Discrete Calculus: Applied Analysis on Graphs for Computational Science

383 Pages·2010·7.55 MB·English

by  Leo J. Grady

Discrete Calculus

Leo J. Grady (cid:2) Jonathan R. Polimeni
Discrete Calculus
Applied Analysis on Graphs
for Computational Science
Dr.LeoJ.Grady Dr.JonathanR.Polimeni
SiemensCorporateResearch AthinoulaA.MartinosCenter
755CollegeRoadEast forBiomedicalImaging
Princeton,NJ08540-6632 DepartmentofRadiology
USA MassachusettsGeneralHospital
[email protected] HarvardMedicalSchool
Charlestown,MA02129
USA
[email protected]
ISBN978-1-84996-289-6 e-ISBN978-1-84996-290-2
DOI10.1007/978-1-84996-290-2
SpringerLondonDordrechtHeidelbergNewYork
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©Springer-VerlagLondonLimited2010
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mittedundertheCopyright,DesignsandPatentsAct1988,thispublicationmayonlybereproduced,
storedortransmitted,inanyformorbyanymeans,withthepriorpermissioninwritingofthepublish-
ers,orinthecaseofreprographicreproductioninaccordancewiththetermsoflicensesissuedbythe
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SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Leo (John) Grady dedicates the book to
his very patient wife, Amy Grady, daughter
AlexandraGrady,sonLeoJamesGrady(who
arrived during the course of writing) and the
memoryofhislatefather,LeoJoseph Grady.

Preface
Thegoalofthisbookistopresentthetopicofdiscretecalculustoscientistsanden-
gineersandtoshowhowthetheorycanbeappliedtosolvingawidevarietyofreal-
worldproblems.Wefeelthatdiscretecalculusallowsustounifymanyapproaches
todataanalysisandcontentextractionwhilebeingaccessibleenoughtobewidely
applied in many fields and disciplines. This project initially began as a tutorial on
discrete calculus and its applications, and we hope that this work can provide an
introductiontodiscretecalculusanddemonstrateitseffectivenessinmanyproblem
domains.
This book benefited enormously from the friends and colleagues who provided
software,data,andtheirtimeinreadingthemanuscript.Inparticular,forsoftware
wewouldliketothankSébastienBougleuxforhelpwithtotalvariationcodeforan
arbitrarygraphandCamilleCouprieforpowerwatershedcode.Foruseoftheirdata,
we wish to thank Jason Bohland, James Fowler, David Gleich, and Robert Sum-
ner. For comments on the manuscript we would like to thank Christopher Alvino,
MukundBalasubramanian,JasonBohland,SébastienBougleux,GaëlleDesbordes,
MathieuDesbrun,NohaEl-Zehiry,GarethFunka-Lea,DavidGleich,OliverHinds,
AnilHirani,HiroshiIshikawa,RobertKotiuga,ElliotSaltzman,DheerajSingaraju,
Ganesh Sundaramoorthi and Enzo Tonti. We thank Eric Schwartz for all his help
and support over the years. We would like to thank Wayne Wheeler, Simon Rees,
andCatherineBrettatSpringerfortheirenthusiasmandforinitiallyproposingthis
project.LeoGradywouldalsoliketothankChenyangXuforconvincinghimtotake
onthisprojectandtoGarethFunka-Leaforhispatienceandsupportthroughoutthe
writingofthisbook.
Princeton,NewJersey LeoGrady
Charlestown,Massachusetts JonathanPolimeni
vii

Contents
1 DiscreteCalculus:HistoryandFuture . . . . . . . . . . . . . . . . . 1
1.1 DiscreteCalculus . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 OriginsofVectorCalculus. . . . . . . . . . . . . . . . . . 2
1.1.2 OriginsofDiscreteCalculus. . . . . . . . . . . . . . . . . 3
1.1.3 Discretevs.Discretized . . . . . . . . . . . . . . . . . . . 4
1.2 ComplexNetworks . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 ContentExtraction . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 OrganizationoftheBook . . . . . . . . . . . . . . . . . . . . . . 8
1.5 IntendedAudience . . . . . . . . . . . . . . . . . . . . . . . . . . 8
PartI ABriefReviewofDiscreteCalculus
2 IntroductiontoDiscreteCalculus . . . . . . . . . . . . . . . . . . . . 13
2.1 TopologyandtheFundamentalTheoremofCalculus . . . . . . . . 14
2.2 DifferentialForms . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 ExteriorAlgebraandAntisymmetricTensors . . . . . . . . 17
2.2.2 DifferentiationandIntegrationofForms . . . . . . . . . . 26
2.2.3 TheHodgeStarOperator . . . . . . . . . . . . . . . . . . 32
2.2.4 DifferentialFormsandLinearPairings . . . . . . . . . . . 37
2.3 DiscreteCalculus . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3.1 DiscreteDomains . . . . . . . . . . . . . . . . . . . . . . 39
2.3.2 DiscreteFormsandtheCoboundaryOperator . . . . . . . 48
2.3.3 PrimalandDualComplexes . . . . . . . . . . . . . . . . . 50
2.3.4 TheRoleofaMetric:theMetricTensor,theDiscrete
HodgeStarOperator,andWeightedComplexes . . . . . . 54
2.3.5 TheDualCoboundaryOperator . . . . . . . . . . . . . . . 59
2.3.6 TheDiscreteLaplace–deRhamOperator . . . . . . . . . . 60
2.4 StructureofDiscretePhysicalLaws . . . . . . . . . . . . . . . . . 63
2.5 ExamplesofDiscreteCalculus . . . . . . . . . . . . . . . . . . . 63
2.5.1 FundamentalTheoremofCalculusandtheGeneralized
Stokes’Theorem . . . . . . . . . . . . . . . . . . . . . . . 65
2.5.2 TheHelmholtzDecomposition . . . . . . . . . . . . . . . 73
ix