AMS Mathematical Review of the book "Géométrie discrète et images numériques" (French)

This is the AMS Mathematical Review for the book: Géométrie discrète et images numériques [Discrete geometry and numerical images] Edited by David Coeurjolly, Annick Montanvert and Jean-Marc Chassery. Traité IC2. Série Signal et Image. [IC2 Treatise. Series Signal and Image] Hermes Science Publications/Lavoisier, Paris, 2007. 401+xii pp.

ISBN: 978-2-7462-1643-3 65D18 (51N99 68U10 94A08) This is a collective book stemming from the French colloquium on Discrete Geometry in Imaging created in 1991, which became international in 1996 under the name of “Discrete Geometry in Computer Science”. The book is structured in five parts. The first part (chapters 2 through 4) presents the general mathematical foundations of discrete geometry and topology: arithmetic (chapter 2, which is later referred to in chapters 6, 7 and 11), topology (chapter 3, which is referred to in chapter 8) and combinatorics (chapter 4, the foundation of chapter 12).

The second part (chapters 5 to 8) introduces discrete geometry and discrete topology, which are the foundations of the next part. The third part (chapters 9 through 12) presents the different representations and their mappings: the medial axis (chapter 9), representations by projections obtained by accumulation (and in particular the Mojette transform, chapter 10), polygonization and polyhedrization (chapter 11) and approximation of surfaces by triangulation (chapter 12).

The medial axis and the Mojette transform are key representations, whose applications are further described in the fifth part. The fourth part (chapters 13 and 14) focuses on image analysis from either original images or their representations introduced in the third part. In the fifth and last part (chapters 15 through 19), the authors present applications of the previous chapters: medial axis based coding techniques for progressive transmission (chapter 15), applications of the Mojette transform to image encoding and protection for storage and transmission (chapter 16), extraction of information from 3D images (plane sections of voxels, chapter 17) and visualization of 3D discrete models (through implict surfaces with skeletons, chapter 18), applications of discrete geometry to medical imaging (chapter 19) and generalization of discrete geometry objects and operators to multiresolution grids (chapter 20).

This book presents very interesting mathematical methodological tools from discrete geometry (and discrete topology) and their applications to digital imaging (analysis, processing and synthesis) in a very clear, didactic and interesting fashion. I enjoyed very much reading this book that brings a very refreshing and inspiring view of digital imaging, that is unfortunately not so widespread.

It should benefit both specialists of imaging and geometers by its methodological contents of high quality, and its many different examples of applications of discrete geometry and topology including the very trendy domain of medical imaging. I would recommend it as a textbook in image analysis and image processing, and especially as an eye opening book presenting an alternative to statistical based digital imaging through strongly motivated methodological tools.

I hope that English and Spanish translations of this book will see light and extend its audience towards this purpose of being a widely spread textbook. While the authors mention both the graph-based approach to discrete topology and the topological approach to digital topology, they cover in much more details the former than the later.

The interested reader can refer to [Kong, T. Y.; Rosenfeld, A. Digital topology: a comparison of the graph-based and topological approaches. Topology and category theory in computer science (Oxford, 1989), 273–289, Oxford Sci. Publ., Oxford Univ. Press, New York, 1991.] for a comparison between the two approaches.