Regularized arrangements of cellular complexes

Abstract: In this paper we propose a novel algorithm to combine two or more cellular complexes, providing a minimal fragmentation of the cells of the resulting complex. We introduce here the idea of arrangement generated by a collection of cellular complexes, producing a cellular decomposition of the embedding space. The algorithm that executes this computation is called \emph{Merge} of complexes. The arrangements of line segments in 2D and polygons in 3D are special cases, as well as the combination of closed triangulated surfaces or meshed models. This algorithm has several important applications, including Boolean and other set operations over large geometric models, the extraction of solid models of biomedical structures at the cellular scale, the detailed geometric modeling of buildings, the combination of 3D meshes, and the repair of graphical models. The algorithm is efficiently implemented using the Linear Algebraic Representation (LAR) of argument complexes, i.e., on sparse representation of binary characteristic matrices of $d$-cell bases, well-suited for implementation in last generation accelerators and GPGPU applications.