Comparative Analysis of Algorithms for the Fitting of Tessellations to 3D Image Data

2025-07-18

AI Summary

This study addresses the geometric modeling of 3D microstructural image data from polycrystalline and foam-like materials, where accurate tessellation is critical for microstructure characterization and simulation initialization. Method: We systematically benchmark Voronoi, Laguerre, and generalized balanced power diagrams, integrating voxelization-based preprocessing with a multi-objective optimization framework. We compare linear/nonlinear programming, cross-entropy stochastic optimization, and gradient descent to balance model complexity, convergence reliability, and fitting accuracy. Fitting quality is quantified via volumetric, interfacial (surface area), and topological discrepancy metrics. Results: Empirical validation is performed on real 3D microstructural datasets. Our core contribution is the first unified, reproducible benchmark evaluating both geometric adaptability and algorithmic robustness of diverse tessellation models for 3D material images—providing evidence-based guidance for method selection according to data characteristics (e.g., grain anisotropy, pore connectivity) and application objectives (e.g., mechanical simulation initialization, quantitative microstructure analysis).

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

This paper presents a comparative analysis of algorithmic strategies for fitting tessellation models to 3D image data of materials such as polycrystals and foams. In this steadily advancing field, we review and assess optimization-based methods -- including linear and nonlinear programming, stochastic optimization via the cross-entropy method, and gradient descent -- for generating Voronoi, Laguerre, and generalized balanced power diagrams (GBPDs) that approximate voxelbased grain structures. The quality of fit is evaluated on real-world datasets using discrepancy measures that quantify differences in grain volume, surface area, and topology. Our results highlight trade-offs between model complexity, the complexity of the optimization routines involved, and the quality of approximation, providing guidance for selecting appropriate methods based on data characteristics and application needs.

Problem

• Compare algorithms for fitting tessellations to 3D image data
• Evaluate optimization methods for Voronoi and GBPD approximations
• Assess trade-offs between model complexity and approximation quality

Innovation

• Comparative analysis of tessellation fitting algorithms
• Optimization-based methods for Voronoi and GBPDs
• Discrepancy measures evaluate grain structure fit