3D Images of Materials Structures: Processing and Analysis

3D Images of Materials Structures......Page 5
Foreword......Page 7
Contents......Page 9
Preface......Page 15
Conventions and Notation......Page 17
1 Introduction......Page 19
2.1.1 Points and Sets in Euclidean Spaces......Page 29
2.1.2 Curvatures......Page 32
2.1.3 Measures and Measurable Spaces......Page 35
2.2.1 The Euler Number and the Integral of Gaussian Curvature......Page 36
2.2.2 The Mean Width and the Integral of the Mean Curvature......Page 38
2.2.3 Intrinsic Volumes of Convex Bodies......Page 40
2.2.4 Additive Extensions on the Convex Ring......Page 42
2.2.5 The Principal Kinematic Formulae of Integral Geometry......Page 43
2.3 Random Sets......Page 44
2.3.1 Definition of Random Sets......Page 45
2.3.2 Characteristics of Random Closed Sets......Page 46
2.3.3 Random Point Fields......Page 48
2.3.4 Random Tessellations......Page 51
2.4.1 Measurable Functions......Page 52
2.4.2 Fourier Transform......Page 54
2.4.3 Bochner's Theorem......Page 58
3.2 Point Lattices, Digitizations and Pixel Configurations......Page 61
3.2.1 Homogeneous Lattices......Page 62
3.2.2 Digitization......Page 63
3.2.3 Pixel Configurations......Page 64
3.3 Adjacency and Euler Number......Page 65
3.3.1 Adjacency Systems......Page 66
3.3.2 Discretization of Sets with Respect to Adjacency......Page 69
3.3.3 Euler Number......Page 70
3.3.4 Complementarity......Page 77
3.3.5 Multi-grid Convergence......Page 78
3.4.1 Counting Nodes in Open Foams......Page 79
3.4.2 Connectivity of the Fibres in Non-woven Materials......Page 81
3.5 Image Data......Page 82
3.5.1 The Inverse Lattice......Page 83
3.5.2 The Nyquist–Shannon Sampling Theorem......Page 84
3.6.1 Volume Rendering......Page 87
3.6.2 Surface Rendering......Page 90
4.1.1 The Discrete Fourier Transform of a Discrete One-Dimensional Signal......Page 97
4.1.2 Fast Fourier Transform......Page 98
4.1.3 Extensions to Higher Dimensions......Page 99
4.2.1 Morphological Transforms of Sets......Page 100
4.2.2 Linear Filters......Page 112
4.2.3 Morphological Filters......Page 120
4.2.4 Rank Value Filters......Page 121
4.2.5 Diffusion Filters......Page 123
4.2.6 Geodesic Morphological Transforms......Page 125
4.2.7 Distance Transforms......Page 129
4.2.8 Skeletonization......Page 134
4.3 Segmentation......Page 138
4.3.1 Binarization......Page 139
4.3.2 Connectedness, Connected Components and Labelling......Page 146
4.3.3 Watershed Transform......Page 161
4.3.4 Further Segmentation Methods......Page 166
5.1 Introduction......Page 167
5.2 Intrinsic Volumes......Page 168
5.2.1 Section Lattices and Translation Lattices......Page 169
5.2.2 Measurement of Intrinsic Volumes......Page 170
5.2.3 Discretization of the Translative Integral......Page 171
5.2.4 Discretization of the Integral over all Subspaces......Page 174
5.2.5 Shape Factors......Page 180
5.2.6 Edge Correction......Page 182
5.3 Intrinsic Volume Densities......Page 184
5.3.1 Estimation of Intrinsic Volume Densities for Macroscopically Homogeneous Random Sets......Page 185
5.3.2 Characterization of Anisotropy......Page 187
5.3.3 Mean Chord Length......Page 188
5.3.4 Structure Model Index......Page 189
5.3.5 Estimation of the Intrinsic Volume Densities for Macroscopically Homogeneous and Isotropic Random Sets......Page 190
5.3.6 Intrinsic Volume Densities of the Solid Matter of Two Natural Porous Structures......Page 194
5.4 Directional Analysis......Page 197
5.4.1 Inverse Cosine Transform......Page 198
5.4.2 Use of Pixel Configurations Carrying Directional Information......Page 200
5.4.3 Gradient and Hessian Matrix......Page 202
5.4.4 Maximum Filter Response......Page 203
5.5 Distances Between Random Sets and Distance Distributions......Page 205
5.5.1 Spherical Contact Distribution Function and Related Quantities......Page 207
5.5.2 Stochastic Dependence of Constituents of Metallic Foams......Page 210
6.1 Introduction......Page 213
6.2 Second-Order Characteristics of a Random Volume Measure......Page 214
6.2.1 Covariance Function and Bartlett Spectrum......Page 215
6.2.2 Power Spectrum......Page 219
6.2.3 Measurement of the Covariance and the Power Spectrum......Page 220
6.2.4 Macroscopic Homogeneity and Isotropy......Page 221
6.2.5 Mean Face Width of an Open Foam......Page 223
6.2.6 Random Packing of Balls......Page 224
6.2.7 Particle Rearrangement During Sintering Processes......Page 225
6.3 Correlations Between Random Structures......Page 226
6.3.1 The Cross-Covariance Function......Page 227
6.3.3 Spatial Cross-Correlation Between Constituents of Metallic Foams......Page 229
6.4 Second-Order Characteristics of Random Surfaces......Page 230
6.4.1 The Random Surface Measure......Page 231
6.4.2 The Bartlett Spectrum......Page 233
6.4.3 Power Spectrum......Page 236
6.4.4 Measurement of the Power Spectrum with Respect to the Surface Measure......Page 238
6.5 Second-Order Characteristics of Random Point Fields......Page 240
6.5.1 Point Fields and Associated Random Functions......Page 241
6.5.2 A Wiener–Khintchine Theorem for Point Fields......Page 242
6.5.3 Estimation of the Pair Correlation Function......Page 244
6.5.4 The Power Spectra of the Centres of Balls in Dense Packings......Page 248
7.1 Introduction, Motivation......Page 251
7.2.1 The Poisson Point Field......Page 252
7.2.3 Finite Point Fields Defined by a Probability Density......Page 253
7.3 Macroscopically Homogeneous Systems of Non-overlapping Particles......Page 257
7.4 Macroscopically Homogeneous Systems of Overlapping Particles......Page 261
7.4.1 Intrinsic Volumes of Boolean Models in Rn......Page 263
7.4.2 Intrinsic Volumes of Boolean Models in R3......Page 266
7.4.3 Structure Model Index for Boolean Models in R3......Page 268
7.5.1 Boolean Cylinder Model......Page 269
7.5.2 PET Stacked Fibre Non-woven Materials......Page 270
7.5.3 Carbon Paper......Page 273
7.6.1 Geometric Properties of Tessellations of R3......Page 274
7.6.2 Voronoï Tessellations......Page 278
7.6.3 Laguerre Tessellations......Page 279
7.6.4 The Weaire–Phelan Foam......Page 283
7.6.5 Mean Values of Geometric Characteristics of Open Foams......Page 285
7.6.6 Modelling a Closed Polymer Foam......Page 288
7.6.7 Modelling an Open Ceramic Foam......Page 294
8.1 Introduction......Page 299
8.2 Effective Conductivity of Polycrystals by Stochastic Homogenization......Page 300
8.3.1 Fundamentals of Linear Elasticity......Page 306
8.3.2 Finite Element Method......Page 309
8.3.3 Effective Stiffness Tensor Random Sets......Page 312
8.3.4 Effective Elastic Moduli of a Porous Alumina Material......Page 314
References......Page 319
Index......Page 337