Polynomial diagrams for microstructure modelling

TL;DR

This paper introduces polynomial diagrams as a flexible extension of power diagrams, enabling algebraic boundary modeling, and presents an efficient GPU-based fitting algorithm applied to steel microstructure images.

Contribution

It develops a novel polynomial diagram framework, connects it to existing diagrams, and creates a GPU-accelerated fitting method for image data.

Findings

Polynomial diagrams generalize power diagrams with algebraic boundaries.

The GPU-accelerated algorithm effectively fits diagrams to electron backscatter images.

The framework includes a comprehensive analysis of the optimization process.

Abstract

We formulate a framework of polynomial diagrams, which are a generalisation of power diagrams (PDs) and anisotropic power diagrams (APDs) allowing for boundaries between cells to be algebraic curves of a prescribed degree. We show that they arise naturally from rephrasing PDs (APDs) as first-degree (second-degree) instances of linear parametrised minimisation diagrams. We also develop an efficient GPU-accelerated framework for fitting polynomial diagrams to image data using Legendre polynomials and by maximising a regularised concave objective function adapted from classical logistic regression literature. A largely self-contained analysis of the optimisation algorithm is also provided, including identification of scale and gauge invariances and the limiting objective function as the regularisation parameter vanishes. We apply the algorithm to fit polynomial diagrams to electron backscatter diffraction images of steel.