Foreword ........................................................ V
Preface ...................................................... XIII
Conventions and Notation ....................................... XV
1 Introduction ................................................. 1
2 Preliminaries ............................................... 11
2.1 General Notation ....................................... 11
2.1.1 Points and Sets in Euclidean Spaces ............. 11
2.1.2 Curvatures ...................................... 14
2.1.3 Measures and Measurable Spaces .................. 17
2.2 Characteristics of Sets ................................ 18
2.2.1 The Euler Number and the Integral of Gaussian
Curvature ....................................... 18
2.2.2 The Mean Width and the Integral of the Mean
Curvature ....................................... 20
2.2.3 Intrinsic Volumes of Convex Bodies .............. 22
2.2.4 Additive Extensions on the Convex Ring .......... 24
2.2.5 The Principal Kinematic Formulae of Integral
Geometry ........................................ 25
2.3 Random Sets ............................................ 26
2.3.1 Definition of Random Sets ....................... 27
2.3.2 Characteristics of Random Closed Sets ........... 28
2.3.3 Random Point Fields ............................. 30
2.3.4 Random Tessellations ............................ 33
2.4 Fourier Analysis ....................................... 34
2.4.1 Measurable Functions ............................ 34
2.4.2 Fourier Transform ............................... 36
2.4.3 Bochner's Theorem ............................... 40
3 Lattices, Adjacency of Lattice Points, and Images ........... 43
3.1 Introduction ........................................... 43
3.2 Point Lattices, Digitizations and Pixel
Configurations ......................................... 43
3.2.1 Homogeneous Lattices ............................ 44
3.2.2 Digitization .................................... 45
3.2.3 Pixel Configurations ............................ 46
3.3 Adjacency and Euler Number ............................. 47
3.3.1 Adjacency Systems ............................... 48
3.3.2 Discretization of Sets with Respect to
Adjacency ....................................... 51
3.3.3 Euler Number .................................... 52
3.3.4 Complementarity ................................. 59
3.3.5 Multi-grid Convergence .......................... 60
3.4 The Euler Number of Microstructure Constituents ........ 61
3.4.1 Counting Nodes in Open Foams .................... 61
3.4.2 Connectivity of the Fibres in Non-woven
Materials ....................................... 63
3.5 Image Data ............................................. 64
3.5.1 The Inverse Lattice ............................. 65
3.5.2 The Nyquist-Shannon Sampling Theorem ............ 66
3.6 Rendering .............................................. 69
3.6.1 Volume Rendering ................................ 69
3.6.1.1 Physical Background .................... 69
3.6.1.2 Transfer function ...................... 70
3.6.1.3 Ray Casting ............................ 71
3.6.1.4 3D Texture Mapping ..................... 72
3.6.2 Surface Rendering ............................... 72
3.6.2.1 Properties of the Reconstructed
Surface ................................ 72
3.6.2.2 Marching Cube Type Algorithms .......... 73
3.6.2.3 The Wrapper Algorithm .................. 75
4 Image Processing ............................................ 79
4.1 Fourier Transform of an Image .......................... 79
4.1.1 The Discrete Fourier Transform of a Discrete
One-Dimensional Signal .......................... 79
4.1.2 Fast Fourier Transform .......................... 80
4.1.3 Extensions to Higher Dimensions ................. 81
4.2 Filtering .............................................. 82
4.2.1 Morphological Transforms of Sets ................ 82
4.2.1.1 Minkowski Addition and Dilation ........ 83
4.2.1.2 Minkowski Subtraction and Erosion ...... 85
4.2.1.3 Mean Co-ordination Number of Sinter
Particles .............................. 86
4.2.1.4 Morphological Opening and Closure ...... 87
4.2.1.5 Top-Hat Transforms ..................... 89
4.2.1.6 Algebraic Opening and Closure .......... 89
4.2.1.7 Aspects of Algorithmic
Implementation ......................... 90
4.2.1.8 Handling of Edge Effects ............... 92
4.2.1.9 Adaptable Morphology ................... 93
4.2.2 Linear Filters .................................. 94
4.2.2.1 Linear Smoothing Filters ............... 94
4.2.2.2 Linear Derivative Filters .............. 98
4.2.3 Morphological Filters .......................... 102
4.2.4 Rank Value Filters ............................. 103
4.2.5 Diffusion Filters .............................. 105
4.2.6 Geodesic Morphological Transforms .............. 107
4.2.6.1 Reconstruction by Erosion ............. 108
4.2.6.2 Reconstruction by Dilation ............ 109
4.2.6.3 Self-Dual Reconstruction .............. 110
4.2.6.4 H-Minima .............................. 111
4.2.7 Distance Transforms ............................ 111
4.2.7.1 Discrete or Chamfer Distance
Transforms ............................ 113
4.2.7.2 Euclidean Distance Transforms ......... 114
4.2.8 Skeletonization ................................ 116
4.3 Segmentation .......................................... 120
4.3.1 Binarization ................................... 121
4.3.1.1 Global Thresholding ................... 121
4.3.1.2 Local Thresholding .................... 123
4.3.1.3 Hysteresis ............................ 125
4.3.1.4 Region Growing ........................ 127
4.3.2 Connectedness, Connected Components and
Labelling ...................................... 128
4.3.2.1 Connectedness ......................... 128
4.3.2.2 Jordan Theorems ....................... 132
4.3.2.3 A Simple Labelling Algorithm .......... 135
4.3.2.4 Advanced Labelling Techniques ......... 141
4.3.3 Watershed Transform ............................ 143
4.3.4 Further Segmentation Methods ................... 148
5 Measurement of Intrinsic Volumes and Related Quantities .... 149
5.1 Introduction .......................................... 149
5.2 Intrinsic Volumes ..................................... 150
5.2.1 Section Lattices and Translation Lattices ...... 151
5.2.2 Measurement of Intrinsic Volumes ............... 152
5.2.3 Discretization of the Translative Integral ..... 153
5.2.4 Discretization of the Integral over all
Subspaces ...................................... 156
5.2.4.1 Simple Quadrature ..................... 156
5.2.4.2 Fourier Expansion ..................... 159
5.2.5 Shape Factors .................................. 162
5.2.6 Edge Correction ................................ 164
5.3 Intrinsic Volume Densities ............................ 166
5.3.1 Estimation of Intrinsic Volume Densities for
Macroscopically Homogeneous Random Sets ........ 167
5.3.2 Characterization of Anisotropy ................. 169
5.3.3 Mean Chord Length .............................. 170
5.3.4 Structure Model Index .......................... 171
5.3.5 Estimation of the Intrinsic Volume Densities
for Macroscopically Homogeneous and Isotropic
Random Sets .................................... 172
5.3.6 Intrinsic Volume Densities of the Solid
Matter of Two Natural Porous Structures ........ 176
5.4 Directional Analysis .................................. 179
5.4.1 Inverse Cosine Transform ....................... 180
5.4.2 Use of Pixel Configurations Carrying
Directional Information ........................ 182
5.4.3 Gradient and Hessian Matrix .................... 184
5.4.4 Maximum Filter Response ........................ 185
5.4.5 Directional Analysis for Fibres in Ultra-
High-Performance Concrete ...................... 187
5.5 Distances Between Random Sets and Distance
Distributions ......................................... 187
5.5.1 Spherical Contact Distribution Function and
Related Quantities ............................. 189
5.5.2 Stochastic Dependence of Constituents of
Metallic Foams ................................. 192
6 Spectral Analysis .......................................... 195
6.1 Introduction .......................................... 195
6.2 Second-Order Characteristics of a Random Volume
Measure ............................................... 196
6.2.1 Covariance Function and Bartlett Spectrum ...... 197
6.2.2 Power Spectrum ................................. 201
6.2.3 Measurement of the Covariance and the Power
Spectrum ....................................... 202
6.2.4 Macroscopic Homogeneity and Isotropy ........... 203
6.2.5 Mean Face Width of an Open Foam ................ 205
6.2.6 Random Packing of Balls ........................ 206
6.2.7 Particle Rearrangement During Sintering
Processes ...................................... 207
6.3 Correlations Between Random Structures ............... 208
6.3.1 The Cross-Covariance Function .................. 209
6.3.2 Measurement of the Cross Covariance Function ... 211
6.3.3 Spatial Cross-Correlation Between
Constituents of Metallic Foams ................. 211
6.4 Second-Order Characteristics of Random Surfaces ....... 212
6.4.1 The Random Surface Measure ..................... 213
6.4.2 The Bartlett Spectrum .......................... 215
6.4.3 Power Spectrum ................................. 218
6.4.4 Measurement of the Power Spectrum with
Respect to the Surface Measure ................. 220
6.5 Second-Order Characteristics of Random Point Fields ... 222
6.5.1 Point Fields and Associated Random Functions ... 223
6.5.2 A Wiener-Khintchine Theorem for Point Fields ... 224
6.5.3 Estimation of the Pair Correlation Function .... 226
6.5.4 The Power Spectra of the Centres of Balls in
Dense Packings ................................. 230
7 Model-based Image Analysis ................................. 233
7.1 Introduction, Motivation .............................. 233
7.2 Point Field Models .................................... 234
7.2.1 The Poisson Point Field ........................ 234
7.2.2 Matern Hard-Core Point Fields .................. 235
7.2.3 Finite Point Fields Defined by a Probability
Density ........................................ 235
7.2.3.1 Simulation of Finite Point Fields:
Metropolis-Hastings ................... 237
7.2.3.2 Simulation of Finite Point Fields:
Spatial Birth-and-Death Processes ..... 238
7.3 Macroscopically Homogeneous Systems of
Non-overlapping Particles ............................. 239
7.4 Macroscopically Homogeneous Systems of Overlapping
Particles ............................................. 243
7.4.1 Intrinsic Volumes of Boolean Models in  n ...... 245
7.4.2 Intrinsic Volumes of Boolean Models in 3 ..... 248
7.4.3 Structure Model Index for Boolean Models in
3 ............................................. 250
7.5 Macroscopically Homogeneous Fibre Systems ............. 251
7.5.1 Boolean Cylinder Model ......................... 251
7.5.2 PET Stacked Fibre Non-woven Materials .......... 252
7.5.3 Carbon Paper ................................... 255
7.6 Tessellations ......................................... 256
7.6.1 Geometric Properties of Tessellations of IR3 ... 256
7.6.1.1 Mean Number of ℓ-Faces Adjacent to
a k-Face .............................. 257
7.6.1.2 The Density of k-Faces ................ 258
7.6.1.3 Mecke's Characteristics ............... 258
7.6.1.4 Cell-Based Characteristics ............ 259
7.6.2 Voronoï Tessellations .......................... 260
7.6.2.1 Poisson Voronoï Tessellation .......... 260
7.6.2.2 Hard-Core Voronoï Tessellation ........ 261
7.6.3 Laguerre Tessellations ......................... 261
7.6.3.1 Poisson-Laguerre Tessellations ........ 264
7.6.3.2 Laguerre Tessellations Generated by
Random Packings of Balls .............. 264
7.6.4 The Weaire-Phelan Foam ......................... 265
7.6.4.1 Random Perturbations of the Weaire-
Phelan Foam ........................... 266
7.6.5 Mean Values of Geometric Characteristics of
Open Foams ..................................... 267
7.6.6 Modelling a Closed Polymer Foam ................ 270
7.6.7 Modelling an Open Ceramic Foam ................. 276
7.6.7.1 Modelling the Polyurethane Core ....... 277
7.6.7.2 Modelling the Coating ........................ 278
8 Simulation of Material Properties .......................... 281
8.1 Introduction .......................................... 281
8.2 Effective Conductivity of Polycrystals by Stochastic
Homogenization ........................................ 282
8.3 Computation of Effective Elastic Moduli of Porous
Media by FEM Simulation ............................... 288
8.3.1 Fundamentals of Linear Elasticity .............. 288
8.3.2 Finite Element Method .......................... 291
8.3.2.1 Discretization ........................ 291
8.3.2.2 Numerical Solution of the Linear
Elastic Problem ....................... 292
8.3.3 Effective Stiffness Tensor Random Sets ......... 294
8.3.4 Effective Elastic Moduli of a Porous Alumina
Material ....................................... 296
References .................................................... 301
Index ......................................................... 319
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