[Paper Review] Voronoi Diagram: The Generator Recognition Problem

M. Montserrat Alonso Ferrero|arXiv (Cornell University)|May 21, 2011

Computational Geometry and Mesh Generation9 references3 citations

TL;DR

This paper proposes a novel method to reconstruct generator points from a given Voronoi diagram using three iterative algorithms (I, II, and III), which estimate generator positions from Voronoi polygon vertices and ray intersections. The key contribution is a stable, fast O(n)-time inversion process with minimal root-mean-squared errors, enabling accurate reconstruction of atomic-scale structures from AFM images in materials science and beyond.

ABSTRACT

For the analysis of systems consisting of small, regular objects, the methods of mathematical morphology applied to images of these systems are well-suited. One of these methods is the use of Voronoi polygons. It was found that the Voronoi tessellation method represents a powerful tool for the analysis of thin film morphology and provides nanostructural information to many multi-particle assemblies. In these notes, several morphological algorithms are analyzed and we study how to join all of them to design a graphical user interface (GUI) that provides as input for the system the "AFM image" and interprets the output of the system in terms of errors and generators coordinates.

Motivation & Objective

• To solve the inverse problem of recovering generator points from a Voronoi diagram in thin film morphology analysis.
• To develop robust, computationally efficient algorithms for generator reconstruction from noisy or imperfectly extracted Voronoi vertices.
• To integrate multiple morphological algorithms into a single graphical user interface (GUI) for practical use with AFM images.
• To enable error quantification and coordinate output for scientific interpretation of nanostructured materials.
• To extend the framework beyond materials science to other fields where Voronoi diagrams model spatial processes.

Proposed method

• Extract Voronoi vertices and adjacency relationships from AFM images of thin films as input.
• Apply Algorithm I: use vertex pairs to compute intersection points of perpendicular bisectors and average them to estimate generator locations.
• Apply Algorithm II: compute all pairwise ray intersections within each Voronoi polygon and average results to improve stability.
• Apply Algorithm III: introduce weighted averaging of intersection points based on stability estimates (via slope perturbation), improving robustness to noise.
• Use a penalty function approach as an alternative, equivalent to minimizing sum of squared distances to intersection points.
• Integrate all algorithms into a unified graphical user interface (GUI) that accepts AFM images and outputs generator coordinates and error metrics.

Experimental results

Research questions

• RQ1 How accurately can generator points be reconstructed from a Voronoi diagram when only vertices and edges are available?
• RQ2 What is the impact of vertex measurement error on the accuracy of generator reconstruction algorithms?
• RQ3 Can a stable, fast, and local algorithm be designed for Voronoi generator inversion using only vertex and ray data?
• RQ4 How can multiple algorithms be combined into a single, user-friendly GUI for practical application in materials science?
• RQ5 To what extent can this framework be generalized to other domains such as ecology, epidemiology, or astronomy where Voronoi diagrams appear?

Key findings

• The proposed algorithms—especially Algorithm III—demonstrate significantly reduced error compared to standard methods, with error stability improved through weighted averaging based on intersection robustness.
• All algorithms operate in O(n) time, making them highly efficient for large-scale reconstruction of generator sets from Voronoi diagrams.
• The root-mean-squared error in generator location is minimal when vertex data is accurate, validating the feasibility of the inversion process.
• Algorithm III outperforms II by incorporating stability estimates from slope perturbations, reducing sensitivity to ill-conditioned ray pairs.
• The integration of all steps into a single GUI enables rapid, interactive analysis of AFM images with real-time output of generator coordinates and error metrics.
• The framework is extensible to any image type suspected to represent a Voronoi diagram, including applications in ecology, meteorology, and astronomy.