Geometric Tomography


by Richard J. Gardner

Cambridge University Press, 2006


Reviewed by:  Arjan Kuijper

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“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 new edition includes numerous updates and improvements, with some 300 new references bringing the total to over 800.”

This text, taken from the back cover of the book gives a true description of the book. It is a comprehensive study, covering the field of geometric tomography from principles to recent state-of-the-art, including problems that are still open.

The main topics are discussed in the text of the 9 chapters. Following each chapter, extensive notes are given of excursions, scientific elaborations, difficulties, and historic facts of prominent researchers.

The book starts with a chapter on background material and then discusses Parallel X-rays of planar convex bodies and Parallel X-rays in n dimensions. This chapters relate to tomography in medical imaging. Here, for instance, the Sepp Logan phantom and the Radon transform are discussed.

Chapters 3 and 4 deal mainly with classical convexity, including Projections and projection functions, and Projection bodies and volume inequalities.

The following two chapters deal with Point X-rays and Chord functions and equichordal problems, which are rather different from the parallel case. The last three chapters discuss various cases related to the ‘core’ chapters 3-6. They discuss Sections, section functions, and point X-rays, Intersection bodies and volume inequalities, and Estimates from projection and section functions.

Three appendices provide more details on Mixed volumes and dual mixed volumes, Inequalities, and Integral transforms.

This book is the second edition, and the differences with the one published 11 years earlier are mainly in the number of references – indicating the active research in the field in geometric tomography. Since most references occur in the notes accompanying the chapters, this doesn’t harm the readability of the text. This edition demonstrates the need for a standard work on the basics of geometric tomography.

The author has been professor of Mathematics at Western Washington University since 1991 and this is clearly visible in the way the chapters are structured: After a short introduction of the topic and the required knowledge, one encounters Definition i.j.k., Theorem i.j.k+1., Lemma i.j.k+2., and, of course, for the latter two cases the corresponding proofs. This makes the book a rigorous treatment of the topic.

For people in computer vision who are interested in geometric tomography and like such a mathematical approach, this book fulfils all their needs. However, it requires the reader to go through some tough and non-trivial mathematics. This book may also serve as a useful reference for scientists and engineers who need to understand concepts of geometrical tomography applied to various fields. Two such examples are: Segmentation of tomographic data without image reconstruction (IEEE Transactions on Medial Imaging, 11, 102-110, 1992) by J.-P. Thirion, or Fundamental stereological formulae based on isotropically orientated probes through fixed points with applications to particle analysis (Journal of Microscopy 153, 249-267, 1989), by E.B. Jensen and H.J.G. Gundersen.