There are different approaches to digital topology. One, developed by Rosenfeld and his followers in the late 1960s and 1970s, is concerned with the adjacency of pixels, or the conditions under which one pixel could be said to be “next to” another one. After such a determination is made, the next step is to describe path connectedness and connected components. This approach has led to some powerful algorithms and theorems describing curves, surfaces, and separation in multi-dimensional digital spaces.

An alternative approach comes from work by Riesz in the early 1900s and Efremovic in the 1950s, which formally defined the properties of “nearness” and proximity in a topological space. It is this approach that forms the basis for Peters’ text, and indeed the author is one of the principal modern contributors to the field. One would expect the text to be authoritative, and to present a clear introduction to the field. Unfortunately, one would be disappointed.

The main problem is that although the mathematics is not new, its use for digital image processing is still a niche area, and there is little published material that explains the uses of this topology in image processing. Confronted with this theory for the first time, it is reasonable to ask: (1) What does this approach provide in terms of efficiency and practicality? (2) Does this approach provide algorithms for which no other approach is suitable? (3) Are there imaging problems for which this approach is the best possible? (4) How easy is it to implement these algorithms? The text answers none of these questions.

Two computer systems are used: MATLAB and Mathematica. These are two of the most mature and widely used mathematical programs, but their use in the text is mostly restricted to elementary and standard imaging. I think the author would have done better to choose an open-source system--for example, Java and its library ImageJ, or Python and its associated imaging libraries--and to provide a complete library of topological algorithms in the chosen language. Even though MATLAB and Mathematica are deservedly popular, they are not used universally, and an open-source solution would allow any reader to experiment. As an aside, MATLAB code is typeset in colored fixed-width font and Mathematica code is typeset in boldface italics. This makes the text appear messy. This is not consistent: some MATLAB code is shown in plain text and some is in italics.

I expect the book is intended as a textbook, as there are exercises throughout. Some of them are confusingly labeled as “thought problems” when they are nothing of the sort. Number 25 on page 162, for example, asks the reader to write a MATLAB script. The author claims that a “thought problem” is meant to encourage thinking “outside the box”; however I can’t see much difference in style between the different problem types. Many of the exercises seem randomly scattered throughout the text, and don’t seem to conform to any consistent pedagogy. I don’t, for example, see any scaffolding from elementary to more difficult.

Most chapters begin with a standard imaging topic: “Edge Detection,” for example, provides a brief introduction to the topic with some examples in MATLAB and Mathematica, and then moves into topology with no explanation of how the topological topic relates to the imaging one.

Similarly, chapter 7, “Separation of Image Regions and Set Patterns,” begins with a discussion of “proximal frameworks” and “proximal framology.” Immediately afterward, there is an introduction to mathematical morphology, in which the author fails to distinguish between the morphology based on Minkowski addition (as developed by Matheron, Serra, and their followers), and MATLAB’s “bwmorph” function, which uses lookup tables. After this whistle-stop tour through some basic morphology, it’s back to separation axioms and Hausdorff spaces, with a few imaging examples to show what some of the definitions might mean in the context of an image. However, there is no actual image processing performed, no MATLAB or Mathematica examples of this theory being applied to images, and no connection between mathematical morphology and the topology that precedes and follows it. This pattern is repeated for every chapter.

The result is that the book seems like two independent texts shuffled together: one a basic imaging text and the other a text on proximal topology. The connections between the two are at most tenuous and at worst nonexistent.

I would also quibble with the author’s style. One example is on page 20, with a discussion of standard MATLAB syntax for accessing elements of an array. These are called “tricks,” which they most definitely are not, and are labeled (trick.1) to (trick.10). Most enumerated lists in the text are of this sort.

It may be that proximal topology as developed in this text will have a profound and lasting impact on digital image processing and image analysis, but if so, this book is not a good advertisement for it.