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Analysis and fitting of random tessellation models [Elektronische Ressource] : applications in telecommunication and cell biology / vorgelegt von Frank Fleischer

233 pages
Universit¨at UlmInstitut fu¨r StochastikAnalysis and Fitting of Random Tessellation ModelsApplications in Telecommunication and Cell BiologyDissertationzur Erlangung des Doktorgrades Dr. rer. nat. der Fakult¨at fu¨rMathematik und Wirtschaftswissenschaften der Universita¨t Ulmvorgelegt vonFrank FleischerausG¨oppingen2007Amtierender Dekan: Prof. Dr. Frank Stehling1. Gutachter: Prof. Dr. Volker Schmidt2. Gutachter: Prof. Dr. Franz Schweiggert3. Gutachter: Prof. Dr. Eva Vedel-Jensen4. Gutachter: Prof. Dr. Michael BeilTag der Promotion: 11. Mai 2007Contents1 Introduction 91.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3 The GeoStoch Library . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Random Tessellations and Basics from Stochastic Geometry 212.1 Random Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Random Point Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.1 Definition of Random Point Processes . . . . . . . . . . . . . . . 232.2.2 Properties of Random Point Processes . . . . . . . . . . . . . . 242.2.3 Palm Distributions and Campbell Measures . . . . . . . . . . . 272.2.4 Poisson Point Processes . . . . . . . . . . . . . . . . . . . . . . 292.2.5 Other Examples of Random Point Processes . . . . . . . . . . . 302.2.6 Marked Point Processes . . . . . . . . . . . . .
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Universit¨at Ulm
Institut fu¨r Stochastik
Analysis and Fitting of Random Tessellation Models
Applications in Telecommunication and Cell Biology
Dissertation
zur Erlangung des Doktorgrades Dr. rer. nat. der Fakult¨at fu¨r
Mathematik und Wirtschaftswissenschaften der Universita¨t Ulm
vorgelegt von
Frank Fleischer
aus
G¨oppingen
2007Amtierender Dekan: Prof. Dr. Frank Stehling
1. Gutachter: Prof. Dr. Volker Schmidt
2. Gutachter: Prof. Dr. Franz Schweiggert
3. Gutachter: Prof. Dr. Eva Vedel-Jensen
4. Gutachter: Prof. Dr. Michael Beil
Tag der Promotion: 11. Mai 2007Contents
1 Introduction 9
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 The GeoStoch Library . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Random Tessellations and Basics from Stochastic Geometry 21
2.1 Random Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Random Point Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 Definition of Random Point Processes . . . . . . . . . . . . . . . 23
2.2.2 Properties of Random Point Processes . . . . . . . . . . . . . . 24
2.2.3 Palm Distributions and Campbell Measures . . . . . . . . . . . 27
2.2.4 Poisson Point Processes . . . . . . . . . . . . . . . . . . . . . . 29
2.2.5 Other Examples of Random Point Processes . . . . . . . . . . . 30
2.2.6 Marked Point Processes . . . . . . . . . . . . . . . . . . . . . . 33
2.2.7 Neveu’s Exchange Formula for Palm Distributions . . . . . . . . 35
2.3 The Boolean Model and Modulated Poisson Point Processes . . . . . . 36
2.3.1 Definition of the Boolean Model . . . . . . . . . . . . . . . . . . 36
2.3.2 Modulated Poisson Point Processes Based on Boolean Models . 38
2.4 Definition of Random Tessellations with Examples . . . . . . . . . . . . 40
2.4.1 Deterministic Tessellations . . . . . . . . . . . . . . . . . . . . . 40
34 Contents
2.4.2 Definition of Random Tessellations . . . . . . . . . . . . . . . . 40
2.4.3 Poisson Line Tessellations . . . . . . . . . . . . . . . . . . . . . 42
2.4.4 Poisson–Voronoi Tessellations . . . . . . . . . . . . . . . . . . . 44
2.4.5 Poisson–Delaunay Tessellations . . . . . . . . . . . . . . . . . . 46
2.4.6 Cox–Voronoi Tessellations Based on a Poisson Line Process . . . 49
2.4.7 Modulated Tessellations . . . . . . . . . . . . . . . . . . . . . . 51
2.4.8 Iterated Tessellations . . . . . . . . . . . . . . . . . . . . . . . . 53
3 Simulation Algorithms for the Typical Cell of Random Tessellations 59
3.1 General Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.1.1 Simulation of Poisson Point Processes . . . . . . . . . . . . . . . 60
3.1.2 Slivnyak’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.1.3 Simulation of the Typical Cell for Poisson–Voronoi Tessellations 62
3.2 Cox–Voronoi Tessellations Based on Poisson Line Processes . . . . . . . 64
3.2.1 Representation of the Typical Cell . . . . . . . . . . . . . . . . . 64
3.2.2 Simulation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2.3 Results of Monte–Carlo Simulations . . . . . . . . . . . . . . . . 67
3.3 Voronoi Tessellations Based on Modulated Poisson Point Processes . . 73
3.3.1 Representation of the Typical Cell . . . . . . . . . . . . . . . . . 74
3.3.2 Simulation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 74
3.3.3 Results of Monte–Carlo Simulations . . . . . . . . . . . . . . . . 79
4 Estimation of Cost Functionals for Random Tessellation Models 83
4.1 Graphs and Shortest Paths . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.1.1 Definition of a Graph . . . . . . . . . . . . . . . . . . . . . . . . 84
4.1.2 Shortest Path Algorithms . . . . . . . . . . . . . . . . . . . . . 86
4.2 Cost Analysis for Hierarchical Models Based on Poisson Line Processes 90
4.2.1 Model Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 90Contents 5
4.2.2 Transformation to Graph Structure . . . . . . . . . . . . . . . . 91
4.2.3 Mean Shortest Path Lengths and Mean Subscriber Line Lengths 92
4.2.4 Application of Neveu’s Formula . . . . . . . . . . . . . . . . . . 94
4.2.5 Efficient Estimation of the Mean Shortest Path Length . . . . . 97
4.2.6 Efficient Estimation of the Mean Subscriber Line Length . . . . 101
4.2.7 Scaling Invariance Properties . . . . . . . . . . . . . . . . . . . 104
4.2.8 Results of Monte–Carlo Simulations . . . . . . . . . . . . . . . . 105
4.3 Average Distances to Nuclei in Modulated Poisson–Voronoi Tessellations 109
4.3.1 Definition of a Cost Functional . . . . . . . . . . . . . . . . . . 110
4.3.2 Application of Neveu’s Formula . . . . . . . . . . . . . . . . . . 111
4.3.3 Estimation Based on Monte-Carlo Simulation . . . . . . . . . . 113
4.3.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . 114
5 Testing Methods for Programs with Random Input or Output 117
5.1 General Principles of Software Testing . . . . . . . . . . . . . . . . . . 118
5.2 Testing Based on Statistical Oracles . . . . . . . . . . . . . . . . . . . . 118
5.2.1 Basic Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.3 Statistical Metamorphic Testing . . . . . . . . . . . . . . . . . . . . . . 123
5.3.1 Basic Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.4 Tests by Comparison to Gold–Standard . . . . . . . . . . . . . . . . . . 127
5.4.1 Basic Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.5 Summary of Testing Methods . . . . . . . . . . . . . . . . . . . . . . . 1296 Contents
6 Concepts of Morphological Image Analysis 131
6.1 Basics of Morphological Image Analysis . . . . . . . . . . . . . . . . . . 132
6.1.1 Digital Grids and Digital Images . . . . . . . . . . . . . . . . . 132
6.1.2 Image Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.2 Skeletonization by Morphological Operators . . . . . . . . . . . . . . . 137
6.3 Morphological Watershed Transformation . . . . . . . . . . . . . . . . . 139
6.4 Other Morphological Operations . . . . . . . . . . . . . . . . . . . . . . 142
6.4.1 Processing of Line Structures . . . . . . . . . . . . . . . . . . . 142
6.4.2 Pruning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.4.3 Merging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7 Fitting Random Tessellation Models to Network Structures 147
7.1 Characteristics of Input Data and Estimation of Global Characteristics 148
7.1.1 Input Data Characteristics . . . . . . . . . . . . . . . . . . . . . 148
7.1.2 Unbiased Estimation . . . . . . . . . . . . . . . . . . . . . . . . 148
7.2 Fitting Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.2.1 Distance Functions . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.2.2 Optimal Model Choice . . . . . . . . . . . . . . . . . . . . . . . 151
7.3 Possible Tessellation Models Considered . . . . . . . . . . . . . . . . . 152
8 ApplicationstoElectronMicroscopyImagesofCytoskeletalNetworks157
8.1 Statistical Analysis of Keratin Filament Structures . . . . . . . . . . . 158
8.1.1 Image Acquisition and Segmentation . . . . . . . . . . . . . . . 159
8.1.2 Analysis of Filament Lengths and Orientations . . . . . . . . . . 160
8.1.3 Fitting of Nested Tessellation Models . . . . . . . . . . . . . . . 167
8.1.4 Summary of the Results . . . . . . . . . . . . . . . . . . . . . . 171
8.2 Model–based Analysis of Actin Filament Networks . . . . . . . . . . . . 176Contents 7
8.2.1 Image Segmentation . . . . . . . . . . . . . . . . . . . . . . . . 176
8.2.2 Fitting of Superposed Tessellation Models . . . . . . . . . . . . 177
8.2.3 Estimation of Actin Network Elasticity . . . . . . . . . . . . . . 179
8.2.4 Summary of the Results . . . . . . . . . . . . . . . . . . . . . . 182
9 Conclusions and Outlook 185
A Basic Mathematical Definitions 191
A.1 Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
A.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
A.3 Measure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
A.4 Probability Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
B Zusammenfassung 199
B.1 Ziele . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
B.2 Gliederung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Bibliography 207
List of Figures 217
List of Tables 221
Nomenclature 223
Acknowledgement 227
Lebenslauf 229
Publikationsliste 231
Erkl¨arung 2338 ContentsChapter 1
Introduction
1.1 Motivation
This thesis has been motivated by two ongoing research projects of the Institute of
Stochastics at Ulm University. In the first project, which is performed in cooperation
with France T´el´ecom R&D, Paris, it is dealt with the modelling and the analysis of
network structures that occur in the field of telecommunication. In particular street
systems, for example of urban regions like Paris, but also telecommunication networks
on nationwide scales are analysed with respect to their geometrical structures and cost
characteristics connected to them. Figure 1.1 shows such an infrastructure system for
the example of Paris, while in Figure 1.2 a magnified cutout is displayed, where the
actual measurements are shown in red and the polygonal connections between them in
black. Figure 1.3a illustrates a modelling approach for telecommunication networks in
an urban area. Here, the streets are represented by random lines. Different types of
telecommunication equipment are then placed on these lines according to a given ran-
dom process. In Figure 1.3b we see a realisation for a modelling approach with respect
to telecommunication networks on a nationwide scale. In this case the network equip-
ment is placed randomly in the plane according to a given random process that reflects
the differences between urban and rural districts by assigning different intensities. The
subscribers are also distributed randomly according to a similar mechanism, thereby
reflecting the underlying population distribution. Aims of such modelling approaches
are, for example, to obtain inference about connected costs like the mean distance from
a subscriber to its nearest telecommunication equipment located along the streets or
about the capacities needed to connect all the subscribers.
In the second project which is performed jointly with co–workers from the Institute
of Internal Medicine 1 at Ulm University, the Central Electron Microscopy Facility at
910 1 Introduction
Figure 1.1: Urban infrastructure of Paris
Figure 1.2: Infrastructure in a region of Paris

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