193,807 research outputs found
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.Comment: revised version submitted to Computer-Aided Geometric Desig
Clear and Compress: Computing Persistent Homology in Chunks
We present a parallelizable algorithm for computing the persistent homology
of a filtered chain complex. Our approach differs from the commonly used
reduction algorithm by first computing persistence pairs within local chunks,
then simplifying the unpaired columns, and finally applying standard reduction
on the simplified matrix. The approach generalizes a technique by G\"unther et
al., which uses discrete Morse Theory to compute persistence; we derive the
same worst-case complexity bound in a more general context. The algorithm
employs several practical optimization techniques which are of independent
interest. Our sequential implementation of the algorithm is competitive with
state-of-the-art methods, and we improve the performance through parallelized
computation.Comment: This result was presented at TopoInVis 2013
(http://www.sci.utah.edu/topoinvis13.html
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