3D extension via face-centered subdivision on cubical grids

Ascertain whether a face-centered subdivision of cubical grids, similar to Carr et al. (2006), achieves comparable geometric accuracy and topological consistency for Morse–Smale complexes in three dimensions when used with the steepest descent discrete gradient method of Robins et al. (2011).

Background

The proposed grid-conversion technique is presented and evaluated in 2D. For 3D data on cubical grids, the authors hypothesize that a face-centered subdivision akin to Carr et al. (2006) could yield similar benefits in geometric embedding while maintaining topological properties when combined with the steepest descent method.

They explicitly leave the verification of this hypothesis to further investigations, indicating an open question about extending their approach to volumetric grids.