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.

翻译：本文通过将质心Voronoi剖分引入细化过程，扩展了几何细化变换，从而增强了对称性、重构精度和数值稳定性。通过在每一细化层级应用Lloyd算法，我们最小化了质心能量，并生成了与函数底层结构更匹配的Voronoi区域。该方法减少了几何畸变，抑制了重构误差，并为自适应细化提供了一个自然框架。我们分析了收敛性，利用基于泰勒展开的估计和Lipschitz连续函数量化了重构误差的降低，并提出了扰动策略以逃离保持对称性的局部极小值。所得变换在医学成像、信号处理和物理模拟等应用中提供了更高的精度，同时保持了原始GRT框架的理论完备性和稳定性保证。