Fast and Accurate Reconstruction of Voronoi Generators in Large Tessellations

TL;DR

This paper introduces a fast, robust algorithm for reconstructing Voronoi generators from large tessellations, achieving high accuracy with linear time complexity.

Contribution

The authors present a novel linear system-based method for inverse Voronoi problems that efficiently reconstructs generators in large diagrams with high precision.

Findings

Average RMSE of 10^{-12} across simulations

Reconstruction error as low as 10^{-8} in worst cases

Linear time complexity O(n) for large diagrams

Abstract

A Voronoi diagram partitions the plane into convex cells, each containing the points closest to a single generator. Given such a tessellation, the inverse Voronoi problem seeks the generator set \( S \) that produced it. Our algorithm selects a single interior cell with \( k \) edges and solves a compact, consistent linear system with \( 2(k+1) \) unknowns and \( 4k \) scalar equations to recover that cell's generator together with the \( k \) generators of its neighbors in one step. The remaining sites follow by successive geometric reflections. The overall running time is \( O(n) \) for a diagram with \( n \) cells. Across \( 10^3 \) Monte Carlo simulations on diagrams of \( 10^4 \) cells, the method achieved an average RMSE of \( 10^{-12} \) and a worst-case individual reconstruction error of \( 10^{-8} \), demonstrating both efficiency and robustness.