Geometric Tessellation & Optimal Transport

• Geometric tessellation is defined by partitioning space using convex, weighted Voronoi cells that solve semi-discrete optimal transport problems.
• The variational and algorithmic frameworks employ Laguerre diagrams and Newton methods to achieve precise mass redistribution and anisotropic mesh alignment.
• Applications include polycrystalline microstructure simulation, adaptive mesh generation, robotic coverage control, and robust image processing techniques.

Geometric tessellation and optimal transport are deeply intertwined through the mathematics of convex geometry, variational optimization, and computational algorithms. The connection is exemplified by the realization that weighted Voronoi diagrams (Laguerre or power tessellations) solve semi-discrete optimal transport problems under quadratic cost, and conversely, optimal transport theory provides the variational and algorithmic framework for generating, inverting, and fitting geometric tessellations in complex applications. This nexus underlies advances in mesh generation, computational materials science, image processing, and multi-agent robotics.

1. Theoretical Foundations: Optimal Transport and Tessellations

Optimal transport (OT) seeks a map $T: X \to Y$ that pushes a source measure $\mu$ onto a target measure $\nu$, minimizing a cost function $c(x,y)$ $1710.02634$. In the Monge problem, $T$ is measurable and mass-preserving, while the Kantorovich relaxation allows for couplings (joint measures). Of particular importance is the semi-discrete regime, where $\mu$ is absolutely continuous (typically Lebesgue measure on a convex domain), and $\nu$ is a sum of Dirac measures at points $\{y_i\}$.

For quadratic cost $c(x, y) = \|x - y\|^2$, the Kantorovich dual reduces to a finite-dimensional optimization in weights $\mu$0 associated to each $\mu$1. The associated Laguerre cells (power diagrams)

$\mu$2

define a convex tessellation of $\mu$3 $\mu$4. Each region is mapped to a target atom, and the measure of each cell is matched to the prescribed mass $\mu$5 of $\mu$6. These geometric partitions are thus natural solutions to semi-discrete OT.

2. Variational and Algorithmic Structure

The equivalence between semi-discrete OT and Laguerre tessellation is formalized through a variational principle: maximizing a concave function (the Kantorovich dual) or minimizing a convex function over the space of weights $\mu$7 (or related "heights" $\mu$8 for polyhedral cell construction). The gradient of this functional is the vector of mass mismatches between the measured cell volumes and the target $\mu$9, and the Hessian captures geometric adjacency through integrals over shared cell faces %%%%2$T: X \to Y$2$\mu$2%%%%2.

Newton or damped Newton methods are the standard computational approach, exploiting the sparsity of the Hessian, which is only nonzero for neighboring cells. Algorithmic advances permit robust solution of large-scale volume-matching problems, including periodic tessellations in the modeling of polycrystalline materials, with global linear and local quadratic convergence rates established for the damped Newton flows $\nu$3.

3. Generalizations: Periodicity, Anisotropy, and Topological Control

Periodic Tessellations

In periodic settings, as for representative volume elements (RVEs) in materials science, the quadratic cost is adapted by minimizing over lattice translations, and Laguerre cells are constructed inside a fundamental domain $\nu$4 $\nu$5. The resulting periodic power diagrams provide efficient geometric descriptors for complex polycrystalline microstructure, and algorithms scale to $\nu$6 with minimal backtracking during Newton optimization.

Anisotropic and Feature-Aligned Meshes

Beyond equidistribution, alignment—optimal mesh element orientation and shape—is captured via the MA (Monge–Ampère) approach, with the mesh mapping $\nu$7 determined by a convex potential $\nu$8 satisfying the nonlinear MA equation. The induced (implicit) metric tensor $\nu$9 as a function of the density $c(x,y)$0 and the Hessian $c(x,y)$1 characterizes mesh anisotropy and alignment to features such as shocks or interfaces. Eigenstructure of $c(x,y)$2 yields strong stretching along tangents to features and compression along normals, with explicit dependence on local density and curvature $c(x,y)$3.

Topological and Quasiconformal Corrections

Standard sd-OT algorithms may alter mesh topology (e.g., Delaunay flips) to maintain convexity. This is incompatible with certain applications (e.g., shape editing, physical deformations) where orientation and connection must be preserved. The quasiconformal OT (QC-OT) approach relaxes the topology-adapting requirements, retaining the original triangulation and applying local quasiconformal corrections to ensure diffeomorphic, low-distortion mappings. The pipeline alternates Newton steps for the relaxed convex energy with Beltrami-coefficient-driven, harmonic patch corrections, validated for mesh parameterization and spatiotemporal mass transport in image editing and biophysical deformation simulations $c(x,y)$4.

4. Inverse Problems and Data-Driven Fitting

Given only volumes and centroids of a tessellation’s cells (possibly noisy or estimated from imaging), one can reconstruct the underlying weights of a Laguerre diagram uniquely, up to translation in the weight space. Bourne–Pearce–Roper provide a convex optimization framework for the inversion and robust fitting of tessellations to observed or target microstructural geometries. A regularized least-squares energy combining volume and centroid discrepancies ensures stability and convergence, with practical initialization heuristics (e.g., choosing seed points as centroids) producing high-quality fits to empirical data, such as electron backscatter diffraction (EBSD) images of steel $c(x,y)$5.

5. Applications in Computational Science and Engineering

Polycrystalline Microstructure Generation

Rapid generation of periodic Laguerre tessellations is essential for simulating polycrystalline materials. The periodic semi-discrete OT method produces large, highly polydisperse 3D tessellations with exact volume constraints, enabling accurate RVEs in a few minutes on standard hardware $c(x,y)$6. The methodology supports both fixed-seed and alternating Lloyd steps for regularization of grain shapes.

Mesh Generation and Adaptation

OT-based mesh adaptation via the Monge–Ampère framework yields meshes that are both equidistributed and aligned to features of the solution, offering optimal anisotropy where needed, and ensuring smooth, orthogonal alignment in the metric induced by the transport map. These properties are theoretically characterized by explicit formulas linking density, curvature, and mesh metric eigenstructure $c(x,y)$7.

Coverage Control in Robotics

Coverage control for robotic swarms is recast as an OT problem, with agents assigned to Laguerre cells that partition the domain according to the target spatial distribution. The OT-based controller (OTCC) generalizes and outperforms classical Voronoi-based methods, achieving lower quadratic coverage cost and better distribution convergence by utilizing ascent–descent dynamics of the OT dual functional $c(x,y)$8.

Image Processing and Surface Parameterization

QC-OT and related geometric OT techniques enable mesh-based area-preserving parameterizations, content-aware mesh refinement, local magnification for image editing, and robust, topology-preserving deformations in medical and computer vision applications. The combination of convex OT optimization with quasiconformal correction achieves flip-free, low-distortion mappings for time-varying and irregular data $c(x,y)$9.

6. Methodological Summary and Algorithmic Structure

The following table summarizes core methodological steps universally present in OT–tessellation algorithms for the quadratic cost case:

Step	Description	References
Measure specification	Define continuous (source) and discrete (target) measures	(Levy et al., 2017, Bourne et al., 2024)
Variational formulation	Derive dual (Kantorovich) functional in weights or heights	(Lei et al., 2017, Bourne et al., 2022)
Cell construction	Build Laguerre/power diagram for current weights	(Levy et al., 2017, Bourne et al., 2024)
Mass/centroid computation	Integrate measure over cells, compute centroids as needed	(Bourne et al., 2024)
Gradient/Hessian calculation	Compute gradient (mass mismatch) and sparse Hessian (face adjacency)	(Lei et al., 2017, Bourne et al., 2022)
Newton (or QC) step	Solve linear system for weight update; or alternate with QC fix if needed	(Bourne et al., 2022, Lv et al., 2 Jul 2025)
Termination	Check convergence in mass, centroid, or distortion	(Bourne et al., 2024, Lv et al., 2 Jul 2025)

Convergence and practical performance rely on the twice-differentiable, sparsely-coupled structure of the objective, and, in topology-sensitive settings, on dynamic correction steps.

7. Significance, Research Directions, and Outlook

The deep equivalence between semi-discrete OT and Laguerre tessellation constitutes a unification of convex geometry, computational geometry, and variational analysis $1710.02634$0[(Bourne et al., 2024)]. This synthesis enables rigorous mathematical results (existence, uniqueness, convexity) and practical algorithms for mesh generation, materials modeling, image analysis, and multi-agent systems.

Current research continues to address significant challenges: incorporating anisotropy and curvature through density-driven mesh metrics, generalizing to non-quadratic costs or non-Euclidean geometries, achieving topological and diffeomorphic guarantees in discrete mesh settings, and scaling robustly to high-dimensional, large-$1710.02634$1 scenarios.

The rapidly growing corpus on arXiv demonstrates the versatility and ongoing innovation in this area, with applications spanning computational science, geometry processing, robotics, and vision. The geometric–OT paradigm provides a principled, efficient, and generalizable framework for both modeling and computation in domains where tessellation and mass redistribution are fundamental.