrasonic Backscattering in Polycrystalline Microstructures

Ultrasonic testing of polycrystalline media relies heavily on simulation of the expected signals in order to detect and correctly interpret deviations due to defects. Many effects disturb ultrasonic waves propagating in polycrystalline media. One of them is scattering due to the granular microstructure of the polycrystal. The thus arising so-called microstructural noise changes with grain size distribution and testing frequency. Here, a method for simulating this noise is introduced. We geometrically model the granular microstructure to determine its influence on the backscattered ultrasonic signal. To this end, we utilize Laguerre tessellations generated by random sphere packings dividing space into convex polytopes—the cells. The cells represent grains in a real polycrystal. Cells are characterized by their volume and act as single scatterers. We compute scattering coefficients cellwise by the Born approximation. We then combine the Generalized Point Source Superposition technique with the backscattered contributions resulting from the cell structure to compute the backscattered ultrasonic signal. Applying this new methodology, we compute the backscattered signals in a pulse-echo experiment for a coarse grain cubic crystallized Inconel-617 and a fine grain hexagonal crystallized titanium. Fitting random Laguerre tessellations to the observed grain structure allows for simulating within multiple realizations of the proposed model and thus to study the variation of the backscattered signal due to microstructural variation.

An improved grain-based numerical manifold method to simulate deformation, damage and fracturing of rocks at the grain size level

An improved grain-based numerical manifold method (NMM) is developed to investigate deformation and damage of intact rocks at the meso‑scale. The grain boundaries are embedded into the numerical manifold method using a random Voronoi tessellation technique to approximate the microstructure of rocks at the meso‑scale. To enhance efficiency, an improved contact loop updating algorithm is proposed, which only preserves the corners of polygonal blocks and deletes the rest of the loop boundary nodes, thus greatly reducing the number of loop nodes involved in contact retrieval. An interface contact model considering cohesion and tensile strength between rock grains is incorporated into the numerical manifold method to simulate fracturing. With the newly developed grain-based numerical manifold method, Brazilian tests and uniaxial compression tests are numerically simulated to validate failure pattern and macroscopic response against laboratory tests. Sensitivity analysis is conducted using the proposed model to further investigate the influence of different number of grains and different stiffness ratio on the macroscopic response of rocks. The results indicate that the improved grain-based numerical manifold method can be effectively used to study deformation, damage and fracturing of rocks at the meso‑scale.

Influence of geometry modifications on the permeability of open‐cell foams

AbstractA stochastic microstructure model based on a random Laguerre tessellation is used to simulate virtual open cell foam structures. Both circular cylindric and concave triangular strut cross section shapes are considered. Additional geometry modifications are introduced by relaxation of the tessellation cells using the Surface Evolver software and by closing a certain percentage of the foam windows. The effect of these modifications on the foams' permeabilities is investigated. In particular, permeability anisotropies resulting from anisotropic closing of the windows are taken into account. The dimensionless permeability (Darcy number) in the different directions is well explained by regression models using porosity, geometric tortuosity, and constrictivity as explanatory variables.