Stereology and Stochastic Geometry

Abstract

Preface. 1: Images and Patterns. 1.1. Introduction: what is stereology about? 1.2. Variables and Invariances. 1.3. Quantitative Attributes of Patterns. 1.4. Images as Slices and Projections. 1.5. Bibliography. 2: Mathematical Preliminaries. 2.1. Review of Some Integral Functions. 2.2. Integral Transforms. 2.3. Probability Theory. 2.4. Some Theoretical Distributions. 2.5. Properties of a Hyper-Sphere. 2.6. Bibliography. 2.7. Problems. 3: Definitions, Sets and Measures. 3.1. Sets, Properties and Mappings. 3.2. Rings of Sets. 3.3. Measure and Related Topics. 3.4. Minkowski Algebra. 3.5. The Connectivity Number. 3.6. Bibliography. 4: Random Probes. 4.1. Randomness, Invariance and Other Matters. 4.2. The Point Probe in One, Two and Three Dimensions. 4.3. The Linear Probe in Two Dimensions. 4.4. The Linear Probe in Three Dimensions. 4.5. The Planar Probe in Three Dimensions. 4.6. Combined Sampling Operations. 4.7. Comments on Alternate Derivations. 4.8. Vertical Sections. 4.9. Problems. 5: General Shape-independent Relationships. 5.1. What is shape-independence. 5.2. The interaction matrix. 5.3. The point probe in one, two and three dimensions. 5.4. The linear probe in one two and three dimensions. 5.5. The planar probe in three dimensions. 5.6. Curvature. 5.7. Summary of relationships. 5.8. Bibliography for Chapters 4 and 5. 5.9. Problems. 6: Shape-dependent Intercept Distributions. 6.1. Introduction. 6.2. Interception of a Linear Probe with Circles. 6.3. Interceptions of a linear probe with stripes. 6.4. Interceptions of a linear probe with spheres. 6.5. Interceptions of a planar probe with spheres. 6.6. Interceptions of a planar probe with disks. 6.7. Interceptions of a linear probe with plates. 6.8. Interception of a planar probe with plates. 6.9. Interception of linear and planar probes with shells. 6.10. Integral Measures of Standard Shapes. 6.11. Bibliography. 6.12. Problems. 7: Relationships for Projected Images. 7.1. Relationships for Parallel-Projected Volumes. 7.2. Relationships for Projected Surfaces, Area and Length. 7.3. Number Per Unit Volume. 7.4. Star Measures. 7.5. Bibliography. 7.6. Problems. 8: Stereological Sampling and Statistics. 8.1. Introduction to Sampling Statistics. 8.2. Average and Variance of a Function of Random Variables. 8.3. Variances in the Estimation of the Moments of a Distribution. 8.4. Variances of Correlated Sums. 8.5. Error in the Estimation of Reciprocal Volume Ratios. 8.6. Estimation of Boundary Area per Unit Volume. 8.7. Estimation of Line Length per Unit Volume. 8.8. Bibliography. 8.9. Problems. 9: Multidimensional Generalizations. 9.1. Introduction. 9.2. Expected Density of Interceptions on a Random Probe. 9.3. Mean Projections of a Body. 9.4. Average Size of an Interception in a Body. 9.5. Superposition of Arrays. 9.6. Crofton's Second Theorem. 9.7. Intercept Distributions in N-Dimensional Spheres. 9.8. Bibliography. 9.9. Problems.10: Topics in Stochastic Geometry. 10.1. Geometric Integrals. 10.2. Translational and Rotational Orientation: Anisotropy Revisited. 10.3. Topological Relationships for Some Tessellated Structures. 10.4. Projections of Curved Surfaces. 10.5. Second Order Stereology. 10.6. Fractals in Practical Stereology. 10.7. Bibliography. 10.8. Non-Problems. Appendix: Additional Topics in Analysis. A.1. Systems of Numbers. A.2. Additional Material on Sets. A.3. Topics Concerning Measure. A.4. Bibliography. Index.