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

139 Pages·2019·29.684 MB·English

by  Rodrigo Rojas Moraleda, Nektarios A. Valous, Wei Xiong, Niels Halama

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
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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