Stereology and Stochastic Geometry

ISBN-13: 9789048164554 / Angielski / Miękka / 2011 / 488 str.

John E. Hilliard;L.R. Lawson

Opis

Somebody had to do it. The Chinese speak of deep water wells called "grandfather wells" because they take three generations of diggers to complete. Imagine the thought of such a well being abandoned incomplete by the third generation. What a loss This book is like a grandfather well except that it has taken only two generations, John Hilliard's and mine, to finish. When I saw his manuscript lying in a heap, I decided that I must spend the time to put it and his notes into a publishable form. Now, it is done. This book is mostly about performing spatial measurements through the statistical sampling of images; it is a text on classical stereology as John Hilliard saw it. His vision of the subject was broad. Consequently, its title is broad too. It presents this subject and some of its modem extensions from the classical perspective of the one of the founders of the field, and my first advisor at Northwestern University, John Hilliard. There is nothing new in this book but much that may have been lost over time. It rediscovers many useful discussions about such subjects as the variances of stereo logical measurements, anisotropy etc. It recovers some of the dialogues between John Hilliard and his students on such topics as fractals and Monte Carlo simulations. It recaptures a little of John Hilliard's unique and subtle wit.

Recenzje

From the reviews:

"Mathematical stereology aims to make inference about the properties of random sets in Euclidean space using numerical characteristics of their lower-dimensional sections or projections. The main tools are based on application of methods from stochastic geometry (in particular point processes), convex geometry and geometric measure theory. ... The book under review aims to provide an introduction to stereology for non-mathematicians." (Ilya S. Molchanov, Zentralblatt MATH, Vol. 1108 (10), 2007)

Spis treści

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. Avera