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

312 Pages·2012·7.21 MB·English

by  Jan Reininghaus

Mathematics and Visualization
SeriesEditors
GeraldFarin
Hans-ChristianHege
DavidHoffman
ChristopherR.Johnson
KonradPolthier
MartinRumpf
Forfurthervolumes:
http://www.springer.com/series/4562
•
Ronald Peikert
Helwig Hauser
Hamish Carr
Raphael Fuchs
Editors
Topological Methods in Data
Analysis and Visualization II
Theory, Algorithms, and Applications
123
Editors
RonaldPeikert HamishCarr
ETHZu¨rich UniversityofLeeds
ComputationalScience SchoolofComputing
Zu¨rich Leeds
Switzerland UnitedKingdom
[email protected] [email protected]
HelwigHauser RaphaelFuchs
UniversityofBergen ETHZu¨rich
Dept.ofInformatics ComputationalScience
Bergen Zu¨rich
Norway Switzerland
[email protected] [email protected]
ISBN978-3-642-23174-2 e-ISBN978-3-642-23175-9
DOI10.1007/978-3-642-23175-9
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2011944972
MathematicalSubjectClassification(2010):37C10,57Q05,58K45,68U05,68U20,76M27
(cid:2)c Springer-VerlagBerlinHeidelberg2012
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Preface
Over the past few decades, scientific research became increasingly dependent
on large-scale numerical simulations to assist the analysis and comprehension of
physicalphenomena.Thisinturnhasledtoanincreasingdependenceonscientific
visualization,i.e.,computationalmethodsforconvertingmassesofnumericaldata
tomeaningfulimagesforhumaninterpretation.
In recentyears, the size of these data sets has increased to scales which vastly
exceed the ability of the human visual system to absorb information, and the
phenomenabeingstudiedhavebecomeincreasinglycomplex.Asaresult,scientific
visualization,andscientificsimulationwhichitassists,havegivenrisetosystematic
approachestorecognizingphysicalandmathematicalfeaturesinthedata.
Of these systematic approaches, one of the most effective has been the use of
a topological analysis, in particular computational topology, i.e., the topological
analysis of discretely sampled and combinatorially represented data sets. As
topologicalanalysishas becomemoreimportantin scientific visualization,a need
forspecializedvenuesforreportinganddiscussingrelatedresearchhasemerged.
Thisbookresultsfromonesuchvenue:theFourthWorkshoponTopologyBased
Methods in Data Analysis and Visualization (TopoInVis 2011), which took place
inZu¨rich,Switzerland,onApril4–6,2011.OriginatinginEuropewithsuccessful
workshops in Budmerice, Slovakia (2005), and Grimma, Germany (2007), this
workshop became truly international with TopoInVis 2009 in Snowbird, Utah,
USA(2009).With43participants,TopoInVis2011continuesthisrunofsuccessful
workshops,and future workshopsare planned in both Europeand North America
undertheauspicesofaninternationalsteeringcommitteeofexpertsintopological
visualization,andadedicatedwebsiteathttp://www.TopoInVis.org/.
The program of TopoInVis 2011 included 20 peer-reviewed presentations and
twokeynotetalksgivenbyinvitedspeakers.MartinRasmussen,ImperialCollege,
London,addressedtheongoingeffortsofourcommunitytoformulateavectorfield
topologyforunsteadyflow.HispresentationAnintroductiontothequalitativethe-
oryofnonautonomousdynamicalsystemswashighlyappreciatedasanillustrative
introduction into a difficult mathematical subject. The second keynote, Looking
for intuition behind discrete topologies, given by Thomas Lewiner, PUC-Rio,
v
vi Preface
RiodeJaneiro,pickedupanothertopicwithinthefocusofcurrentresearch,namely
combinatorialmethods,forwhichhistalkgavestrongmotivation.Attheendofthe
workshop,DominicSchneiderandhiscoauthorsweregiventheawardforthebest
paperbyajury.
NineteenofthepaperspresentedatTopoInVis2011wererevisedand,inasecond
roundofreviewing,acceptedforpublicationinthisbook.Basedonthemajortopics
covered,thepapershavebeengroupedintofourparts.
ThefirstpartofthebookisconcernedwithcomputationaldiscreteMorsetheory,
bothin2Dandin3D.In2D,ReininghausandHotzapplieddiscreteMorsetheory
todivergence-freevectorfields. Incontrast,Gu¨ntheretal.presentacombinatorial
algorithm to construct a hierarchy of combinatorial gradient vector fields in 3D,
whileGyulassyandPascucciprovideanalgorithmthatcomputesthedistinctcellsof
theMScomplexconnectingtwocriticalpoints.Finally,aninterestingcontribution
is also made by Reich et al. who developeda combinatorialvector field topology
in3D.
In Part2,hierarchicalmethodsforextractingandvisualizingtopologicalstruc-
tures such as the contour tree and Morse-Smale complex were presented. Weber
et al. propose an enhanced method for contour trees that is able to visualize two
additionalscalarattributes.Harveyetal.introduceanewclustering-basedapproach
to approximatethe Morse–Smalecomplex.Finally,Wagner et al. describe how to
efficientlycomputepersistenthomologyofcubicaldatainarbitrarydimensions.
Thethirdpartofthebookdealswiththevisualizationofdynamicalsystems,vec-
torandtensorfields. Tricocheet al.visualize chaoticstructuresinarea-preserving
maps. The same problem was studied by Sanderson et al. in the context of an
application,namelythestructureofmagneticfieldlinesintokamaks,withafocuson
thedetectionofislandsofstability.Jadhavetal.presentacompleteanalysisofthe
possiblemappingsfrominflowboundariestooutflowboundariesintriangularcells.
Anovelalgorithmforpathlineplacementwithcontrolledintersectionsisdescribed
by Weinkauf et al., while Wiebel et al. propose glyphs for the visualization of
nonlinearvectorfieldsingularities.Asaninterestingresultintensorfieldtopology,
Linetal.presentanextensiontoasymmetric2Dtensorfields.
The final part is dedicated to the topological visualization of unsteady flow.
Kastenetal.analyzefinite-timeLyapunovexponents(FTLE)andproposealterna-
tiverealizationsofLagrangiancoherentstructures(LCS).Schindleretal.investigate
the flux through FTLE ridges and propose an efficient, high-quality alternative
to height ridges. Pobitzer et al. present a technique for detecting and removing
falsepositivesinLCScomputation.Schneideretal.proposeanFTLE-likemethod
capable of handling uncertain velocity data. Sadlo et al. investigate the time
parameterin the FTLE definitionand providea lower bound.Finally,Fuchset al.
explorescale-spaceapproachestoFTLEandFTLEridgecomputation.
Acknowledgements TopoInVis 2011 was organized by the Scientific Visualization Group of
ETH Zurich, the Visualization Group at the University of Bergen, and the Visualization and
VirtualRealityGroupattheUniversityofLeeds.WeacknowledgethesupportfromETHZurich,
particularlyforallowingustousetheprestigiousSemperAulainthemainbuilding.TheEvento
Preface vii
teamprovided valuablesupport bysettinguptheregistration webpageand promptlyresolving
issues withon-line payments. We are grateful to Marianna Berger, Katharina Schuppli, Robert
Carnecky,andBenjaminSchindlerfortheiradministrativeandorganizationalhelp.Wealsowish
to thank the TopoInVis steering committee for their advice and their help with advertising the
event.TheprojectSemSeg–4DSpace-TimeTopologyforSemanticFlowSegmentationsupported
TopoInVis2011inseveralways,mostnotablybyoffering12youngresearcherspartialrefunding
of their travel costs. The project SemSeg acknowledges the financial support of the Future
and Emerging Technologies (FET) programme within the Seventh Framework Programme for
ResearchoftheEuropeanCommission,underFET-Opengrantnumber226042.
WearelookingforwardtothenextTopoInVisworkshop,whichisplannedtotakeplacein2013
inNorthAmerica.
RonaldPeikert
HelwigHauser
HamishCarr
RaphaelFuchs
•
Contents
PartI DiscreteMorseTheory
ComputationalDiscreteMorseTheoryforDivergence-Free
2DVectorFields.................................................................. 3
JanReininghausandIngridHotz
EfficientComputationofaHierarchyofDiscrete3DGradient
VectorFields...................................................................... 15
DavidGu¨nther,JanReininghaus,SteffenProhaska,TinoWeinkauf,
andHans-ChristianHege
ComputingSimply-ConnectedCellsinThree-Dimensional
Morse-SmaleComplexes ........................................................ 31
AttilaGyulassyandValerioPascucci
CombinatorialVectorFieldTopologyinThreeDimensions ................ 47
WielandReich,DominicSchneider,ChristianHeine,
AlexanderWiebel,GuoningChen,GerikScheuermann
PartII Hierarchical Methods for Extracting
andVisualizingTopologicalStructures
TopologicalCacti:VisualizingContour-BasedStatistics .................... 63
GuntherH.Weber,Peer-TimoBremer,andValerioPascucci
EnhancedTopology-SensitiveClusteringbyReebGraphShattering...... 77
W.Harvey,O.Ru¨bel,V.Pascucci,P.-T.Bremer,andY.Wang
EfficientComputationofPersistentHomologyforCubicalData .......... 91
HubertWagner,ChaoChen,andEraldVuc¸ini
ix