A computational framework for tracking grain boundaries in 3D image data: Quantifying boundary curvatures and velocities in polycrystalline materials

Diffraction-based methods for 3D microstructure mapping have revolutionized the investigation of grain growth phenomena in polycrystalline materials. These techniques provide unprecedented experimental access to the location, shape and migration of individual grain boundaries in real samples, together with other relevant boundary parameters, such as misorientation and inclination. However, achieving the ultimate goal of such studies — the identification of physical mechanisms responsible for the observed evolution of microstructure — is impeded by the voxel-based representation of microstructure intrinsic to 3D mapping. The staircase-like discretization of grain boundaries complicates the determination of boundary positions and local curvatures, motivating a transition from a discrete to a continuous representation of the boundary network. This paper introduces an approach based on smoothing thin-plate splines with radial basis functions to represent grain boundaries as continuous surfaces. The method can approximate arbitrarily shaped boundaries while preserving local mean curvatures and, for time-resolved series of images, local boundary velocities. Its accuracy is validated against two datasets: a multiplicatively weighted Laguerre tessellation, where mean curvatures are known analytically, and a Reuleaux tetrahedron evolving under a phase field model, for which local boundary velocities can be determined exactly. The applicability of the method to experimental data is demonstrated using time-resolved 3D maps of grain growth in an Al–Mg alloy.