A simple and fast algorithm for computing discrete Voronoi, Johnson-Mehl or Laguerre diagrams of points

TL;DR

This paper introduces a simple, fast algorithm for computing digital Voronoi, Johnson-Mehl, or Laguerre diagrams in any dimension, optimized for high-resolution microstructure imaging and compatible with FFT-based homogenization.

Contribution

It presents a novel, efficient algorithm that simplifies the computation of various Voronoi-type diagrams with optimal performance and broad applicability.

Findings

Algorithm operates in O(N_v log N_s) time

Suitable for high-resolution microstructure imaging

Compatible with FFT-based homogenization methods

Abstract

This article presents an algorithm to compute digital images of Voronoi, Johnson-Mehl or Laguerre diagrams of a set of punctual sites, in a domain of a Euclidean space of any dimension. The principle of the algorithm is, in a first step, to investigate the voxels in balls centred around the sites, and, in a second step, to process the voxels remaining outside the balls. The optimal choice of ball radii can be determined analytically or numerically, which allows a performance of the algorithm in $O(N_v \ln N_s$), where $N_v$ is the total number of voxels of the domain and $N_s$ the number of sites of the tessellation. Periodic and non-periodic boundary conditions are considered. A major advantage of the algorithm is its simplicity which makes it very easy to implement. This makes the algorithm suitable for creating high resolution images of microstructures containing a large number of cells, in particular when calculations using FFT-based homogenisation methods are then to be applied to the simulated materials.