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Computational Topology for Biomedical Image and Data Analysis: Theory and Applications PDF

139 Pages·2019·29.684 MB·English
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Computational Topology for Biomedical Image and Data Analysis Theory and Applications CRCPressFocusSeriesinMedicalPhysicsandBiomedical Engineering SeriesEditors: MagdalenaS.StoevaandTae-SukSuh Recentbooksintheseries: ComputationalTopologyforBiomedicalImageandDataAnalysis: TheoryandApplications RodrigoRojasMoraleda,NektariosA.Valous,WeiXiong,and NielsHalama Computational Topology for Biomedical Image and Data Analysis Theory and Applications Rodrigo Rojas Moraleda Nektarios A. Valous Wei Xiong Niels Halama CRCPress Taylor&FrancisGroup 6000BrokenSoundParkwayNW,Suite300 BocaRaton,FL33487-2742 (cid:13)c 2020byTaylor&FrancisGroup,LLC CRCPressisanimprintofTaylor&FrancisGroup,anInformabusiness NoclaimtooriginalU.S.Governmentworks Printedonacid-freepaper InternationalStandardBookNumber-13:978-1-138-33634-6(Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and infor- mation,buttheauthorandpublishercannotassumeresponsibilityforthevalid- ityofallmaterialsortheconsequencesoftheiruse.Theauthorsandpublishers haveattemptedtotracethecopyrightholdersofallmaterialreproducedinthis publicationandapologizetocopyrightholdersifpermissiontopublishinthis formhasnotbeenobtained.Ifanycopyrightmaterialhasnotbeenacknowledged pleasewriteandletusknowsowemayrectifyinanyfuturereprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including pho- tocopying,microfilming,andrecording,orinanyinformationstorageorretrieval system,withoutwrittenpermissionfromthepublishers. Forpermissiontophotocopyorusematerialelectronicallyfromthiswork,please access www.copyright.com (http://www.copyright.com/) or contact the Copy- right Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400.CCCisanot-for-profitorganizationthatprovideslicensesandreg- istrationforavarietyofusers.Fororganizationsthathavebeengrantedaphoto- copylicensebytheCCC,aseparatesystemofpaymenthasbeenarranged. TrademarkNotice:Productorcorporatenamesmaybetrademarksorregistered trademarks,andareusedonlyforidentificationandexplanationwithoutintent toinfringe. VisittheTaylor&FrancisWebsiteat http://www.taylorandfrancis.com andtheCRCPressWebsiteat http://www.crcpress.com with love to Emily Karx Contents SeriesPreface xi Foreword xv Preface xvii Contributors xix Authors xxi SectionI Theoreticalfoundations Chapter 1(cid:4) Elementsoftopologyandhomology 3 1.1 ELEMENTSOFGROUPTHEORY 4 1.1.1 Set 4 1.1.2 Group 7 1.1.3 Ringandfield 10 1.1.4 Homomorphism 13 1.1.5 Modulararithmetic 15 1.2 TOPOLOGICALSPACES 18 1.2.1 Connectedtopologicalspaces 21 1.2.2 Continuityandcontinuousmaps 22 1.3 METRICSPACES 24 1.3.1 Topologyinametricspace 24 vii viii (cid:4) Contents 1.4 ELEMENTSOFAFFINEVECTORSPACES 26 1.4.1 AffinevectorspaceA 26 1.5 SIMPLICIALCOMPLEXES 29 1.5.1 Geometricsimplicialcomplexes 29 1.5.2 Orientedsimplicialcomplexes 31 1.5.3 Unorientedsimplicialcomplexes 33 1.5.4 Abstractsimplicialcomplexes 35 1.6 HOMOLOGY 36 1.6.1 Homologyoforientedsimplicial complexes 37 1.6.2 Homology of unoriented simplicial complexes 41 1.6.3 Homologyofabstractsimplicial complexes 43 1.7 CHAPTERFIGURES 47 Chapter 2(cid:4) Persistenthomologyofimages 51 2.1 INTRODUCTION 51 2.2 DIGITALIMAGES 52 2.3 ANEIGHBORHOODSYSTEMFORIMAGES 53 2.3.1 Spatial-basedneighborhood 54 2.3.2 Intensity-basedneighborhood 54 2.4 COMPUTINGHOMOLOGYFROMIMAGES 57 2.4.1 Thedigitalimagefunctionφ 57 2.4.2 Projectedumbra 58 2.4.3 Inclusiontreeofumbraprojections 60 2.4.4 Algorithmsforcomputinghomology 62 2.4.5 Construction of abstract simplicial complexes 64 2.5 PERSISTENTHOMOLOGY 66 2.5.1 Filtration 66 2.5.2 Persistence 67 Contents (cid:4) ix 2.5.3 Computationofpersistenthomology 68 2.5.4 Persistencediagram 73 2.6 COMPUTATIONALCOST 75 2.7 CHAPTERFIGURES 77 SectionII Casestudies Chapter 3(cid:4) Recognizingnoise 87 3.1 SYNTHETICIMAGES 87 3.2 WORKFLOW 88 3.3 RESULTS 89 3.4 CHAPTERFIGURES 91 Chapter 4(cid:4) Imagesegmentation 93 4.1 INTRODUCTION 94 4.2 MATERIALSANDIMAGEACQUISITION 95 4.3 FEATURESPACE 95 4.4 IMAGEDISMANTLINGANDINCLUSIONTREE 96 4.5 FILTRATION 96 4.6 HOMOLOGYANDPERSISTENCEDIAGRAM 96 4.7 STATISTICALINFERENCE 97 4.8 MASKSEGMENTATION 98 4.9 VALIDATIONWITHCASESTUDYDATA 98 4.10 VALIDATIONWITHILASTIK 99 4.11 RESULTSWITHCASESTUDYDATA 99 4.12 RESULTSWITHILASTIK 99 4.13 CHAPTERFIGURES 101 Chapter 5(cid:4) Pointcloudcharacterization 103 5.1 INTRODUCTION 103 5.2 POINTCLOUD 104

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