This book takes the novel approach of considering the geometry of image processing by the visual cortex system of the brain. The editors emphasize that it is not the geometry of the shape of the brain per se that the researchers contributing to this text are interested in, but rather the underlying (differential) geometry of the neural connections of the brain necessary for the human processing of visual optics. Readers expect many contributing chapters in a compendium book on a given topic. At nearly 400 pages, this book has lot of content but only eight chapters. A number of the authors of the individual chapters are quite famous in their respective fields.

The utilization of differential geometry to understand perception and image processing has a longer history than one would imagine. The editors collect a number of key efforts in this regard, such as the famous early works of Koenderink, who studied perceptual spaces using differential geometry during the 1980s. Hoffman’s research [1] is also of note; many don’t realize that he started exploring the concept in the 1960s. I would recommend reading Lappin and van de Grind’s chapter on visual forms in space-time [2] for an extensive treatment of this interesting history. To this I would like to add that the novelty of incorporating differential geometry for understanding how computation works in general has also been studied. See, for example, Soare [3], who shows the surprising connection that computability theory has with differential geometry, and further solved a problem in algebraic geometry based on recursion theory. This latter result should motivate researchers in many diverse fields to pursue understanding their fundamental concepts based on differential geometry.

Chapter 1, by Petitot, provides a comprehensive survey of the perceptual geometry literature with four subsections concentrating on the 1970s, 1980s, 1990s, and 2000s, respectively. Chapter 2, written by the famous research duo Koenderink and van Doorn (whose names need no introduction in this field), presents an approach that analyzes shading cues, which avoids the problems that previous research has encountered. In chapter 3, Kunsberg and Zucker study the famous “shape from shading” problem found in computer vision (a field for which author Zucker is well known), and parameterize the local matching shapes that could be consistent with a given shaded patch perceived. Their approach allows for the simplification of the equations derived and a reduction of ambiguity in the results obtained.

The editors themselves wrote chapter 4. They start from the earliest known structures, such as Lie groups with sub-Riemannian metrics, to mathematically model the underlying architecture of the primary visual cortex. A marriage between neurology and vision is presented using a probabilistic approach, and is compared to a statistical account of the co-occurrence of edges in following natural images. Finally, a nonlinear principal component analysis performs a spectral decomposition of the fundamental operator in order to explain perceptual theory. Chapter 5 is a highly mathematical chapter investigating cuspless sub-Riemannian geodesics within a Euclidean motion group, which is then applied to modeling in neuropsychology.

As a student of perceptual organization and gestalt psychology, I enjoyed chapter 6, which discusses psychophysics, gestalts, and Turing test games. One such game is to present the subject with an alignment detection procedure and elicit responses from the subject to trick it into a false alarm. To these authors, this is a visual version of the Turing test, which seeks to determine whether a human can detect whether a machine or a human provided a given response. Chapter 7 is an extensive treatment on invariance of visual cues in mammalian visual systems. While the treatment is very mathematical, the concepts are fundamental and explained nicely. Chapter 8 surveys a modeling notion that a simple set of general neural mechanisms is sufficient to explain the seemingly complex relationship between the geometry, statistics, and visual perception provided by the primary visual cortex. The author has co-authored a well-known book in the field [4].

This book is an essential read for anyone conducting serious research in the field of neurogeometry of visual perception.