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The generalised balanced power diagram: flat sections, affine transformations and an improved rendering algorithm

Felix Ballani

TL;DR

The paper formalizes the generalized balanced power diagram (GBPD) as a Voronoi-like tessellation built from seeds with anisotropy $M$ and Laguerre weight $w$, and studies how GBPDs behave under affine transformations and flat sections. It extends an efficient rendering approach originally developed for standard power diagrams to GBPDs, enabling fast determination of pixel-to-cell memberships even when cell boundaries are curved or disconnected. A detailed complexity analysis is provided in the Poisson-process setting, showing that the improved rendering algorithm achieves near-Logarithmic scaling in the number of seeds, with explicit bounds and optimal parameter choices. The results offer practical guidance for stochastic modelling and rendering of GBPD-based microstructures, including how linear transforms affect generator intensity and how sectional geometry can be handled computationally.

Abstract

The generalised balanced power diagram (GBPD) is regarded in the literature as a suitable geometric model for describing polycrystalline microstructures with curved grain boundaries. This article compiles properties of GBPDs with regard to affine transformations and flat sections. Furthermore, it extends an algorithm known for power diagrams for generating digital images, which is more efficient than the usual brute force approach, on GBPDs.

The generalised balanced power diagram: flat sections, affine transformations and an improved rendering algorithm

TL;DR

The paper formalizes the generalized balanced power diagram (GBPD) as a Voronoi-like tessellation built from seeds with anisotropy and Laguerre weight , and studies how GBPDs behave under affine transformations and flat sections. It extends an efficient rendering approach originally developed for standard power diagrams to GBPDs, enabling fast determination of pixel-to-cell memberships even when cell boundaries are curved or disconnected. A detailed complexity analysis is provided in the Poisson-process setting, showing that the improved rendering algorithm achieves near-Logarithmic scaling in the number of seeds, with explicit bounds and optimal parameter choices. The results offer practical guidance for stochastic modelling and rendering of GBPD-based microstructures, including how linear transforms affect generator intensity and how sectional geometry can be handled computationally.

Abstract

The generalised balanced power diagram (GBPD) is regarded in the literature as a suitable geometric model for describing polycrystalline microstructures with curved grain boundaries. This article compiles properties of GBPDs with regard to affine transformations and flat sections. Furthermore, it extends an algorithm known for power diagrams for generating digital images, which is more efficient than the usual brute force approach, on GBPDs.
Paper Structure (18 sections, 43 equations, 2 figures)

This paper contains 18 sections, 43 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definition
  3. Transformations
  4. Translation
  5. Rotation
  6. Scaling
  7. Linear distortion
  8. Flat sections of GBPDs
  9. Section with a hyperplane
  10. Section with a $q$-dimensional affine space, $1\leq q\leq d-2$
  11. Rendering of digital images
  12. The ‘brute force’ algorithm
  13. The algorithm of Moulinec2022 adapted to GBPDs
  14. Details for the section with a hyperplane
  15. Complexity estimation for a marked Poisson process
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Laguerre tessellation (left) and GBPD (right) with respect to the same set of seed points (black dots) and each the same weights. Even in Laguerre tessellations there might be seeds with an empty cell or lying outside their cell. In GBPDs cells can be non-convex and even disconnected.
  • Figure 2: Assigned memberships to seed points after the first step of the improved algorithm for three different distances $t$ (growing from left to right). For each pixel farther than $t$ away from any seed point, i. e. outside the coloured region, the 'brute force' algorithm has to be applied.