Topological Methods in Data Analysis and Visualization: Theory, Algorithms, and Applications

270 Pages·2011·5.78 MB·English

by  Kirk E. Jordan

Mathematics and Visualization
Series Editors
Gerald Farin
Hans-Christian Hege
David Hoffman
Christopher R. Johnson
Konrad Polthier
Martin Rumpf
For further volumes:
http://www.springer.com/series/4562
.
Valerio Pascucci
Xavier Tricoche
Hans Hagen
Julien Tierny
Editors
Topological Methods
in Data Analysis
and Visualization
Theory, Algorithms,
and Applications
With 146 Figures, 122 in Color
123
Editors
Valerio Pascucci Hans Hagen
University of Utah – School of Computing Universität Kaiserslautern
Scientific Computing and Imaging Fachbereich Informatik
Institute (SCI) 67653 Kaiserslautern
S. Central Campus Dr. 50 Germany
Salt Lake City, UT 84132 [email protected]
USA
[email protected]
Xavier Tricoche Julien Tierny
Purdue University CNRS
Department of Computer Science LTCI - Telecom ParisTech
N. University Street 305 46 rue Barrault
West Lafayette, IN 47907-2107 Paris Cedex 13
USA France
[email protected] [email protected]
ISBN 978-3-642-15013-5 e-ISBN 978-3-642-15014-2
DOI 10.1007/978-3-642-15014-2
Springer Heidelberg Dordrecht London New York
Mathematics Subject Classification (2010): 57Q05, 68U05, 68U20, 76M27
© Springer-Verlag Berlin Heidelberg 2011
Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis
concerned,specifically therightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,
reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication
orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,
1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare
liabletoprosecutionundertheGermanCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply,
evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws
andregulationsandthereforefreeforgeneraluse.
Coverdesign:deblik, Berlin
Printedonacid-freepaper
Springer is part of Springer Science+Business Media (www.springer.com)
Preface
The deluge of data observed throughout research and industry has turned the analy-
sis of the resulting informationinto the primarylimiting factor for the rapid progress
of science, engineering,and medicine. The field of visualization strives to tackle this
data analysis challenge by devising visual representations that afford users an effec-
tive interfacewith theirdatasets. Drivenbythe explosionin data size and complexity
experienced over the last decade, a prominent trend in today’s visualization research
applies a data abstraction approach to yield high-level depictions emphasizing vari-
ous salient properties of the phenomenon considered. Topology-based methods have
proved especially compelling in this regard,as the topologicalabstraction providesa
commonmathematicallanguageto identifyremarkablestructuresin a broadrangeof
applicationsandsemanticcontexts.Topologicalconceptsandmetaphorsaretherefore
permeatingthe visualizationliteratureand they are the focusof a significantresearch
effortspanningtheoretical,algorithmic,andpracticalaspects.
ThisbookdescribestheresearchthatwaspresentedduringthethirdWorkshopon
TopologicalMethodsinDataAnalysisandVisualization,whichtookplaceinSnowbird,
Utah, on February 23-24, 2009. Following two successful, seminal TopoInVis work-
shopsheldinEurope,the2009editionwasorganizedintheUnitedStatesinresponse
tothegrowinginternationalinterestintopologicalmethods.Asinpreviousyears,this
eventofferedinternationalexpertstheopportunitytopresenttheirongoingresearchin
an informal and inspiring atmosphere, as well as to discuss the emerging trends and
open challengesof the field. A defining feature of the 2009 edition was the attention
paidtoapplications,reflectingtheimportanceoftopologicaltechniquesinpracticaldata
analysisscenariosandtheincreasedprominenceofproblem-drivenresearch.Thework-
shop featured two eminent invited speakers (Herbert Edelsbrunner, Duke University,
andJackieChen,SandiaNationalLaboratory),whogaveexcitinglectureshighlighting
significantaccomplishmentsandpromisingavenuesforbothfundamentalandapplied
research on topological methods. All in all, TopoInVis ’09 was a resounding success
thankstotheexcellentcontributionsofover60participants.
Eachofthe20researchpaperscontainedinthisbookwasacceptedforpresentation
attheworkshopaftercarefulpeer-reviewbytheinternationalprogramcommittee.The
contentsareorganizedin5mainthemesthatcorrespondtomajorresearchdirections.
The first part is concerned with the theoretical foundations of the topological
approach. Jordan et al. apply topological concepts to the precise visualization of
1-manifolds, while Ka¨lberer et al. employ the topologyof a tubular surface to define
agloballyconsistentstripetexture.Edelsbrunnerandhisco-authorsprovethestability
ofapparentcontours(i.e.silhouettes)undertheerosiondistanceandDillardetal.study
thetopologicallyconsistentreconstructionofcellcomplexesfromcross-sectionimages.
Thesecondpartofthebookfocusesonhierarchicaltopologicaldatastructures,as
requiredbytheanalysisofverylargedatasets.Vivodtzevetal.presentanewtechnique
for the topology-preservingsimplification of very large tetrahedral meshes. Guylassy
et al. propose several improvements to the computation of Morse-Smale complexes
inthecontextoffeaturedetectionalgorithms,whileComic´ andDeFlorianiintroduce
VI Preface
two simplification operatorsfor Morse complexesapplicable in arbitrary dimensions.
SuthambharaandNatarajan considerthe simplificationof Jacobisets froma levelset
andoffsetmanifoldperspective,andReininghausandHotzdescribeanovelcombina-
torialframeworkforthetopologicalanalysisof2Dvectorfields.
The third part deals with the algorithmic extraction of topological structures of
interestinvectorandtensorfields.Kastenetal.putforwardaGalileaninvariantnotion
ofcriticalpointsthatcapturessignificantpatternsinturbulentflows,whileObermaier
and his co-authors propose an algorithm for the segmentation of three-dimensional
grid-less flow simulations. Sreevalsan-Nair et al. study the impact of the interpola-
tionschemeonthetopologicalstructureof2Deigenvectorfields. Finally,Sadloetal.
presenta newmethodforthe efficientcomputationofLagrangiancoherentstructures
inunsteadyflows.
The following part is dedicated to practical applications of topological methods
in data analysis and visualization. Grottel et al. consider defect detection in crystal
structures;Kellerandhisco-authorsapplyauser-assistedmulti-scaletechniquetothe
detectionofsalientfeaturesinLiDaRdatasets.Wiebeletal.reportontheshortcomings
ofexistingtopologicalapproachesintheanalysisofrotation-mediatedcellaggregation
andproposeanovelsolutiontothisproblem.Szymczakappliescategorytheorytothe
robustsegmentationofairwaysfromCTscans,whileBajajandhisco-authorsvisualize
thecomplementaryspaceofcomplexgeometricmodelstoresolvesubtlestructureswith
applicationstouncertaintyanddynamicsvisualization.
Thebookconcludeswithtwopapersthatspecificallyconsiderthechallengesposed
bythetopologicalanalysisofverylargedatasetsproducedbystateoftheartcombus-
tion simulations. Mascarenhas et al. present a combinatorial streaming algorithm for
theefficientcharacterizationoftopologicalfeaturesinlarge-scaledatasets,whichthey
applytothecomparisonofterascalecombustionsimulations.Weberandhisco-authors
proposeanoveltechnique,leveragingReeb graphstostudythetemporalevolutionof
burningregionsinsimulatedflames.
Acknowledgments
TopoInVis’09wasorganizedbytheScientificComputingandImagingInstituteatthe
UniversityofUtahinSaltLakeCity.Wegratefullyacknowledgethegeneroussupport
ofoursponsors:theTechnicalUniversityofKaiserslautern,theSciDACVisualization
and Analytics Center of Enabling Technologies, and NVIDIA Corporation. We also
wishtoexpressourthanksfortheinvaluablelogisticalassistanceprovidedbyDeborah
Zemek, who was instrumentalin the successful organizationof the workshop.Please
joinusinZurich,Switzerland,forthe2011editionofTopoInVis!
ValerioPascucci
XavierTricoche
HansHagen
JulienTierny
Table of Contents
Foundations ofComputational Topology ....................... 1
GeometricTopology&Visualizing1-Manifolds.......................... 1
Kirk E.Jordan,LanceE.Miller, ThomasJ.Peters, andAlexanderC.
Russell
StripeParameterizationofTubularSurfaces ............................. 13
FelixKa¨lberer,MatthiasNieser,andKonradPolthier
TheStabilityoftheApparentContourofanOrientable2-Manifold........... 27
HerbertEdelsbrunner,DmitriyMorozov,andAmitPatel
ReconstructingCellComplexesFromCross-sections ...................... 43
ScottE.Dillard,DanThoma,andBerndHamann
Hierarchical TopologicalData-Structures ...................... 55
SubstructureTopologyPreservingSimplificationofTetrahedralMeshes ....... 55
FabienVivodtzev,Georges-PierreBonneau,StefanieHahmann,and
HansHagen
PracticalConsiderationsinMorse-SmaleComplexComputation............. 67
AttilaGyulassy,Peer-TimoBremer,BerndHamann,andValerioPascucci
ModelingandSimplifyingMorseComplexesinArbitraryDimensions ........ 79
LidijaCˇomic´ andLeilaDeFloriani
SimplificationofJacobiSets.......................................... 91
SuthambharaNandVijayNatarajan
Combinatorial2DVectorFieldTopologyExtractionandSimplification ....... 103
JanReininghausandIngridHotz
TopologicalFeature ExtractionAlgorithms .................... 115
OntheExtractionofLong-livingFeaturesinUnsteadyFluidFlows........... 115
JensKasten,IngridHotz,BerndR.Noack,andHans-ChristianHege
StreamVolumeSegmentationofGrid-LessFlowSimulation ................ 127
Harald Obermaier, Jo¨rg Kuhnert, Martin Hering-Bertram, and
HansHagen
VIII TableofContents
Eigenvector-basedInterpolationandSegmentationof2DTensorFields........ 139
JayaSreevalsan-Nair,CorneliaAuer,BerndHamann,andIngridHotz
Time-DependentVisualizationofLagrangianCoherentStructuresbyGrid
Advection ........................................................ 151
FilipSadlo,AlessandroRigazzi,andRonaldPeikert
ApplicationsinScientific DataAnalysisand Visualization...... 167
TopologicalExtractionandTrackingofDefectsinCrystalStructures ......... 167
SebastianGrottel,CarlosA.Dietrich,Joa˜oL.D.Comba,andThomasErtl
ExtractingandVisualizingStructuralFeaturesinEnvironmentalPointCloud
LiDaRDataSets .................................................. 179
PatricKeller,OliverKreylos,MarekVanco,MartinHering-Bertram,Eric
S.Cowgill,LouiseH.Kellogg,BerndHamann,andHansHagen
TopologicalFlowStructuresinaMathematicalModelforRotation-Mediated
CellAggregation................................................... 193
AlexanderWiebel, RaymondChan,Christina Wolf, AndreaRobitzki,
AngelaStevens,andGerikScheuermann
ACategoricalApproachtoContour,SplitandJoinTreeswithApplication
toAirwaySegmentation ............................................. 205
AndrzejSzymczak
ComplementarySpaceforEnhancedUncertaintyandDynamicsVisualization .. 217
ChandrajitBajaj,AndrewGillette,SamratGoswami,BongJuneKwon,
andJoseRivera
TopologicalAnalysisofLarge-Scale Scientific Data ............ 229
TopologicalFeatureExtractionforComparisonofTerascaleCombustion
SimulationData ................................................... 229
AjithMascarenhas,RayW.Grout,Peer-TimoBremer,EvattR.Hawkes,
ValerioPascucci,andJacquelineH.Chen
FeatureTrackingUsingReebGraphs................................... 241
Gunther Weber, Peer-Timo Bremer, Marcus Day, John Bell,
andValerioPascucci
Author Index................................................... 255
Subject Index .................................................. 257
.