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Stereology and Stochastic Geometry: (v. 28 Computational Imaging and Vision)

Stereology and Stochastic Geometry: (v. 28 Computational Imaging and Vision)


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About the Book

Written for the scientist-practitioner, this text presents in a concise, step-by-step form, the derivations of all the formulas of classical stereology ("quantitative microscopy") along with those of such modern constructs as star volume and the disector. Anatomists, histologists, materials scientists and geo-scientists should find this work a readable explanation of the theory underlying their procedures. It covers the formulas for methods based on sectioning as well as those based on plane projections as used in transmission electron microscopy and point projections as used in photogrammetry. It reintroduces the useful Cahn-Hilliard estimators for the variances of stereological measurements, originally published by the National Bureau of Standards. To give the reader a perspective on their accuracy, new Monte Carlo bench tests of their performance have been included. It provides a short discussion of the practical aspects of fractal geometry for the stereologist.

Table of Contents:
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.

Review :
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)


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Product Details
  • ISBN-13: 9781402016875
  • Publisher: Kluwer Academic Publishers
  • Publisher Imprint: Kluwer Academic Publishers
  • Height: 240 mm
  • Returnable: N
  • Weight: 1022 gr
  • ISBN-10: 1402016875
  • Publisher Date: 01 Nov 2003
  • Binding: Hardback
  • Language: English
  • Series Title: v. 28 Computational Imaging and Vision
  • Width: 160 mm


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