Discrete Calculus：Applied Analysis on Graphs for Computational Science

Thefieldofdiscretecalculus,alsoknownasdiscreteexteriorcalculus,focusesonfindingapropersetofdefinitionsanddifferentialoperatorsthatmakeitpossibletooperatethemachineryofmultivariatecalculusonafinite,discretespace.Incontrasttotraditionalgoalsoffindinganaccuratediscretizationofconventionalmultivariatecalculus,discretecalculusestablishesaseparate,equivalentcalculusthatoperatespurelyinthediscretespacewithoutanyreferencetoanunderlyingcontinuousprocess.Thisuniquetextbringstogetherintoasingleframeworkcurrentresearchinthethreeareasofdiscretecalculus,complexnetworks,andalgorithmiccontentextraction.Althoughtherehavebeenafewintersectionsintheliteraturebetweenthesedisciplines,theyhavedevelopedlargelyindependentlyofoneanother,yetresearchersworkinginanyoneofthesethreeareascanstronglybenefitfromthetoolsandtechniquesbeingusedintheothers.Manyexampleapplicationsfromseveralfieldsofcomputationalscienceareprovidedtodemonstratetheusefulnessofthisframeworktoabroadrangeofproblems.Readersareassumedtobefamiliarwiththebasicsofvectorcalculus,graphtheory,andlinearalgebra.Topicsandfeatures:presentsathoroughreviewofdiscretecalculus,withafocusonkeyconceptsrequiredforsuccessfulapplication;unifiesmanystandardimageprocessingalgorithmsintoacommonframeworkforviewingawidevarietyofstandardalgorithmsinfiltering,clustering,andmanifoldlearningthatmaybeappliedtoprocessingdataassociatedwithagraphornetwork;explainshowdiscretecalculusprovidesanaturaldefinitionoflow-frequencyonagraph,whichthenyieldsfilteringanddenoisingalgorithms;discusseshowfilteringalgorithmscangiverisetoclusteringalgorithms,whichcanbeusedtodevelopmanifoldlearninganddatadiscoverymethods;examinesrankingalgorithms,aswellasalgorithmsforanalyzingthestructureofanetwork.Graduatestudentsandresearchersinterestedindiscretecalculus,complexnetworks,imageprocessingandcomputergraphicswillfindthistext/referenceaclearintroductiontothefoundationsofdiscretecalculusaswellasausefulguidetohavereadilyavailablefortheirwork.Dr.LeoJ.GradyisaSeniorResearchScientistwithSiemensCorporateResearchinPrinceton,NewJersey,USA.Dr.JonathanR.PolimeniisaResearchFellowattheMassachusettsGeneralHospitalinBoston,Massachusetts,USA,andInstructorinRadiologyatHarvardMedicalSchool,Boston,Massachusetts,USA.