Geometric Tomography
Geometric Tomography
Geometric tomography deals with the retrieval of information about a geometric object from data concerning its projections (shadows) on planes or cross-sections by planes. It is a geometric relative of computerized tomography, which reconstructs an image from X-rays of a human patient. The subject overlaps with convex geometry and employs many tools from that area, including some formulas from integral geometry. It also has connections to discrete tomography, geometric probing in robotics and to stereology. This comprehensive study provides a rigorous treatment of the subject. Although primarily meant for researchers and graduate students in geometry and tomography, brief introductions, suitable for advanced undergraduates, are provided to the basic concepts. More than 70 illustrations are used to clarify the text. The book also presents 66 unsolved problems. Each chapter ends with extensive notes, historical remarks, and some biographies. This edition includes numerous updates and improvements, with some 300 new references bringing the total to over 800.
- Over 70 illustrations
- Comprehensive bibliography has over 800 references
- Well-written and clear; the opening chapters are accessible to advanced undergraduates
Reviews & endorsements
praise for the first edition … 'The work is written in a very clear style, is well organized, and is amply provided with helpful illustrations and diagrams.' Wm. J. Firey, SIAM Review
'This is an extremely useful monograph, collecting together all the currently available material on geometric tomography … The book is well organized and easy to read, with many illustrations, and copious notes to each chapter … I recommend the book highly as a valuable introduction and continuing reference.' Peter McMullen, Bulletin of the London Mathematical Society
'… A valuable reference source. Libraries should have this book.' A. G. Ramm, Inverse Problems Newsletter
'This well-written book describes the recent theory … it includes copious notes and a comprehensive bibliography.' Mathematika
'The book is self contained, readable and can be warmly recommended not only for specialists.' European Mathematical Society Newsletter
'… this book is a must for all those who are or plan to work on geometric tomography …' Acta Scientiarum Mathematicarum
Product details
September 2006Hardback
9780521866804
516 pages
234 × 156 × 33 mm
0.857kg
Available
Table of Contents
- Preface to the second edition
- Preface
- 0. Background material
- 1. Parallel X-rays of planar convex bodies
- 2. Parallel X-rays in n dimensions
- 3. Projections and projection functions
- 4. Projection bodies and volume inequalities
- 5. Point X-rays
- 6. Chord functions and equichordal problems
- 7. Sections, section functions, and point X-rays
- 8. Intersection bodies and volume inequalities
- 9. Estimates from projection and section functions
- Appendix A. Mixed volumes and dual mixed volumes
- Appendix B. Inequalities
- Appendix C. Integral transforms
- References
- Notation
- Author index
- Subject index.
