An Analytical Representation of the 2d Generalized Balanced Power Diagram

TL;DR

This paper presents an analytical method for representing and computing 2D generalized balanced power diagrams, enabling more flexible modeling of microstructures with curved and anisotropic grains.

Contribution

It introduces an explicit analytical representation of the diagram's vertices and edges and a new algorithm for its computation, advancing beyond previous discretized approaches.

Findings

Derived explicit formulas for vertices and edges.

Proposed a novel algorithm for diagram computation.

Enhanced modeling flexibility for anisotropic microstructures.

Abstract

Tessellations are an important tool to model the microstructure of cellular and polycrystalline materials. Classical tessellation models include the Voronoi diagram and Laguerre tessellation whose cells are polyhedra. Due to the convexity of their cells, those models may be too restrictive to describe data that includes possibly anisotropic grains with curved boundaries. Several generalizations exist. The cells of the generalized balanced power diagram are induced by elliptic distances leading to more diverse structures. So far, methods for computing the generalized balanced power diagram are restricted to discretized versions in the form of label images. In this work, we derive an analytic representation of the vertices and edges of the generalized balanced power diagram in 2d. Based on that, we propose a novel algorithm to compute the whole diagram.