Marked Nested Tessellation: Hierarchical Structures

• Marked nested tessellation is a hierarchical tessellation model that integrates random geometric divisions with additional stochastic marks across multiple scales.
• It leverages frameworks like Laguerre, Voronoi, and Markov processes to accurately represent complex structures in materials science and fluid dynamics.
• Advanced statistical fitting and simulation methods validate these models by aligning synthetic tessellations with observed cell features and interactions.

Searching arXiv for recent and foundational papers on marked nested tessellations and closely related tessellation frameworks. {"query":"all:\"marked nested tessellation\" OR ti:\"nested tessellation\" OR ti:\"Laguerre tessellation\" tessellation marks stochastic geometry", "max_results": 10} {"query":"ti:\"Random marked nested tessellations\" OR ti:\"Shape-Driven Nested Markov Tessellations\" OR ti:\"Fitting three-dimensional Laguerre tessellations\"", "max_results": 10} Marked nested tessellation can be understood as a hierarchical structure in which a random tessellation of space is generated by underlying random generators, additional random attributes are attached to cells, faces, edges, or generators, and the stochastic structure may be multilevel or nested, for example through multiple scales, interaction layers, or tessellations within tessellations (Kornijčuk et al., 18 Jul 2025). In contemporary work, the concept appears both explicitly, as in random Laguerre grain tessellations further subdivided into twin lamellae and matrix subcells, and implicitly, as in hierarchical marked point process models for 3D Laguerre tessellations or level-set/component/Voronoi decompositions whose regions carry statistical summaries (Seitl et al., 2021).

1. Core definition and geometric setting

A tessellation in $\mathbb{R}^3$ is a locally finite family of closed sets $T=\{C_i:i\in\mathcal I\}$ whose interiors are pairwise disjoint and whose union covers space or an observation window. In a nested framework, a mother cell is a cell of an initial tessellation, and each mother cell is further subdivided into subcells. Following Schreiber–Thäle’s terminology, a nested tessellation $T_N$ is obtained by starting from a tessellation $T_L=\{C_i\}$, constructing a tessellation of each $C_i$ into subcells, and taking the total collection $\{S_{i,k}:i\in\mathcal I,\ k\in K_i\}$ (Kornijčuk et al., 18 Jul 2025).

A marked tessellation is a collection of pairs $(C_i,M_i)$, where $C_i$ are cells and $M_i$ are marks in a mark space. In the literature considered here, the marks include radii or weights for Laguerre generators, crystallographic orientations in $SO/\mathcal O$, birth times of facets, and region-level statistical descriptors such as histograms, moments, Gaussian mixture models, and kernel density estimates (Kornijčuk et al., 18 Jul 2025).

Laguerre tessellations supply a central geometric substrate. Given weighted generators $T=\{C_i:i\in\mathcal I\}$0, the Laguerre cell is

$T=\{C_i:i\in\mathcal I\}$1

and the nonempty cells form a tessellation. In the marked-point formulation, one writes generators as $T=\{C_i:i\in\mathcal I\}$2, defines the power distance

$T=\{C_i:i\in\mathcal I\}$3

and obtains the cell $T=\{C_i:i\in\mathcal I\}$4 by comparing power distances to all generators. If all radii are equal, the induced partition coincides with the standard Voronoi tessellation (Seitl et al., 2021).

A common misconception is that marked nested tessellation is synonymous with a weighted Voronoi or Laguerre model. The available work does not support that restriction. In deformation twinning, the nested tessellation is explicitly not Laguerre, because lamellae slice through mother cells in a way that breaks the face-to-face property typical of Laguerre and Voronoi tessellations (Kornijčuk et al., 18 Jul 2025).

2. Principal hierarchical constructions

Several distinct constructions realize marked nested tessellation in current arXiv literature. They differ in whether nesting is induced by stochastic cell splitting, hierarchical conditioning, or deterministic restriction by level sets.

Framework	Nested structure	Marks
Random marked nested tessellation for deformation twinning	Laguerre grains $T=\{C_i:i\in\mathcal I\}$5 lamellae and interlamellar matrix subcells	orientations, propensity for twinning, twin-volume fraction
Hierarchical Laguerre fitting	points $T=\{C_i:i\in\mathcal I\}$6 $T=\{C_i:i\in\mathcal I\}$7 marks $T=\{C_i:i\in\mathcal I\}$8 $T=\{C_i:i\in\mathcal I\}$9 tessellation	radii and tessellation characteristics
Level set restricted Voronoi tessellation	$T_N$0	region-level statistical summaries
Shape-driven nested Markov tessellation	recursive binary cell splits over time	birth times of facets

In the deformation-twinning model, the mother tessellation is a random marked Laguerre tessellation $T_N$1, and each grain $T_N$2 may be subdivided into parallel twin lamellae and complementary matrix regions. The subcells inherit either the mother orientation or a twin orientation $T_N$3, so the nested object is marked at both mother-cell and subcell levels (Kornijčuk et al., 18 Jul 2025).

In the hierarchical Laguerre framework, the observed data are a finite marked pattern $T_N$4 inside a rectangular window $T_N$5, extended periodically to define the tessellation in a stationary way. The hierarchy has two layers: a stationary point process $T_N$6 for grain-center locations, modeled as a parametric Gibbs point process, and a conditional model for marks $T_N$7, where the sufficient statistics are functions of the resulting Laguerre tessellation. The tessellation is then a deterministic functional of $T_N$8 (Seitl et al., 2021).

In the level-set restricted Voronoi construction, the hierarchy is explicit: $T_N$9 The volume is first partitioned into isobands $T_L=\{C_i\}$0, then into connected components $T_L=\{C_i\}$1, and finally into restricted Voronoi regions $T_L=\{C_i\}$2 inside each component using graph-geodesic distance constrained by level-set and block boundaries. The authors do not explicitly use the term “marked nested tessellation,” but they organize the decomposition as a coherent nested hierarchy and attach statistical summaries to each region (Neuroth et al., 2022).

At a more abstract level, shape-driven nested Markov tessellations are stationary random tessellations of $T_L=\{C_i\}$3 constructed by a spatio-temporal random recursive split dynamics governed by a family of Markovian split kernels. In this setting, nestedness is temporal: facets are marked by birth times, and the tessellation at time $T_L=\{C_i\}$4 is obtained by retaining only facets born no later than $T_L=\{C_i\}$5 (Schreiber et al., 2011).

3. Marks and tessellation observables

The mark space depends on the application. In polycrystalline deformation twinning, the primary cell mark is the crystallographic orientation $T_L=\{C_i\}$6, where $T_L=\{C_i\}$7 is the rotation group and $T_L=\{C_i\}$8 is the $T_L=\{C_i\}$9-element subgroup of cubic crystal symmetries. Derived marks include the Schmid factor $C_i$0, the propensity for twinning $C_i$1, the twin normal $C_i$2, and the twin-volume fraction $C_i$3. Lamellar subcells are marked by the twin orientation $C_i$4, while matrix subcells retain the mother orientation (Kornijčuk et al., 18 Jul 2025).

In hierarchical Laguerre models, the primary marks are radii $C_i$5, but the model is expressly tessellation-aware because the sufficient statistics are functions of cell and face characteristics. The observed or modeled characteristics include cell volume $C_i$6, surface area $C_i$7, number of faces $C_i$8, total edge length $C_i$9, sphericity $\{S_{i,k}:i\in\mathcal I,\ k\in K_i\}$0, and face-level quantities such as absolute volume difference $\{S_{i,k}:i\in\mathcal I,\ k\in K_i\}$1, face area, face perimeter, and number of face edges. Candidate sufficient statistics include beta-type radii terms, total number of faces, total surface area, sum of squared volumes, and sums over neighboring volume contrasts (Seitl et al., 2021).

In large-scale spatial statistical analysis, the marks are region-level statistical attributes. For each Voronoi cell, connected component, or isoband, the recorded summaries include histograms, Gaussian distributions, Gaussian mixture models, covariance, coskewness, cokurtosis, kernel density estimates, and conditional statistics such as conditional means and standard deviations. The resulting object is a region hierarchy in which each node carries a feature vector of statistical descriptors (Neuroth et al., 2022).

The combinatorial analysis of spatial STIT tessellations provides additional categorical marks. Vertices come in two types, T and X, with probabilities $\{S_{i,k}:i\in\mathcal I,\ k\in K_i\}$2 and $\{S_{i,k}:i\in\mathcal I,\ k\in K_i\}$3. Edges can be classified as $\{S_{i,k}:i\in\mathcal I,\ k\in K_i\}$4, $\{S_{i,k}:i\in\mathcal I,\ k\in K_i\}$5, or $\{S_{i,k}:i\in\mathcal I,\ k\in K_i\}$6 according to endpoint types, with approximate probabilities $\{S_{i,k}:i\in\mathcal I,\ k\in K_i\}$7, $\{S_{i,k}:i\in\mathcal I,\ k\in K_i\}$8, and $\{S_{i,k}:i\in\mathcal I,\ k\in K_i\}$9. They can also be classified by how many plate sides or cell ridges coincide with the edge. The paper explicitly treats these labels as natural mark-candidates on vertices and edges (Thaele et al., 2011).

4. Stochastic and dynamic foundations

The most general probabilistic foundation in the cited literature is the split-kernel formulation of nested Markov tessellations. For a tessellation-valued Markov process $(C_i,M_i)$0, the evolution is governed by a family of split kernels $(C_i,M_i)$1, and the infinite-volume generator is

$(C_i,M_i)$2

The spatial Markov property is equivalent to $(C_i,M_i)$3, meaning that the splitting behavior of a cell depends only on that cell and not on the outside environment (Schreiber et al., 2011).

Shape-driven nested Markov tessellations impose canonical split intensities $(C_i,M_i)$4, time homogeneity, spatial homogeneity under scaling, and isometry invariance. Under the regularity assumptions [SKR1]–[SKR2], there exists a unique time-consistent whole-space tessellation-valued Markov process of stationary random tessellations compatible with the given split kernels. The process satisfies stationarity, the scaling relation

$(C_i,M_i)$5

and time consistency $(C_i,M_i)$6 for $(C_i,M_i)$7 (Schreiber et al., 2011).

STIT tessellations are a special case. They are spatially homogeneous random tessellations that are stable under iteration in $(C_i,M_i)$8, arise as outcomes of a spatio-temporal process of subsequent cell division, and are not facet-to-facet. Their significance for marked nested tessellation lies in the fact that iteration already provides a canonical nesting mechanism, while birth times, vertex types, and edge classes furnish natural marks (Thaele et al., 2011).

These results also delimit the concept. Nestedness need not arise from cellwise subdivision alone. In hierarchical Laguerre fitting, nestedness is induced by conditioning points, then marks, then the tessellation functional; in level-set restricted Voronoi analysis, it is induced by successive restriction from isobands to components to restricted Voronoi cells (Seitl et al., 2021).

5. Statistical fitting, simulation, and computation

The most developed inferential treatment is the hierarchical marked point process model for 3D Laguerre tessellations. For locations, the point process density is modeled as a multiscale pairwise interaction process $(C_i,M_i)$9, with a nested sequence $C_i$0. For fixed interaction radii, the model is an exponential family in $C_i$1, so the log pseudolikelihood is concave and can be maximized via Newton–Raphson. For the conditional mark model, the pseudolikelihood is again concave in the canonical parameter vector $C_i$2, and one-dimensional integrals are approximated numerically for each mark update (Seitl et al., 2021).

Model checking proceeds in two steps. For the point process, candidate models are fitted, simulated, and compared by global rank envelopes for $C_i$3, $C_i$4, and $C_i$5. The Poisson model is rejected very strongly, the Strauss model is still borderline with $C_i$6, and the multiscale model with $C_i$7 gives $C_i$8, with observed functions inside the $C_i$9 global envelopes. For marks, candidate exponential-family models are compared by maximized pseudolikelihood and then by global envelopes for kernel-smoothed density estimates of $M_i$0. The final model, denoted beta+nof+dvol, gives a global envelope test $M_i$1, with observed density curves entirely inside the envelopes (Seitl et al., 2021).

Simulation likewise follows the hierarchy. Point patterns are generated by a birth–death–move Metropolis–Hastings algorithm, and radii are updated by Metropolis-within-Gibbs using full conditional densities and recomputation of affected Laguerre cells via Voro++. In the NiTi application, the fitted point-process parameters are $M_i$2, $M_i$3, $M_i$4, $M_i$5, and $M_i$6 $M_i$7m, while the fitted mark model has $M_i$8, $M_i$9, $SO/\mathcal O$0, and $SO/\mathcal O$1 (Seitl et al., 2021).

The level-set restricted centroidal Voronoi tessellation is realized algorithmically by a modified Lloyd procedure: StratifiedRandomSiteDistribution(), RestrictedGeodesicVoronoiDecomposition(), RestrictedGeodesicallyWeightedUpdate(), and a final GeodesicVoronoiDecomposition(). The implementation is GPU-parallel within blocks and MPI-parallel across blocks. On Summit with V100 GPUs, for a dataset of size $SO/\mathcal O$2 and site density about $SO/\mathcal O$3 voxels per site, the reported worst-case node times are about $SO/\mathcal O$4 s on $SO/\mathcal O$5 GPUs and about $SO/\mathcal O$6 s on $SO/\mathcal O$7 GPUs to reach mean $SO/\mathcal O$8 voxel widths (Neuroth et al., 2022).

6. Scientific applications

The materials-science applications are the most direct realizations of marked nested tessellation. In 3D Laguerre fitting, the data come from a polycrystalline NiTi alloy studied via 3D X-ray diffraction microscopy. A sub-window $SO/\mathcal O$9 $T=\{C_i:i\in\mathcal I\}$00m is extracted, a periodic extension is used to avoid boundary artifacts, and cells with centers in $T=\{C_i:i\in\mathcal I\}$01 and nonempty Laguerre cells yield a dataset with $T=\{C_i:i\in\mathcal I\}$02. The fitted hierarchical model generates synthetic tessellations that mimic observed distributions of grain volumes, shapes, numbers of neighbors, surface areas, and neighboring-volume differences, and simulations under the fitted model may substitute expensive laboratory experiments (Seitl et al., 2021).

The deformation-twinning model moves to a finer scale. Grains are modeled by a random Laguerre tessellation with experimentally motivated grain volumes, crystallographic orientation marks, and a twinning rule based on the propensity $T=\{C_i:i\in\mathcal I\}$03 and a Hall–Petch type threshold $T=\{C_i:i\in\mathcal I\}$04. Twinning cells are subdivided into parallel lamellae subject to spacing, thickness, and total twin-volume constraints. The resulting marked nested tessellation is meshed and passed to finite-element analysis, where anisotropic linear elasticity, isotropic von Mises plasticity with hardening, and eigenstrains representing twinning are used to compute stress, strain, and strain energy density fields. Sensitivity studies vary texture $T=\{C_i:i\in\mathcal I\}$05 and macroscopic strain $T=\{C_i:i\in\mathcal I\}$06, and compare independent marking with moving-average marking (Kornijčuk et al., 18 Jul 2025).

In large-scale simulation analysis, the level-set restricted Voronoi hierarchy is applied to turbulent combustion and turbulent channel flow. For combustion, isobands and connected components are derived from a progress variable based on the normalized mass fraction of $T=\{C_i:i\in\mathcal I\}$07, and region marks summarize species mass fractions, temperature, and local strain or stretch rates. For channel flow, level sets of $T=\{C_i:i\in\mathcal I\}$08 and distance-to-vortex fields organize the domain around vortical structures. In both cases, the hierarchy supports top-down analysis through multiple levels of detail linking phase-space statistics with spatial features (Neuroth et al., 2022).

7. Assumptions, misconceptions, and extensions

The literature makes several assumptions explicit. In hierarchical Laguerre fitting, the point process is stationary on a torus, the conditional mark distribution is homogeneous, dependence is restricted to pairwise interactions at finitely many distance ranges, and the primary marks are radii. The authors also note possible bias in maximum pseudolikelihood estimates due to discretization effects and the computational burden of repeated tessellation recomputation in 3D (Seitl et al., 2021).

In deformation twinning, the mother tessellation uses planar-faced Laguerre cells from a homogeneous Poisson point process, orientations are either independent or moving-average correlated, twins are strictly parallel planar lamellae within each grain, and nesting occurs at one level only. Time evolution and dynamic twinning are not modeled; the morphology is static at a given macroscopic strain (Kornijčuk et al., 18 Jul 2025).

In level-set restricted Voronoi analysis, the decomposition depends on the choice of scalar field, isovalues, site-density parameter $T=\{C_i:i\in\mathcal I\}$09, attraction parameter $T=\{C_i:i\in\mathcal I\}$10, and region-level statistical models. The restricted CVT optimality is heuristic, and the current implementation computes the tessellation per block rather than globally (Neuroth et al., 2022).

A further misconception is that marked nested tessellation must be facet-to-facet. The cited works repeatedly show otherwise. Spatial STIT tessellations are not facet-to-facet because sequential cell division creates new vertices and edges inside existing facets, and deformation-twinning subcells likewise break the face-to-face property typical of Laguerre and Voronoi tessellations (Thaele et al., 2011).

The principal extension directions stated in the literature are consistent across frameworks: other point-process models with higher-order interactions, more complex tessellation characteristics in mark models, additional mark types such as crystallographic orientation, multi-level nested tessellations beyond grains and lamellae, alternative optimization schemes for restricted CVT, and automated or temporal pipelines built on region-level marks (Kornijčuk et al., 18 Jul 2025). These directions suggest that marked nested tessellation is best viewed not as a single model class but as a family of hierarchical stochastic-geometric and statistical constructions in which geometry and marks are coupled across levels.