15 research outputs found
LIPIcs
Given a finite set A ⊂ ℝ^d, let Cov_{r,k} denote the set of all points within distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow larger when r increases or k decreases, called the multicover bifiltration. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors as well. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness
Computing the Multicover Bifiltration
Given a finite set , let Cov denote the set of
all points within distance to at least points of . Allowing and
to vary, we obtain a 2-parameter family of spaces that grow larger when
increases or decreases, called the \emph{multicover bifiltration}.
Motivated by the problem of computing the homology of this bifiltration, we
introduce two closely related combinatorial bifiltrations, one polyhedral and
the other simplicial, which are both topologically equivalent to the multicover
bifiltration and far smaller than a \v Cech-based model considered in prior
work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid
tiling of Edelsbrunner and Osang, and can be efficiently computed using a
variant of an algorithm given by these authors. Using an implementation for
dimension 2 and 3, we provide experimental results. Our simplicial construction
is useful for understanding the polyhedral construction and proving its
correctness.Comment: 25 pages, 8 figures, 4 tables. Extended version of a paper accepted
to the 2021 Symposium on Computational Geometr
Sparse Higher Order ?ech Filtrations
For a finite set of balls of radius r, the k-fold cover is the space covered by at least k balls. Fixing the ball centers and varying the radius, we obtain a nested sequence of spaces that is called the k-fold filtration of the centers. For k = 1, the construction is the union-of-balls filtration that is popular in topological data analysis. For larger k, it yields a cleaner shape reconstruction in the presence of outliers. We contribute a sparsification algorithm to approximate the topology of the k-fold filtration. Our method is a combination and adaptation of several techniques from the well-studied case k = 1, resulting in a sparsification of linear size that can be computed in expected near-linear time with respect to the number of input points
Geometric Inference on Kernel Density Estimates
We show that geometric inference of a point cloud can be calculated by
examining its kernel density estimate with a Gaussian kernel. This allows one
to consider kernel density estimates, which are robust to spatial noise,
subsampling, and approximate computation in comparison to raw point sets. This
is achieved by examining the sublevel sets of the kernel distance, which
isomorphically map to superlevel sets of the kernel density estimate. We prove
new properties about the kernel distance, demonstrating stability results and
allowing it to inherit reconstruction results from recent advances in
distance-based topological reconstruction. Moreover, we provide an algorithm to
estimate its topology using weighted Vietoris-Rips complexes.Comment: To appear in SoCG 2015. 36 pages, 5 figure
Delaunay Bifiltrations of Functions on Point Clouds
The Delaunay filtration of a point cloud is a central tool of computational topology. Its use is justified
by the topological equivalence of and the offset
(i.e., union-of-balls) filtration of . Given a function , we introduce a Delaunay bifiltration
that satisfies an analogous topological
equivalence, ensuring that topologically
encodes the offset filtrations of all sublevel sets of , as well as the
topological relations between them. is of size
, which for odd matches the worst-case
size of . Adapting the Bowyer-Watson algorithm for
computing Delaunay triangulations, we give a simple, practical algorithm to
compute in time . Our implementation, based on CGAL, computes
with modest overhead compared to computing
, and handles tens of thousands of points in
within seconds.Comment: 28 pages, 7 figures, 8 tables. To appear in the proceedings of SODA2
Detection of Small Holes by the Scale-Invariant Robust Density-Aware Distance (RDAD) Filtration
A novel topological-data-analytical (TDA) method is proposed to distinguish,
from noise, small holes surrounded by high-density regions of a probability
density function whose mass is concentrated near a manifold (or more generally,
a CW complex) embedded in a high-dimensional Euclidean space. The proposed
method is robust against additive noise and outliers. In particular, sample
points are allowed to be perturbed away from the manifold. Traditional TDA
tools, like those based on the distance filtration, often struggle to
distinguish small features from noise, because of their short persistence. An
alternative filtration, called Robust Density-Aware Distance (RDAD) filtration,
is proposed to prolong the persistence of small holes surrounded by
high-density regions. This is achieved by weighting the distance function by
the density in the sense of Bell et al. Distance-to-measure is incorporated to
enhance stability and mitigate noise due to the density estimation. The utility
of the proposed filtration in identifying small holes, as well as its
robustness against noise, are illustrated through an analytical example and
extensive numerical experiments. Basic mathematical properties of the proposed
filtration are proven.Comment: 47 pages, 60 figures, GitHub repo: https://github.com/c-siu/RDA
IST Austria Thesis
In this thesis we study persistence of multi-covers of Euclidean balls and the geometric structures underlying their computation, in particular Delaunay mosaics and Voronoi tessellations.
The k-fold cover for some discrete input point set consists of the space where at least k balls of radius r around the input points overlap. Persistence is a notion that captures, in some sense, the topology of the shape underlying the input. While persistence is usually computed for the union of balls, the k-fold cover is of interest as it captures local density,
and thus might approximate the shape of the input better if the input data is noisy. To compute persistence of these k-fold covers, we need a discretization that is provided by higher-order Delaunay mosaics.
We present and implement a simple and efficient algorithm for the computation of higher-order Delaunay mosaics, and use it to give experimental results for their combinatorial properties. The algorithm makes use of a new geometric structure, the rhomboid tiling. It contains the higher-order Delaunay mosaics as slices, and by introducing a filtration
function on the tiling, we also obtain higher-order α-shapes as slices. These allow us to compute persistence of the multi-covers for varying radius r; the computation for varying k is less straight-foward and involves the rhomboid tiling directly. We apply our algorithms to experimental sphere packings to shed light on their structural properties. Finally, inspired by periodic structures in packings and materials, we propose and implement an algorithm for periodic Delaunay triangulations to be integrated into the Computational Geometry Algorithms Library (CGAL), and discuss
the implications on persistence for periodic data sets