2,625 research outputs found
Gap Processing for Adaptive Maximal Poisson-Disk Sampling
In this paper, we study the generation of maximal Poisson-disk sets with
varying radii. First, we present a geometric analysis of gaps in such disk
sets. This analysis is the basis for maximal and adaptive sampling in Euclidean
space and on manifolds. Second, we propose efficient algorithms and data
structures to detect gaps and update gaps when disks are inserted, deleted,
moved, or have their radius changed. We build on the concepts of the regular
triangulation and the power diagram. Third, we will show how our analysis can
make a contribution to the state-of-the-art in surface remeshing.Comment: 16 pages. ACM Transactions on Graphics, 201
The Topology ToolKit
This system paper presents the Topology ToolKit (TTK), a software platform
designed for topological data analysis in scientific visualization. TTK
provides a unified, generic, efficient, and robust implementation of key
algorithms for the topological analysis of scalar data, including: critical
points, integral lines, persistence diagrams, persistence curves, merge trees,
contour trees, Morse-Smale complexes, fiber surfaces, continuous scatterplots,
Jacobi sets, Reeb spaces, and more. TTK is easily accessible to end users due
to a tight integration with ParaView. It is also easily accessible to
developers through a variety of bindings (Python, VTK/C++) for fast prototyping
or through direct, dependence-free, C++, to ease integration into pre-existing
complex systems. While developing TTK, we faced several algorithmic and
software engineering challenges, which we document in this paper. In
particular, we present an algorithm for the construction of a discrete gradient
that complies to the critical points extracted in the piecewise-linear setting.
This algorithm guarantees a combinatorial consistency across the topological
abstractions supported by TTK, and importantly, a unified implementation of
topological data simplification for multi-scale exploration and analysis. We
also present a cached triangulation data structure, that supports time
efficient and generic traversals, which self-adjusts its memory usage on demand
for input simplicial meshes and which implicitly emulates a triangulation for
regular grids with no memory overhead. Finally, we describe an original
software architecture, which guarantees memory efficient and direct accesses to
TTK features, while still allowing for researchers powerful and easy bindings
and extensions. TTK is open source (BSD license) and its code, online
documentation and video tutorials are available on TTK's website
Metric systolicity and two-dimensional Artin groups
We introduce the notion of metrically systolic simplicial complexes. We study
geometric and large-scale properties of such complexes and of groups acting on
them geometrically. We show that all two-dimensional Artin groups act
geometrically on metrically systolic complexes. As direct corollaries we obtain
new results on two-dimensional Artin groups and all their finitely presented
subgroups: we prove that the Conjugacy Problem is solvable, and that the Dehn
function is quadratic. We also show several large-scale features of finitely
presented subgroups of two-dimensional Artin groups, lying background for
further studies concerning their quasi-isometric rigidity.Comment: final preprint version, to appear in Math. An
Cuneiform Detection in Vectorized Raster Images
Documents written in cuneiform script are one of the largest sources about ancient history. The script is written by imprinting wedges (Latin: cunei) into clay tablets and was used for almost four millennia. This three-dimensional script is typically transcribed by hand with ink on paper. These transcriptions are available in large quantities as raster graphics by online sources like the Cuneiform Database Library Initative (CDLI). Within this article we present an approach to extract Scalable Vector Graphics (SVG) in 2D from raster images as we previously did from 3D models. This enlarges our basis of data sets for tasks like word-spotting. In the first step of vectorizing the raster images we extract smooth outlines and a minimal graph representation of sets of wedges, i.e., main components of cuneiform characters. Then we discretize these outlines followed by a Delaunay triangulation to extract skeletons of sets of connected wedges. To separate the sets into single wedges we experimented with different conflict resolution strategies and candidate pruning. A thorough evaluation of our methods and its parameters on real word data shows that the wedges are extracted with a true positive rate of 0.98. At the same time the false positive rate is 0.2, which requires future extension by using statistics about geometric configurations of wedge sets
Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition
We provide efficient constant factor approximation algorithms for the
problems of finding a hierarchical clustering of a point set in any metric
space, minimizing the sum of minimimum spanning tree lengths within each
cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of
cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can
also be used to provide a pants decomposition, that is, a set of disjoint
simple closed curves partitioning the plane minus the input points into subsets
with exactly three boundary components, with approximately minimum total
length. In the Euclidean case, these curves are squares; in the hyperbolic
case, they combine our Euclidean square pants decomposition with our tree
clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now
Lemma 5.2, as the previous proof was erroneou
Courbure discrète : théorie et applications
International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor
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