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Multicore Homology. with Ryan Lewis.
Full Manuscript, 2012.
We design and implement a framework for parallel computation of homology of cellular spaces over
field coefficients. We begin by cutting a space into local pieces. We then compute the homology of
each piece in parallel using the persistence algorithm. Finally we
glue the pieces together by constructing the MayerVietoris blowup
complex and computing its homology. Theoretically, we show that the
first step, optimal decomposition into local pieces is NPHARD. In practice we use a decomposition
based on graph partitions, which produces decompositions with a simple structure and bounded overlap. We implement our algorithms for multicore computers, and demonstrate their efficacy with a suite
of experiments. For example, we achieve roughly an 8x speedup of the homology computations on a
3dimensional complex with about 10 million simplices using 11 cores.




Constructing Simplicial Complexes over Topological Spaces. with Milka Doktorova.
Full Manuscript, 2012.
Topological data analysis usually begins by constructing a combinatorial structure, such as a simplicial
complex, to approximate the lost topology of a sampled point set.
In this paper, we present an oraclebased framework and algorithms that construct
highdimensional simplicial complexes over arbitrary topological spaces. Using the minimumsized
representation for the simplicial complexes, we design a novel topdown algorithm and analyze it both
theoretically and experimentally. We compare its performance to other algorithmic approaches, building
up to 27dimensional complexes on a standard desktop machine. Finally, we apply our framework to
problems from three domains: Google search, word composition, and protein structure. 



Advances in Applied and Computational Topology (Editor)
American Mathematical Society, 2012.
Amazon
Barnes and Noble
What is the shape of data? How do we describe flows? Can we count by integrating?
How do we plan with uncertainty? What is the most compact representation? These questions, while unrelated,
become similar when recast into a computational setting. Our input is a set of finite, discrete, noisy samples
that describes an abstract space. Our goal is to compute qualitative features of the unknown space.
It turns out that topology is sufficiently tolerant to provide us with robust tools.
This is the proceedings of the
Short Course on Computational Topology at the
2011 Joint Mathematics Meeting, organized by myself.




Digitizing 18thCentury French Literature: Comparing transcription
methods for a critical edition text. with Ann Irvine and
Laure Marcellesi.
Workshop on Computational Linguistics for Literature, 2012.
Joséphine
de Monbart: Lettres taïtiennes,
Laure Marcellesi (Editor), 2012.
We compare four methods for transcribing early printed texts. Our comparison is through
a casestudy of digitizing an eighteenthcentury French novel for a new critical edition:
the 1784 Lettres taïtiennes by Joséphine de Monbart. We provide a detailed
error analysis of transcription by optical character recognition (OCR),
nonexpert humans, and expert humans and weigh each technique based on
accuracy, speed, cost and the need for scholarly overhead.




Topological Data Analysis
Advances in Applied and Computational Topology, 2012.
Scientific data is often in the form of a finite set of noisy
points, sampled from an unknown space, and embedded in a
highdimensional space.
Topological data analysis focuses on recovering the topology
of the sampled space.
In this chapter, we look at methods for constructing combinatorial
representations of point sets, as well as theories and algorithms
for effective computation of robust topological invariants.
Throughout, we maintain a computational view by applying our
techniques to a dataset representing the conformation space of a
small molecule.




The Tidy Set: A Minimal Simplicial Set for Computing Homology of
Clique Complexes
Full Manuscript, 2010. (Invited, then rejected from CGTA)
26th ACM Symposium on
Computational Geometry, Snowbird, UT, 2010.
We introduce the tidy set, a minimal simplicial set
that captures the topology of a simplicial complex.
The tidy set is particularly effective for computing the
homology of clique complexes.
This family of complexes include the VietorisRips
complex and the weak witness complex, methods that are popular in
topological data analysis.
The key feature of our approach is that it skips constructing
the clique complex.
We give algorithms for constructing tidy sets, implement them, and present
experiments.
Our preliminary results show that tidy sets are orders of magnitude
smaller than clique complexes, giving us a homology engine with small
memory requirements.




Fast Construction of the VietorisRips Complex
Computers & Graphics, 2010.
(Invited)
– doi:10.1016/j.cag.2010.03.007
Shape Modeling International, AixenProvence, France, 2010.
The VietorisRips complex characterizes the topology of a point
set.
This complex is popular in topological data analysis as
its construction extends easily to higher dimensions.
We formulate a twophase approach for its construction that separates
geometry from topology.
We survey methods for the first phase, give three algorithms for the
second phase, implement all algorithms, and present experimental
results.
Our software can also be used for constructing any clique complex,
such as the weak witness complex.




Computing Multidimensional Persistence.
with Gunnar Carlsson and Gurjeet Singh.
Journal of Computational
Geometry, 2010.
20th International Symposium on Algorithms and Computation, Waikiki, HI, 2009.
The theory of multidimensional persistence captures the topology of a
multifiltration – a multiparameter family of increasing spaces.
Multifiltrations arise naturally in the topological analysis of
scientific data.
In this paper, we give a polynomial time algorithm for computing
multidimensional persistence.
We recast this computation as a problem within computational
commutative
algebra and utilize algorithms from this area to solve it.
While the resulting problem is EXPSPACEcomplete and the standard
algorithms take doublyexponential time, we exploit the structure
inherent withing multifiltrations to yield practical algorithms.
We implement all algorithms in the paper and provide statistical
experiments to demonstrate their feasibility.




Computational Topology
Algorithms and Theory of Computation
Handbook, 2010.
According to the Oxford English Dictionary, the word topology is
derived of topos (τοπος)
meaning place, and
logy (λογια),
a variant of the verb λεγειν, meaning
to speak.
As such, topology speaks about places: how local neighborhoods connect to
each other to form a space.
Computational topology, in turn, undertakes the challenge of studying
topology using a computer.
This is Chapter 3 of Volume 2 of the second edition of the
Algorithms and
Theory of Computation Handbook, edited by
Mikhail
J. Atallah and
Marina Blanton.




Colon Polyp Detection using Smoothed Shape Operators: Preliminary
Results.
with Padma Sundaram, Christopher Beaulieu, and Sandy Napel.
Medical Image Analysis, 2008.
– doi:10.1016/j.media.2007.08.001
United States Patent 8,055,047, November 8, 2011.
Computeraided detection (CAD) algorithms identify locations in Computed
Tomographic (CT) images of the colon that are most likely to contain
polyps.
In this paper, we present the Smoothed Shape Operators method
(SSO), which uses a geometry processing approach.
We extract a triangle
mesh representation of the colon surface, and estimate curvature on this
surface using the shape operator. We then smooth the shape operators on
the surface iteratively.
We evaluate our algorithm on patient data and
provide freeresponse receiver operating characteristic performance
analysis over all size ranges of polyps. We also provide confidence
intervals for our performance estimates. We compare our performance with
the Surface Normal Overlap (SNO) method for the same data.




On the Local Behavior of Spaces of Natural Images.
with Gunnar Carlsson, Tigran Ishkhanov, and Vin de Silva.
International Journal of Computer Vision, 2008.
(ps.gz)
– doi:10.1007/s112630070056x
In this study, we concentrate on qualitative topological analysis of the
local behavior of the space of natural images. To this end, we use a space
of 3 by 3 highcontrast patches M studied by Mumford et
al. We develop a theoretical model for the highdensity 2dimensional
submanifold of M showing that it has the topology of the Klein
bottle. Using our topological software package PLEX we experimentally
verify our theoretical conclusions. We use polynomial representation to
give coordinatization to various subspaces of M. We find the
bestfitting embedding of the Klein bottle into the ambient space of
M. Our results are currently being used in developing a
compression algorithm based on a Klein bottle dictionary.




Persistent voids: a new structural metric for membrane fusion.
with Peter M. Kasson, Sanghyun Park, Nina Singhal, Leonidas J. Guibas, and
Vijay S. Pande.
Bioinformatics, 2007.
(ps.gz)
– doi:10.1093/bioinformatics/btm250
Membrane fusion constitutes a key stage in cellular processes such as
synaptic neurotransmission and infection by enveloped viruses.
We introduce a novel structural
measurement of vesicle topology and fusion geometry: persistent voids.
We use persistent voids to
compute dynamic relationships between hemifusion neck widening and
formation of a full fusion pore in our simulation data.
Our findings suggest that rapid fusion between small vesicles proceeds
via a small hemifusion diaphragm rather than a fullexpanded one.




Localized Homology.
with Gunnar Carlsson.
Computational Geometry: Theory & Applications, 2008.
(ps.gz)
– doi:10.1016/j.comgeo.2008.02.003
Shape Modeling International, Lyon, France, 2007.
(ps.gz)
Supplemental package: Localizing within a 3D Solid
In this paper, we provide a complete theoretical foundation and an
effective algorithm for localizing topological attributes.
Unlike previous work that focused on 2manifolds with restricted geometry,
our theory is general and localizes arbitrarydimensional attributes in
arbitrary spaces.
We implement our algorithm to validate our approach in practice.




The Theory of Multidimensional Persistence.
with Gunnar Carlsson.
Discrete and Computational Geometry, 2009 (Invited).
(ps.gz)
– doi:10.1007/s0045400991760
23rd ACM Symposium on Computational Geometry, Gyeongju, South Korea,
2007.
(ps.gz)
Persistent homology captures the topology of a filtration –
a oneparameter
family of increasing spaces – in terms of a complete discrete invariant.
This invariant is a multiset of intervals that denote the lifetimes of the
topological entities within the filtration.
In many applications of topology, we need to study a multifiltration:
a family of spaces parameterized along multiple geometric dimensions.
In this paper, we show that no complete discrete invariant exists for
multidimensional persistence.
We also propose a discrete invariant, the rank invariant, for the
robust estimation of Betti numbers in a multifiltration, and prove its
completeness in one dimension.




Geometric Filtering of Pairwise Atomic Interactions
Applied to the Design of Efficient Statistical
Potentials. with Leonidas J. Guibas and Patrice Koehl.
Computer Aided Geometric
Design, 2006.
(ps.gz) – doi:10.1016/j.cagd.2006.03.002
Distancedependent, pairwise, statistical potentials are
based on the concept that the packing observed in known protein
structures can be used as a reference for comparing different
3D models for a protein.
We show that we can filter the list of all interactions in a protein to
generate a much smaller subset of pairs that retains most of the
structural information
contained in proteins.




The Conformal Alpha Shape Filtration.
with Frederic Cazals, Joachim Giesen, and Mark Pauly.
The Visual Computer, 2006 (Invited).
(ps.gz)
– doi:10.1007/s0037100600271
Symposium on PointBased Graphics, Stony Brook, NY, 2005.
(ps.gz)
We define a new filtration of the Delaunay triangulation of a
finite set of points in R^{d}, similar to the alpha shape
filtration. The new filtration is parameterized by a local scale
parameter instead of the global scale parameter in
alpha shapes. Since our approach shares many properties with the
alpha shape filtration and the local scale parameter conforms to
the local geometry we call it conformal alpha shape filtration.
The local scale parameter is motivated from applications and previous
algorithms in surface reconstruction. We show how conformal alpha
shapes can be used for surface reconstruction of nonuniformly sampled
surfaces, which is not possible with alpha shapes.
Note: Conference version has the title "Conformal Alpha Shapes".




Persistence Barcodes for Shapes.
with Gunnar Carlsson, Anne Collins, and Leonidas J. Guibas.
International Journal of Shape Modeling, 2005.
(ps.gz)
– doi:10.1142/S0218654305000761
Symposium on Geometry Processing, Nice, France, 2004.
(ps.gz)
In this paper, we initiate a study of shape description and
classification via the application of persistent homology to two
tangential constructions on geometric objects.
The homology of our first construction, the
tangent complex, can distinguish between topologically identical
shapes with different "sharp" features, such as corners.
To capture "soft" curvaturedependent features, we define a second
complex, the filtered tangent complex, obtained by parametrizing a
family of increasing subcomplexes of the tangent complex.
Applying persistent homology, we obtain a shape descriptor, called a
barcode, that is a finite union of intervals.
We define a metric over the space of such
intervals, arriving at a continuous invariant that reflects the
geometric properties of shapes.




A Barcode Shape Descriptor for Curve Point Cloud Data.
with Anne Collins, Gunnar Carlsson, and Leonidas J. Guibas.
Computers and Graphics, 2004 (Invited).
(ps.gz)
– doi:10.1016/j.cag.2004.08.015
Symposium on PointBased Graphics, Zürich, Switzerland, 2004
(ps.gz)
Video (Quicktime)
In this paper, we present a complete computational pipeline for extracting
a compact shape descriptor for curve point cloud data.
Our shape descriptor, called a barcode, is based on a blend of
techniques from differential geometry and algebraic topology.
We also provide a metric over the space of barcodes, enabling fast
comparison of PCDs for shape recognition and clustering.
To demonstrate the feasibility of our approach, we have implemented our
pipeline and provide experimental evidence in shape classification and
parametrization.




Computing Persistent Homology.
with Gunnar Carlsson
Discrete and Computational Geometry, 2005
(ps.gz)
(Errata)
– doi:10.1007/s004540041146y
20th ACM Symposium on Computational Geometry, Brooklyn, NY, 2004
(ps.gz)
Application Video: Witness Complexes  The Mumford Dataset (AVI)
We study the homology of a filtered ddimensional
simplicial complex K as a single algebraic entity and
establish a correspondence that
provides a simple description over fields.
Our analysis enables us to derive a natural
algorithm for computing persistent homology over
an arbitrary field in any dimension.
Our study also implies the lack of a simple
classification over nonfields.
Instead, we give an algorithm for computing individual
persistent homology groups over an arbitrary PIDs
in any dimension.




Computing and Comprehending Topology: Persistence and Hierarchical Morse Complexes. Ph.D. Thesis. Urbana, Illinois, October 2001.
OneSided
(Hyperlinked)
TwoSided
(Hyperlinked)
Technical Report UIUCDCSR20012240
The fully hyperlinked versions include bookmarks, thumbnails, and
backreferences, and are appropriate for online viewing.
The twosided versions have different odd and even margins for duplex printing
and binding.




Computing Linking Numbers of a Filtration.
with Herbert Edelsbrunner
Homology, Homotopy, and Applications, 2003 (Invited).
1st Workshop on Algorithms in BioInformatics, Århus, Denmark, 2001
We develop fast algorithms for computing the linking number
of a simplicial complex within a filtration.
We give experimental results in applying our work toward the detection
of nontrivial tangling in biomolecules, modeled as alpha complexes.




Hierarchical MorseSmale Complexes for Piecewise Linear 2Manifolds.
with Herbert Edelsbrunner and John Harer
French translation/interpretation
by Sara Deriviere, 2006
Discrete and Computational Geometry, 2003 (Invited).
–
doi:10.1007/s0045400329265
17th ACM Symposium on Computational Geometry, Medford, MA, 2001
We present algorithms for constructing a hierarchy of increasingly
coarse Morse complexes that decompose a piecewise linear 2manifold.
While Morse complexes are defined only in the smooth category,
we extend the construction to the piecewise linear category
by ensuring structural integrity and simulating differentiability.
We then simplify Morse complexes by cancelling pairs of
critical points in order of increasing persistence.
Note: Conference version refers to "Morse complexes". We changed the
title according to a reviewer's suggestion.




Topological Persistence and Simplification. with Herbert
Edelsbrunner and David Letscher
Discrete and Computational Geometry, 2002 (Invited).
–
doi:10.1007/s0045400228852
41st Symposium on Foundations of Computer Science, Redondo Beach, CA, 2000
We formalize a notion of topological simplification within
the framework of a filtration,
which is the history of a growing complex.
We classify a topological change that happens during growth
as either a feature or noise depending on its lifetime
or persistence within the filtration.
We give fast algorithms for computing persistence
and experimental evidence for their speed and utility.




Fast Software for Box Intersection. with Herbert
Edelsbrunner
International Journal of Computational Geometry and Applications, 2002
(Invited).
–
doi:10.1142/S0218195902000785
16th ACM Symposium on Computational Geometry,
Hong Kong, 2000
We present fast implementations of a hybrid algorithm
for reporting box and cube intersections. Our algorithm initially
takes a divideandconquer approach and switches to simpler
algorithms for low numbers of boxes. We use our implementations as engines
to solve problems about geometric primitives. We look at two such problems
in the category of quality analysis of surface triangulations.




ContextFree Language Induction by Evolution of Deterministic
PushDown Automata Using Genetic Programming.
AAAI Fall Symposium.
Boston, 1995.
The process of learning often consists of Inductive Inference, making
generalizations from samples. The problem here is finding generalizations
(Grammars) for Formal Languages from finite sets of positive and negative
sample sentences. The focus of this paper is on ContextFree Languages
(CFL's) as defined by ContextFree Grammars (CFG's), some of which are
accepted by Deterministic PushDown Automata (DPDA). This paper describes
a metalanguage for constructing DPDA's. This language is then combined
with Genetic Programming to evolve DPDA's which accept languages. The
technique is illustrated with two favorite CFL's.

 