Centroidal Voronoi Refinement in the Geometric Refinement Transform: Symmetry, Stability, and Optimal Reconstruction
Zachary Mullaghy
TL;DR
This work addresses reconstruction accuracy and stability in multiscale geometric refinements by integrating centroidal Voronoi tessellations into the Geometric Refinement Transform. It combines Lloyd's algorithm for CVT generation, a Lipschitz-based error framework, and symmetry-breaking perturbations to enhance accuracy and robustness. A key contribution is an energy-minimization perspective showing CVT refinements reduce reconstruction error relative to arbitrary refinements, with explicit per-cell error considerations. The resulting framework enables adaptive, geometry-aware multiscale reconstruction with broad implications for medical imaging, physical simulations, materials science, and geometry-aware signal processing.
Abstract
We extend the Geometric Refinement Transform (GRT) by introducing centroidal Voronoi tessellations (CVTs) into the refinement process, enhancing symmetry, reconstruction accuracy, and numerical stability. By applying Lloyds algorithm at each refinement level, we minimize centroidal energy and generate Voronoi regions that better align with the functions underlying structure. This approach reduces geometric distortion, suppresses reconstruction error, and provides a natural framework for adaptive refinement. We analyze convergence properties, quantify the reduction in reconstruction error using Taylor-based estimates and Lipschitz continuous functions, and propose perturbation strategies to escape symmetry-preserving local minima. The resulting transform offers improved accuracy for applications in medical imaging, signal processing, and physics simulations, while preserving the theoretical completeness and stability guarantees of the original GRT framework.