arXiv subjects

Alberto Paoluzzi

Publications and source records attributed to Alberto Paoluzzi.

6 recordsLinked to original sources

arXiv2020

Local congruence of chain complexes

The object of this paper is to transform a set of local chain complexes to a single global complex using an equivalence relation of congruence of cells, solving topologically the numerical inaccuracies of floating-point arithmetics. While computing the space arrangement generated by a collection of cellular complexes, one may start from independently and efficiently computing the intersection of each single input 2-cell with the others. The topology of these intersections is codified within a set of (0-2)-dimensional chain complexes. The target of this paper is to merge the local chains by using the equivalence relations of ε-congruence between 0-, 1-, and 2-cells (elementary chains). In particular, we reduce the block-diagonal coboundary matrices [Δ_0] and [Δ_1], used as matrix accumulators of the local coboundary chains, to the global matrices [δ_0] and [δ_1], representative of congruence topology, i.e., of congruence quotients between all 0-,1-,2-cells, via elementary algebraic operations on their columns. This algorithm is codified using the Julia porting of the SuiteSparse:GraphBLAS implementation of the GraphBLAS standard, conceived to efficiently compute algorithms on large graphs using linear algebra and sparse matrices [1, 2].

cs.CG↗

arXiv2019

Topological computing of arrangements with (co)chains

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.

cs.CG↗

arXiv2019

Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays

The goal of this paper is to introduce a new method in computer-aided geometry of solid modeling. We put forth a novel algebraic technique to evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with regularized operators of union, intersection, and difference, i.e., any CSG tree. The result is obtained in three steps: first, by computing an independent set of generators for the d-space partition induced by the input; then, by reducing the solid expression to an equivalent logical formula between Boolean terms made by zeros and ones; and, finally, by evaluating this expression using bitwise operators. This method is implemented in Julia using sparse arrays. The computational evaluation of every possible solid expression, usually denoted as CSG (Constructive Solid Geometry), is reduced to an equivalent logical expression of a finite set algebra over the cells of a space partition, and solved by native bitwise operators.

cs.CG↗

arXiv2017

Geometric Computing with Chain Complexes: Design and Features of a Julia Package

Geometric computing with chain complexes allows for the computation of the whole chain of linear spaces and (co)boundary operators generated by a space decomposition into a cell complex. The space decomposition is stored and handled with LAR (Linear Algebraic Representation), i.e. with sparse integer arrays, and allows for using cells of a very general type, even non convex and with internal holes. In this paper we discuss the features and the merits of this approach, and describe the goals and the implementation of a software package aiming at providing for simple and efficient computational support of geometric computing with any kind of meshes, using linear algebra tools with sparse matrices. The library is being written in Julia, the novel efficient and parallel language for scientific computing. This software, that is being ported on hybrid architectures (CPU+GPU) of last generation, is yet under development.

cs.CG↗

arXiv2017

Regularized arrangements of cellular complexes

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.

cs.CG↗

arXiv2008

Chain-Based Representations for Solid and Physical Modeling

In this paper we show that the (co)chain complex associated with a decomposition of the computational domain, commonly called a mesh in computational science and engineering, can be represented by a block-bidiagonal matrix that we call the Hasse matrix. Moreover, we show that topology-preserving mesh refinements, produced by the action of (the simplest) Euler operators, can be reduced to multilinear transformations of the Hasse matrix representing the complex. Our main result is a new representation of the (co)chain complex underlying field computations, a representation that provides new insights into the transformations induced by local mesh refinements. Our approach is based on first principles and is general in that it applies to most representational domains that can be characterized as cell complexes, without any restrictions on their type, dimension, codimension, orientability, manifoldness, connectedness.

cs.CG↗