Centroidal Voronoi Refinement in the Geometric Refinement Transform: Symmetry, Stability, and Optimal Reconstruction

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.