An algorithm to compute CVTs for finitely generated Cantor distributions

TL;DR

This paper presents an algorithm for computing centroidal Voronoi tessellations (CVTs) on finitely generated Cantor sets, extending CVT computation to fractal measures generated by self-similar mappings.

Contribution

The paper introduces a novel algorithm to compute CVTs with arbitrary generators and levels for Cantor sets generated by specific self-similar mappings, applicable to any compatible probability distribution.

Findings

Algorithm successfully computes CVTs for Cantor sets with various parameters.

The method generalizes CVT computation to fractal measures.

Results demonstrate the algorithm's effectiveness across different self-similar structures.

Abstract

Centroidal Voronoi tessellations (CVTs) are Voronoi tessellations of a region such that the generating points of the tessellations are also the centroids of the corresponding Voronoi regions with respect to a given probability measure. CVT is a fundamental notion that has a wide spectrum of applications in computational science and engineering. In this paper, an algorithm is given to obtain the CVTs with $n$-generators to level $m$, for any positive integers $m$ and $n$, of any Cantor set generated by a pair of self-similar mappings given by $S_1(x)=r_1x$ and $S_2(x)=r_2x+(1-r_2)$ for $x\in \mathbb R$, where $r_1, r_2>0$ and $r_1+r_2<1$, with respect to any probability distribution $P$ such that $P=p_1 P\circ S_1^{-1}+p_2 P\circ S_2^{-1}$, where $p_1, p_2>0$ and $p_1+p_2=1$.