Centroidal Voronoi Tessellations (CVT)

• CVTs are tessellations where each generator coincides with the centroid of its cell, ensuring minimized distortion and optimal partitioning of convex domains.
• They establish an equivalence between centroidal energy minimization and electrostatic equilibrium, supporting innovative global optimization and metastable state recovery.
• Applications span adaptive meshing, materials science, and biological pattern formation, leveraging methodologies like thermodynamic annealing for robust results.

Centroidal Voronoi Tessellations (CVT) are a class of tessellations of a compact domain in which each generator coincides with the mass centroid of its Voronoi cell. Beyond their classical geometric and quantization interpretations, recent research has established a deep equivalence between CVTs and electrostatic equilibria in convex domains, motivating new algorithms for global optimization and metastable state enumeration. This article synthesizes mathematical foundations, variational principles, optimization schemes, convergence phenomena, and emerging extensions with reference to (Mullaghy, 3 Apr 2025).

1. Mathematical Definition and Centroidal Condition

Let $\Omega \subset \mathbb{R}^n$ be a compact convex domain, and consider a set of $N$ generators $\{x_i\}_{i=1}^N \subset \Omega$. The Voronoi cell associated to $x_i$ is

$$V_i = \{x \in \Omega : \|x - x_i\| \le \|x - x_j\|, \forall j \ne i\}.$$

A Centroidal Voronoi Tessellation (CVT) is defined by the condition that each generator coincides with the centroid of its cell: $$x_i = \frac{1}{\mathrm{Vol}(V_i)}\int_{V_i} y\, dy.$$ This equality arises as the stationarity condition for the classical centroidal energy functional: $$E_{\mathrm{centroid}}(\{x_i\}) = \sum_{i=1}^N \int_{V_i} \|x - x_i\|^2 dx.$$ The minimizers of $E_{\mathrm{centroid}}$ under reasonable regularity requirements satisfy the centroidal equations $x_i = \mathrm{centroid}(V_i)$, giving rise to the CVT structure.

2. Electrostatic Energy Model and Variational Equivalence

A generalized Thomson problem establishes a correspondence between CVTs and electrostatic equilibria within a convex domain. The configuration consists of $N$ unit charges $\{x_i\}$ inside $\Omega$ and a continuous boundary charge $\sigma$ on $\partial \Omega$ such that $\int_{\partial \Omega} \sigma(y)\, dS(y) = N$. The electrostatic energy functional is given by

$$U(\{x_i\}) = \sum_{i=1}^N \sum_{j \ne i} \frac{1}{\|x_i - x_j\|} + \sum_{i=1}^N \int_{\partial \Omega} \frac{\sigma(y)}{\|x_i - y\|} dS(y).$$

Stationarity (vanishing first variation) yields the equilibrium equations

$$\sum_{j \ne i} \frac{-(x_i - x_j)}{\|x_i - x_j\|^3} - \int_{\partial \Omega} \frac{(x_i - y)}{\|x_i - y\|^3} \sigma(y) dS(y) = 0,\quad i=1,...,N.$$

The main theorem demonstrated is that local minima of $U$ coincide with CVT configurations. This is substantiated via:

• An edge-based approximation of $E_{\mathrm{centroid}}$:

  $$\widetilde E_{\mathrm{centroid}} = \sum_{i<j} \ell_{ij}\, \|x_i - x_j\|^2,$$

which shares (up to smooth positive scaling) second variation properties with the true $E_{\mathrm{centroid}}$ near CVT states.

• At a CVT configuration, the Hessian of $U$ matches that of $\widetilde E_{\mathrm{centroid}}$ (again up to positive weights), so every CVT is a strict local minimizer of the electrostatic energy.

3. Global Optimization via Thermodynamic Annealing

To recover global CVTs and escape local minima, a stochastic optimization procedure rooted in thermodynamic annealing is employed. The ensemble follows the Boltzmann-Gibbs distribution: $P_T(\{x_i\}) \propto \exp(-U(\{x_i\})/T).$ A Metropolis–Hastings random walk is performed as follows:

1. Randomly select $i$; propose $x_i' = x_i + \delta$, $\delta \sim \mathcal{N}(0,\sigma^2 I)$.
2. Evaluate $$\Delta U = U(\ldots, x_i', \ldots) - U(\ldots, x_i, \ldots)$$.
3. Accept update with probability $1$ if $\Delta U \le 0$, or $\exp(-\Delta U/T)$ if $\Delta U > 0$.
4. Update temperature via a schedule, e.g., $T(t) = T_0 \alpha^t$, $0<\alpha<1$.

The temperature schedule governs the likelihood of escaping local minima; slower cooling ($\alpha \rightarrow 1$) increases global optimum accessibility but at the expense of computational effort. The algorithm's complexity per energy evaluation is naively $O(N^2)$, but fast multipole or tree-based methods can greatly accelerate calculations for large $N$.

4. Convergence Analysis and Symmetry Breaking

• Theoretically, a cooling schedule $T(t) \sim O(1/\log t)$ guarantees convergence but is seldom used due to practical inefficiency.
• The annealing algorithm is particularly advantageous in symmetric domains, where many CVTs are energetically equivalent (multiple local minima).
• Step-size $\sigma^2$ must be tuned appropriately to balance move acceptance rates with domain coverage.
• The method supports recovering multiple distinct CVT minima by varying the annealing timescale, with longer $\tau$ preferentially reaching lower-energy states (energy gap $\Delta E$ relates $\tau \propto 1/\Delta E$).

5. Lattice Anchoring and Metastable State Organization

Following annealing, "lattice-anchored annealing mapping" (LAAM) augments discovered CVT minima with external periodic lattice extensions $A_k$. By re-optimizing with anchor $A_k$, the interior configuration returns to its associated CVT. This mapping pairs each metastable configuration with a structural continuation, categorizing the landscape of CVTs and associated minima. Applications include:

• Materials science (grain-boundary enumeration)
• Adaptive meshing (robust partitioning)
• Biological pattern formation (organization of cell arrangements)

This pairing constructs a customizable framework for systematic minimum seeking and stability analysis.

6. Broader Implications for Quantization and Geometric Analysis

The demonstrated equivalence between centroidal energy minimization and electrostatic equilibrium positions CVTs as a central object bridging geometric quantization, optimization, and physical modeling. CVTs minimize not only classical geometric distortion metrics but also naturally arise as energetic equilibria under repulsive physical constraints. Embedding CVT search in physical models enables principled exploration of tessellation spaces, robustly cataloguing metastable and ground-state partitions without relying on purely local optimization.

In summary, CVTs provide a rigorous geometric and variational framework for optimal partitioning under both mathematical and physical constraints. The electrostatic formulation facilitates robust global optimization and metastable state recovery, advancing their utility in modeling, simulation, and computational geometry (Mullaghy, 3 Apr 2025).