NIKOS NIKOLAIDIS, PhD, is a senior researcher in the Artificial Intelligence and Information Analysis Laboratory, Department of Informatics, Aristotle University of Thessaloniki, Greece. IOANNIS PITAS, PhD, is a professor in the Department of Informatics, Aristotle University of Thessaloniki, Greece.

"Explains numerous 3-D image processing, analysis, and visualization techniques, such as volume filtering, skeletonization and registration, and visualization." (SciTech Book News Vol. 25, No. 2 June 2001)

• Foreword VPreface XIIIConventions and Notation XV1 Introduction 12 Preliminaries 112.1 General Notation 112.1.1 Points and Sets in Euclidean Spaces 112.1.2 Curvatures 142.1.3 Measures and Measurable Spaces 172.2 Characteristics of Sets 182.2.1 The Euler Number and the Integral of Gaussian Curvature 182.2.2 The Mean Width and the Integral of the Mean Curvature 202.2.3 Intrinsic Volumes of Convex Bodies 222.2.4 Additive Extensions on the Convex Ring 242.2.5 The Principal Kinematic Formulae of Integral Geometry 252.3 Random Sets 262.3.1 Definition of Random Sets 272.3.2 Characteristics of Random Closed Sets 282.3.3 Random Point Fields 302.3.4 Random Tessellations 332.4 Fourier Analysis 342.4.1 Measurable Functions 342.4.2 Fourier Transform 362.4.3 Bochner’s Theorem 403 Lattices, Adjacency of Lattice Points, and Images 433.1 Introduction 433.2 Point Lattices, Digitizations and Pixel Configurations 433.2.1 Homogeneous Lattices 443.2.2 Digitization 453.2.3 Pixel Configurations 463.3 Adjacency and Euler Number 473.3.1 Adjacency Systems 483.3.2 Discretization of Sets with Respect to Adjacency 513.3.3 Euler Number 523.3.4 Complementarity 593.3.5 Multi-grid Convergence 603.4 The Euler Number of Microstructure Constituents 613.4.1 Counting Nodes in Open Foams 613.4.2 Connectivity of the Fibres in Non-woven Materials 633.5 Image Data 643.5.1 The Inverse Lattice 653.5.2 The Nyquist–Shannon Sampling Theorem 663.6 Rendering 693.6.1 Volume Rendering 693.6.1.1 Physical Background 693.6.1.2 Transfer function 703.6.1.3 Ray Casting 713.6.1.4 3D Texture Mapping 723.6.2 Surface Rendering 723.6.2.1 Properties of the Reconstructed Surface 723.6.2.2 Marching Cube Type Algorithms 733.6.2.3 The Wrapper Algorithm 753.6.2.4 Merging and Simplification of Surface Meshes 774 Image Processing 794.1 Fourier Transform of an Image 794.1.1 The Discrete Fourier Transform of a Discrete One-Dimensional Signal 794.1.2 Fast Fourier Transform 804.1.3 Extensions to Higher Dimensions 814.2 Filtering 824.2.1 Morphological Transforms of Sets 824.2.1.1 Minkowski Addition and Dilation 834.2.1.2 Minkowski Subtraction and Erosion 854.2.1.3 Mean Co-ordination Number of Sinter Particles 864.2.1.4 Morphological Opening and Closure 874.2.1.5 Top-Hat Transforms 894.2.1.6 Algebraic Opening and Closure 894.2.1.7 Aspects of Algorithmic Implementation 904.2.1.8 Handling of Edge Effects 924.2.1.9 Adaptable Morphology 934.2.2 Linear Filters 944.2.2.1 Linear Smoothing Filters 944.2.2.2 Linear Derivative Filters 984.2.3 Morphological Filters 1024.2.4 Rank Value Filters 1034.2.5 Diffusion Filters 1054.2.6 Geodesic Morphological Transforms 1074.2.6.1 Reconstruction by Erosion 1084.2.6.2 Reconstruction by Dilation 1094.2.6.3 Self-Dual Reconstruction 1104.2.6.4 H-Minima 1114.2.7 Distance Transforms 1114.2.7.1 Discrete or Chamfer Distance Transforms 1134.2.7.2 Euclidean Distance Transforms 1144.2.8 Skeletonization 1164.3 Segmentation 1204.3.1 Binarization 1214.3.1.1 Global Thresholding 1214.3.1.2 Local Thresholding 1234.3.1.3 Hysteresis 1254.3.1.4 Region Growing 1274.3.2 Connectedness, Connected Components and Labelling 1284.3.2.1 Connectedness 1284.3.2.2 Jordan Theorems 1324.3.2.3 A Simple Labelling Algorithm 1354.3.2.4 Advanced Labelling Techniques 1414.3.3 Watershed Transform 1434.3.4 Further Segmentation Methods 1485 Measurement of Intrinsic Volumes and Related Quantities 1495.1 Introduction 1495.2 Intrinsic Volumes 1505.2.1 Section Lattices and Translation Lattices 1515.2.2 Measurement of Intrinsic Volumes 1525.2.3 Discretization of the Translative Integral 1535.2.4 Discretization of the Integral over all Subspaces 1565.2.4.1 Simple Quadrature 1565.2.4.2 Fourier Expansion 1595.2.5 Shape Factors 1625.2.6 Edge Correction 1645.3 Intrinsic Volume Densities 1665.3.1 Estimation of Intrinsic Volume Densities for Macroscopically Homogeneous Random Sets 1675.3.2 Characterization of Anisotropy 1695.3.3 Mean Chord Length 1705.3.4 Structure Model Index 1715.3.5 Estimation of the Intrinsic Volume Densities for Macroscopically Homogeneous and Isotropic Random Sets 1725.3.6 Intrinsic Volume Densities of the Solid Matter of Two Natural Porous Structures 1765.4 Directional Analysis 1795.4.1 Inverse Cosine Transform 1805.4.2 Use of Pixel Configurations Carrying Directional Information 1825.4.3 Gradient and Hessian Matrix 1845.4.4 Maximum Filter Response 1855.4.5 Directional Analysis for Fibres in Ultra-High-Performance Concrete 1875.5 Distances Between Random Sets and Distance Distributions 1875.5.1 Spherical Contact Distribution Function and Related Quantities 1895.5.2 Stochastic Dependence of Constituents of Metallic Foams 1926 Spectral Analysis 1956.1 Introduction 1956.2 Second-Order Characteristics of a Random Volume Measure 1966.2.1 Covariance Function and Bartlett Spectrum 1976.2.2 Power Spectrum 2016.2.3 Measurement of the Covariance and the Power Spectrum 2026.2.4 Macroscopic Homogeneity and Isotropy 2036.2.5 Mean Face Width of an Open Foam 2056.2.6 Random Packing of Balls 2066.2.7 Particle Rearrangement During Sintering Processes 2076.3 Correlations Between Random Structures 2086.3.1 The Cross-Covariance Function 2096.3.2 Measurement of the Cross Covariance Function 2116.3.3 Spatial Cross-Correlation Between Constituents of Metallic Foams 2116.4 Second-Order Characteristics of Random Surfaces 2126.4.1 The Random Surface Measure 2136.4.2 The Bartlett Spectrum 2156.4.3 Power Spectrum 2186.4.4 Measurement of the Power Spectrum with Respect to the Surface Measure 2206.5 Second-Order Characteristics of Random Point Fields 2226.5.1 Point Fields and Associated Random Functions 2236.5.2 A Wiener–Khintchine Theorem for Point Fields 2246.5.3 Estimation of the Pair Correlation Function 2266.5.4 The Power Spectra of the Centres of Balls in Dense Packings 2307 Model-based Image Analysis 2337.1 Introduction,Motivation 2337.2 Point Field Models 2347.2.1 The Poisson Point Field 2347.2.2 Matern Hard-Core Point Fields 2357.2.3 Finite Point Fields Defined by a Probability Density 2357.2.3.1 Simulation of Finite Point Fields: Metropolis–Hastings 2377.2.3.2 Simulation of Finite Point Fields: Spatial Birth-and-Death Processes 2387.3 Macroscopically Homogeneous Systems of Non-overlapping Particles 2397.4 Macroscopically Homogeneous Systems of Overlapping Particles 2437.4.1 Intrinsic Volumes of Boolean Models in Rn 2457.4.2 Intrinsic Volumes of Boolean Models in R3 2487.4.3 Structure Model Index for Boolean Models in R3 2507.5 Macroscopically Homogeneous Fibre Systems 2517.5.1 Boolean Cylinder Model 2517.5.2 PET Stacked Fibre Non-woven Materials 2527.5.3 Carbon Paper 2557.6 Tessellations 2567.6.1 Geometric Properties of Tessellations of R3 2567.6.1.1 Mean Number of `-Faces Adjacent to a k-Face 2577.6.1.2 The Density of k-Faces 2587.6.1.3 Mecke’s Characteristics 2587.6.1.4 Cell-Based Characteristics 2597.6.2 Voronoi Tessellations 2607.6.2.1 Poisson Voronoi Tessellation 2607.6.2.2 Hard-Core Voronoi Tessellation 2617.6.3 Laguerre Tessellations 2617.6.3.1 Poisson–Laguerre Tessellations 2647.6.3.2 Laguerre Tessellations Generated by Random Packings of Balls 2647.6.4 The Weaire–Phelan Foam 2657.6.4.1 Random Perturbations of the Weaire–Phelan Foam 2667.6.5 Mean Values of Geometric Characteristics of Open Foams 2677.6.6 Modelling a Closed Polymer Foam 2707.6.7 Modelling an Open Ceramic Foam 2767.6.7.1 Modelling the Polyurethane Core 2777.6.7.2 Modelling the Coating 2788 Simulation of Material Properties 2818.1 Introduction 2818.2 Effective Conductivity of Polycrystals by Stochastic Homogenization 2828.3 Computation of Effective Elastic Moduli of Porous Media by FEM Simulation 2888.3.1 Fundamentals of Linear Elasticity 2888.3.2 Finite Element Method 2918.3.2.1 Discretization 2918.3.2.2 Numerical Solution of the Linear Elastic Problem 2928.3.3 Effective Stiffness Tensor Random Sets 2948.3.4 Effective Elastic Moduli of a Porous AluminaMaterial 296References 301Index 319

NIKOS NIKOLAIDIS, PhD, is a senior researcher in the Artificial Intelligence and Information Analysis Laboratory, Department of Informatics, Aristotle University of Thessaloniki, Greece. IOANNIS PITAS, PhD, is a professor in the Department of Informatics, Aristotle University of Thessaloniki, Greece.

"Explains numerous 3-D image processing, analysis, and visualization techniques, such as volume filtering, skeletonization and registration, and visualization." (SciTech Book News Vol. 25, No. 2 June 2001)

• Foreword VPreface XIIIConventions and Notation XV1 Introduction 12 Preliminaries 112.1 General Notation 112.1.1 Points and Sets in Euclidean Spaces 112.1.2 Curvatures 142.1.3 Measures and Measurable Spaces 172.2 Characteristics of Sets 182.2.1 The Euler Number and the Integral of Gaussian Curvature 182.2.2 The Mean Width and the Integral of the Mean Curvature 202.2.3 Intrinsic Volumes of Convex Bodies 222.2.4 Additive Extensions on the Convex Ring 242.2.5 The Principal Kinematic Formulae of Integral Geometry 252.3 Random Sets 262.3.1 Definition of Random Sets 272.3.2 Characteristics of Random Closed Sets 282.3.3 Random Point Fields 302.3.4 Random Tessellations 332.4 Fourier Analysis 342.4.1 Measurable Functions 342.4.2 Fourier Transform 362.4.3 Bochner’s Theorem 403 Lattices, Adjacency of Lattice Points, and Images 433.1 Introduction 433.2 Point Lattices, Digitizations and Pixel Configurations 433.2.1 Homogeneous Lattices 443.2.2 Digitization 453.2.3 Pixel Configurations 463.3 Adjacency and Euler Number 473.3.1 Adjacency Systems 483.3.2 Discretization of Sets with Respect to Adjacency 513.3.3 Euler Number 523.3.4 Complementarity 593.3.5 Multi-grid Convergence 603.4 The Euler Number of Microstructure Constituents 613.4.1 Counting Nodes in Open Foams 613.4.2 Connectivity of the Fibres in Non-woven Materials 633.5 Image Data 643.5.1 The Inverse Lattice 653.5.2 The Nyquist–Shannon Sampling Theorem 663.6 Rendering 693.6.1 Volume Rendering 693.6.1.1 Physical Background 693.6.1.2 Transfer function 703.6.1.3 Ray Casting 713.6.1.4 3D Texture Mapping 723.6.2 Surface Rendering 723.6.2.1 Properties of the Reconstructed Surface 723.6.2.2 Marching Cube Type Algorithms 733.6.2.3 The Wrapper Algorithm 753.6.2.4 Merging and Simplification of Surface Meshes 774 Image Processing 794.1 Fourier Transform of an Image 794.1.1 The Discrete Fourier Transform of a Discrete One-Dimensional Signal 794.1.2 Fast Fourier Transform 804.1.3 Extensions to Higher Dimensions 814.2 Filtering 824.2.1 Morphological Transforms of Sets 824.2.1.1 Minkowski Addition and Dilation 834.2.1.2 Minkowski Subtraction and Erosion 854.2.1.3 Mean Co-ordination Number of Sinter Particles 864.2.1.4 Morphological Opening and Closure 874.2.1.5 Top-Hat Transforms 894.2.1.6 Algebraic Opening and Closure 894.2.1.7 Aspects of Algorithmic Implementation 904.2.1.8 Handling of Edge Effects 924.2.1.9 Adaptable Morphology 934.2.2 Linear Filters 944.2.2.1 Linear Smoothing Filters 944.2.2.2 Linear Derivative Filters 984.2.3 Morphological Filters 1024.2.4 Rank Value Filters 1034.2.5 Diffusion Filters 1054.2.6 Geodesic Morphological Transforms 1074.2.6.1 Reconstruction by Erosion 1084.2.6.2 Reconstruction by Dilation 1094.2.6.3 Self-Dual Reconstruction 1104.2.6.4 H-Minima 1114.2.7 Distance Transforms 1114.2.7.1 Discrete or Chamfer Distance Transforms 1134.2.7.2 Euclidean Distance Transforms 1144.2.8 Skeletonization 1164.3 Segmentation 1204.3.1 Binarization 1214.3.1.1 Global Thresholding 1214.3.1.2 Local Thresholding 1234.3.1.3 Hysteresis 1254.3.1.4 Region Growing 1274.3.2 Connectedness, Connected Components and Labelling 1284.3.2.1 Connectedness 1284.3.2.2 Jordan Theorems 1324.3.2.3 A Simple Labelling Algorithm 1354.3.2.4 Advanced Labelling Techniques 1414.3.3 Watershed Transform 1434.3.4 Further Segmentation Methods 1485 Measurement of Intrinsic Volumes and Related Quantities 1495.1 Introduction 1495.2 Intrinsic Volumes 1505.2.1 Section Lattices and Translation Lattices 1515.2.2 Measurement of Intrinsic Volumes 1525.2.3 Discretization of the Translative Integral 1535.2.4 Discretization of the Integral over all Subspaces 1565.2.4.1 Simple Quadrature 1565.2.4.2 Fourier Expansion 1595.2.5 Shape Factors 1625.2.6 Edge Correction 1645.3 Intrinsic Volume Densities 1665.3.1 Estimation of Intrinsic Volume Densities for Macroscopically Homogeneous Random Sets 1675.3.2 Characterization of Anisotropy 1695.3.3 Mean Chord Length 1705.3.4 Structure Model Index 1715.3.5 Estimation of the Intrinsic Volume Densities for Macroscopically Homogeneous and Isotropic Random Sets 1725.3.6 Intrinsic Volume Densities of the Solid Matter of Two Natural Porous Structures 1765.4 Directional Analysis 1795.4.1 Inverse Cosine Transform 1805.4.2 Use of Pixel Configurations Carrying Directional Information 1825.4.3 Gradient and Hessian Matrix 1845.4.4 Maximum Filter Response 1855.4.5 Directional Analysis for Fibres in Ultra-High-Performance Concrete 1875.5 Distances Between Random Sets and Distance Distributions 1875.5.1 Spherical Contact Distribution Function and Related Quantities 1895.5.2 Stochastic Dependence of Constituents of Metallic Foams 1926 Spectral Analysis 1956.1 Introduction 1956.2 Second-Order Characteristics of a Random Volume Measure 1966.2.1 Covariance Function and Bartlett Spectrum 1976.2.2 Power Spectrum 2016.2.3 Measurement of the Covariance and the Power Spectrum 2026.2.4 Macroscopic Homogeneity and Isotropy 2036.2.5 Mean Face Width of an Open Foam 2056.2.6 Random Packing of Balls 2066.2.7 Particle Rearrangement During Sintering Processes 2076.3 Correlations Between Random Structures 2086.3.1 The Cross-Covariance Function 2096.3.2 Measurement of the Cross Covariance Function 2116.3.3 Spatial Cross-Correlation Between Constituents of Metallic Foams 2116.4 Second-Order Characteristics of Random Surfaces 2126.4.1 The Random Surface Measure 2136.4.2 The Bartlett Spectrum 2156.4.3 Power Spectrum 2186.4.4 Measurement of the Power Spectrum with Respect to the Surface Measure 2206.5 Second-Order Characteristics of Random Point Fields 2226.5.1 Point Fields and Associated Random Functions 2236.5.2 A Wiener–Khintchine Theorem for Point Fields 2246.5.3 Estimation of the Pair Correlation Function 2266.5.4 The Power Spectra of the Centres of Balls in Dense Packings 2307 Model-based Image Analysis 2337.1 Introduction,Motivation 2337.2 Point Field Models 2347.2.1 The Poisson Point Field 2347.2.2 Matern Hard-Core Point Fields 2357.2.3 Finite Point Fields Defined by a Probability Density 2357.2.3.1 Simulation of Finite Point Fields: Metropolis–Hastings 2377.2.3.2 Simulation of Finite Point Fields: Spatial Birth-and-Death Processes 2387.3 Macroscopically Homogeneous Systems of Non-overlapping Particles 2397.4 Macroscopically Homogeneous Systems of Overlapping Particles 2437.4.1 Intrinsic Volumes of Boolean Models in Rn 2457.4.2 Intrinsic Volumes of Boolean Models in R3 2487.4.3 Structure Model Index for Boolean Models in R3 2507.5 Macroscopically Homogeneous Fibre Systems 2517.5.1 Boolean Cylinder Model 2517.5.2 PET Stacked Fibre Non-woven Materials 2527.5.3 Carbon Paper 2557.6 Tessellations 2567.6.1 Geometric Properties of Tessellations of R3 2567.6.1.1 Mean Number of `-Faces Adjacent to a k-Face 2577.6.1.2 The Density of k-Faces 2587.6.1.3 Mecke’s Characteristics 2587.6.1.4 Cell-Based Characteristics 2597.6.2 Voronoi Tessellations 2607.6.2.1 Poisson Voronoi Tessellation 2607.6.2.2 Hard-Core Voronoi Tessellation 2617.6.3 Laguerre Tessellations 2617.6.3.1 Poisson–Laguerre Tessellations 2647.6.3.2 Laguerre Tessellations Generated by Random Packings of Balls 2647.6.4 The Weaire–Phelan Foam 2657.6.4.1 Random Perturbations of the Weaire–Phelan Foam 2667.6.5 Mean Values of Geometric Characteristics of Open Foams 2677.6.6 Modelling a Closed Polymer Foam 2707.6.7 Modelling an Open Ceramic Foam 2767.6.7.1 Modelling the Polyurethane Core 2777.6.7.2 Modelling the Coating 2788 Simulation of Material Properties 2818.1 Introduction 2818.2 Effective Conductivity of Polycrystals by Stochastic Homogenization 2828.3 Computation of Effective Elastic Moduli of Porous Media by FEM Simulation 2888.3.1 Fundamentals of Linear Elasticity 2888.3.2 Finite Element Method 2918.3.2.1 Discretization 2918.3.2.2 Numerical Solution of the Linear Elastic Problem 2928.3.3 Effective Stiffness Tensor Random Sets 2948.3.4 Effective Elastic Moduli of a Porous AluminaMaterial 296References 301Index 319

NIKOS NIKOLAIDIS, PhD, is a senior researcher in the Artificial Intelligence and Information Analysis Laboratory, Department of Informatics, Aristotle University of Thessaloniki, Greece. IOANNIS PITAS, PhD, is a professor in the Department of Informatics, Aristotle University of Thessaloniki, Greece.

"Explains numerous 3-D image processing, analysis, and visualization techniques, such as volume filtering, skeletonization and registration, and visualization." (SciTech Book News Vol. 25, No. 2 June 2001)

• Foreword VPreface XIIIConventions and Notation XV1 Introduction 12 Preliminaries 112.1 General Notation 112.1.1 Points and Sets in Euclidean Spaces 112.1.2 Curvatures 142.1.3 Measures and Measurable Spaces 172.2 Characteristics of Sets 182.2.1 The Euler Number and the Integral of Gaussian Curvature 182.2.2 The Mean Width and the Integral of the Mean Curvature 202.2.3 Intrinsic Volumes of Convex Bodies 222.2.4 Additive Extensions on the Convex Ring 242.2.5 The Principal Kinematic Formulae of Integral Geometry 252.3 Random Sets 262.3.1 Definition of Random Sets 272.3.2 Characteristics of Random Closed Sets 282.3.3 Random Point Fields 302.3.4 Random Tessellations 332.4 Fourier Analysis 342.4.1 Measurable Functions 342.4.2 Fourier Transform 362.4.3 Bochner’s Theorem 403 Lattices, Adjacency of Lattice Points, and Images 433.1 Introduction 433.2 Point Lattices, Digitizations and Pixel Configurations 433.2.1 Homogeneous Lattices 443.2.2 Digitization 453.2.3 Pixel Configurations 463.3 Adjacency and Euler Number 473.3.1 Adjacency Systems 483.3.2 Discretization of Sets with Respect to Adjacency 513.3.3 Euler Number 523.3.4 Complementarity 593.3.5 Multi-grid Convergence 603.4 The Euler Number of Microstructure Constituents 613.4.1 Counting Nodes in Open Foams 613.4.2 Connectivity of the Fibres in Non-woven Materials 633.5 Image Data 643.5.1 The Inverse Lattice 653.5.2 The Nyquist–Shannon Sampling Theorem 663.6 Rendering 693.6.1 Volume Rendering 693.6.1.1 Physical Background 693.6.1.2 Transfer function 703.6.1.3 Ray Casting 713.6.1.4 3D Texture Mapping 723.6.2 Surface Rendering 723.6.2.1 Properties of the Reconstructed Surface 723.6.2.2 Marching Cube Type Algorithms 733.6.2.3 The Wrapper Algorithm 753.6.2.4 Merging and Simplification of Surface Meshes 774 Image Processing 794.1 Fourier Transform of an Image 794.1.1 The Discrete Fourier Transform of a Discrete One-Dimensional Signal 794.1.2 Fast Fourier Transform 804.1.3 Extensions to Higher Dimensions 814.2 Filtering 824.2.1 Morphological Transforms of Sets 824.2.1.1 Minkowski Addition and Dilation 834.2.1.2 Minkowski Subtraction and Erosion 854.2.1.3 Mean Co-ordination Number of Sinter Particles 864.2.1.4 Morphological Opening and Closure 874.2.1.5 Top-Hat Transforms 894.2.1.6 Algebraic Opening and Closure 894.2.1.7 Aspects of Algorithmic Implementation 904.2.1.8 Handling of Edge Effects 924.2.1.9 Adaptable Morphology 934.2.2 Linear Filters 944.2.2.1 Linear Smoothing Filters 944.2.2.2 Linear Derivative Filters 984.2.3 Morphological Filters 1024.2.4 Rank Value Filters 1034.2.5 Diffusion Filters 1054.2.6 Geodesic Morphological Transforms 1074.2.6.1 Reconstruction by Erosion 1084.2.6.2 Reconstruction by Dilation 1094.2.6.3 Self-Dual Reconstruction 1104.2.6.4 H-Minima 1114.2.7 Distance Transforms 1114.2.7.1 Discrete or Chamfer Distance Transforms 1134.2.7.2 Euclidean Distance Transforms 1144.2.8 Skeletonization 1164.3 Segmentation 1204.3.1 Binarization 1214.3.1.1 Global Thresholding 1214.3.1.2 Local Thresholding 1234.3.1.3 Hysteresis 1254.3.1.4 Region Growing 1274.3.2 Connectedness, Connected Components and Labelling 1284.3.2.1 Connectedness 1284.3.2.2 Jordan Theorems 1324.3.2.3 A Simple Labelling Algorithm 1354.3.2.4 Advanced Labelling Techniques 1414.3.3 Watershed Transform 1434.3.4 Further Segmentation Methods 1485 Measurement of Intrinsic Volumes and Related Quantities 1495.1 Introduction 1495.2 Intrinsic Volumes 1505.2.1 Section Lattices and Translation Lattices 1515.2.2 Measurement of Intrinsic Volumes 1525.2.3 Discretization of the Translative Integral 1535.2.4 Discretization of the Integral over all Subspaces 1565.2.4.1 Simple Quadrature 1565.2.4.2 Fourier Expansion 1595.2.5 Shape Factors 1625.2.6 Edge Correction 1645.3 Intrinsic Volume Densities 1665.3.1 Estimation of Intrinsic Volume Densities for Macroscopically Homogeneous Random Sets 1675.3.2 Characterization of Anisotropy 1695.3.3 Mean Chord Length 1705.3.4 Structure Model Index 1715.3.5 Estimation of the Intrinsic Volume Densities for Macroscopically Homogeneous and Isotropic Random Sets 1725.3.6 Intrinsic Volume Densities of the Solid Matter of Two Natural Porous Structures 1765.4 Directional Analysis 1795.4.1 Inverse Cosine Transform 1805.4.2 Use of Pixel Configurations Carrying Directional Information 1825.4.3 Gradient and Hessian Matrix 1845.4.4 Maximum Filter Response 1855.4.5 Directional Analysis for Fibres in Ultra-High-Performance Concrete 1875.5 Distances Between Random Sets and Distance Distributions 1875.5.1 Spherical Contact Distribution Function and Related Quantities 1895.5.2 Stochastic Dependence of Constituents of Metallic Foams 1926 Spectral Analysis 1956.1 Introduction 1956.2 Second-Order Characteristics of a Random Volume Measure 1966.2.1 Covariance Function and Bartlett Spectrum 1976.2.2 Power Spectrum 2016.2.3 Measurement of the Covariance and the Power Spectrum 2026.2.4 Macroscopic Homogeneity and Isotropy 2036.2.5 Mean Face Width of an Open Foam 2056.2.6 Random Packing of Balls 2066.2.7 Particle Rearrangement During Sintering Processes 2076.3 Correlations Between Random Structures 2086.3.1 The Cross-Covariance Function 2096.3.2 Measurement of the Cross Covariance Function 2116.3.3 Spatial Cross-Correlation Between Constituents of Metallic Foams 2116.4 Second-Order Characteristics of Random Surfaces 2126.4.1 The Random Surface Measure 2136.4.2 The Bartlett Spectrum 2156.4.3 Power Spectrum 2186.4.4 Measurement of the Power Spectrum with Respect to the Surface Measure 2206.5 Second-Order Characteristics of Random Point Fields 2226.5.1 Point Fields and Associated Random Functions 2236.5.2 A Wiener–Khintchine Theorem for Point Fields 2246.5.3 Estimation of the Pair Correlation Function 2266.5.4 The Power Spectra of the Centres of Balls in Dense Packings 2307 Model-based Image Analysis 2337.1 Introduction,Motivation 2337.2 Point Field Models 2347.2.1 The Poisson Point Field 2347.2.2 Matern Hard-Core Point Fields 2357.2.3 Finite Point Fields Defined by a Probability Density 2357.2.3.1 Simulation of Finite Point Fields: Metropolis–Hastings 2377.2.3.2 Simulation of Finite Point Fields: Spatial Birth-and-Death Processes 2387.3 Macroscopically Homogeneous Systems of Non-overlapping Particles 2397.4 Macroscopically Homogeneous Systems of Overlapping Particles 2437.4.1 Intrinsic Volumes of Boolean Models in Rn 2457.4.2 Intrinsic Volumes of Boolean Models in R3 2487.4.3 Structure Model Index for Boolean Models in R3 2507.5 Macroscopically Homogeneous Fibre Systems 2517.5.1 Boolean Cylinder Model 2517.5.2 PET Stacked Fibre Non-woven Materials 2527.5.3 Carbon Paper 2557.6 Tessellations 2567.6.1 Geometric Properties of Tessellations of R3 2567.6.1.1 Mean Number of `-Faces Adjacent to a k-Face 2577.6.1.2 The Density of k-Faces 2587.6.1.3 Mecke’s Characteristics 2587.6.1.4 Cell-Based Characteristics 2597.6.2 Voronoi Tessellations 2607.6.2.1 Poisson Voronoi Tessellation 2607.6.2.2 Hard-Core Voronoi Tessellation 2617.6.3 Laguerre Tessellations 2617.6.3.1 Poisson–Laguerre Tessellations 2647.6.3.2 Laguerre Tessellations Generated by Random Packings of Balls 2647.6.4 The Weaire–Phelan Foam 2657.6.4.1 Random Perturbations of the Weaire–Phelan Foam 2667.6.5 Mean Values of Geometric Characteristics of Open Foams 2677.6.6 Modelling a Closed Polymer Foam 2707.6.7 Modelling an Open Ceramic Foam 2767.6.7.1 Modelling the Polyurethane Core 2777.6.7.2 Modelling the Coating 2788 Simulation of Material Properties 2818.1 Introduction 2818.2 Effective Conductivity of Polycrystals by Stochastic Homogenization 2828.3 Computation of Effective Elastic Moduli of Porous Media by FEM Simulation 2888.3.1 Fundamentals of Linear Elasticity 2888.3.2 Finite Element Method 2918.3.2.1 Discretization 2918.3.2.2 Numerical Solution of the Linear Elastic Problem 2928.3.3 Effective Stiffness Tensor Random Sets 2948.3.4 Effective Elastic Moduli of a Porous AluminaMaterial 296References 301Index 319