Voxel graph operators: Topological voxelization, graph generation, and derivation of discrete differential operators from voxel complexes

Abstract

This paper presents a novel algebraic workflow for topological voxelization of spatial objects, construction of voxel connectivity graphs & hyper-graphs, and derivation of partial differential and multiple integral operators. Discretization of models of spatial domains is central to many analytic applications in such application areas as medical imaging, geometric modelling, computer graphics, engineering optimization, geospatial analysis, and scientific simulations. Whilst in some medical applications raster data models of spatial objects based on voxels arise naturally, e.g. in CT Scan and MRI imaging, in engineering applications the so-called boundary representations or vector data models based on points are far more common. The presented methodology puts forward a complete alternative geometry processing pipeline on par with the conventional vector-based geometry processing pipelines but far more elegant and advantageous for parallelization due to its explicit algebraic nature: effectively, by creating a mapping of geometric models from $R^3$ to $Z^3$ to $N^3$ and eventually to an index space created by Morton Codes in $N$ while ensuring the topological validity of the voxel models; namely their topological thinness and their geometrical consistency. The set of differential and integral operators presented in this paper generalizes beyond graphs and hyper-graphs constructed out of voxel models and provides an unprecedented complete set of algebraic differential operators for the discretization of digital simulations based on PDEs and advanced analyses using Spectral Graph Theory and Spectral Mesh Processing.