Topological computing of arrangements with (co)chains

Abstract

In many areas of applied geometric/numeric computational mathematics, including geo-mapping, computer vision, computer graphics, finite element analysis, medical imaging, geometric design, and solid modeling, one has to compute incidences, adjacencies and ordering of cells, generally using disparate and often incompatible data structures and algorithms. This paper introduces computational topology algorithms to discover the 2D/3D space partition induced by a collection of geometric objects of dimension 1D/2D, respectively. Methods and language are those of basic geometric and algebraic topology. Only sparse vectors and matrices are used to compute both spaces and maps, i.e., the chain complex, from dimension zero to three.

Local congruence of chain complexes

“…In this paper we have discussed an efficient algebraic algorithm to create the [δ 0 ] and [δ 1 ] sparse coboundary matrices, encoding the topological congruences between a set of chain complexes. In the algorithmic pipeline to construct the arrangement of Euclidean 3-space [6], local chain complexes are generated independently from single fragmented input 2-cells, starting from a collection of cellular 2-or 3-complexes in 3D. The correctness of results, that might depend on numeric approximations of floating-point arithmetics, is checked by testing the matrix constraint [δ 1 ][δ 0 ] = [0].…”

“…The chain complex congruence (CCC) enabling algorithm introduced here was inplemented in Julia using the package SuiteSparseGraphBLAS.jl, and it is a key component of a computational pipeline to produce solid models of complex geometric scenes, using robust Boolean algebra methods for next-generation image understanding. 18,18,19,19,20,20,21,21,22,22,23,23,24,24], Int8[-1, -1, 1, -1, -1, 1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1]); 1 julia> Delta_1 = SparseArrays.sparse([1, 1, 1, 1, 2, 2, 2, 2, 3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6], [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,...…”

“…An important problem in computational geometry is to asses the arrangement [4,6] of Euclidean 3space, i.e., the partition of E 3 produced by a collection of cellular 3-complexes. The simplest computational topology solution is provided by the computation of the chain complex…”

“…Within the space decomposition pipeline to compute the arrangement [4] generated by a collection of cellular complexes [5,6], the intermediate step is the assessment of the ∂ 2 operator in 3D, after independent fragmentation of 2-faces in 2D, and before the topological gift wrapping (TGW) algorithm in 3D.…”

“…In particular, the (co)boundary chain is produced by merging a set of local coboundary matrices computed independently from each other on every fragmented input 2-cell. In paper [6] this computational pipeline is discussed in full detail. In the present manuscript we provide the reduction of a chain complex to its "quotient topology" through congruence quotient computation.…”