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Digital geometry in image processing
Mukhopadhyay J., Das P., Chattopadhyay S., Bhowmick P., Chatterji B., Taylor & Francis, Inc., Boca Raton, FL, 2013. 318 pp. Type: Book (978-1-466505-67-4)
Date Reviewed: Dec 13 2013

I wanted to like this book very much: the topic is fascinating, and deserves all the exposure it can get. The “gold standard” is the splendid tome by Klette and Rosenfeld [1], but the field could do with another text to provide a different, maybe more up-to-date, introduction to the field.

However, this book is not it. It covers a brave spread of topics, but is let down by confusing definitions, a lack of appropriate explanations and useful diagrams, and a confusion of purpose. It is published in the “IIT Kharagpur Research Monograph Series,” in which it is the fifth volume, but each chapter ends with exercises, which seem to push it toward being a textbook. If the latter, then the explanations and examples need to be clearer and better scaffolded; if the former, then it could be significantly tightened. The text as a whole could have benefited from some more thorough editing.

There are seven chapters: “Digital Topology: Fundamentals,” “Distance Functions in Digital Geometry,” “Digitization of Straight Lines and Planes,” “Digital Straightness and Polygonal Approximation,” “Parametric Curve Estimation and Reconstruction,” “Medial Axis Transform,” and “Modeling of a Voxelated Surface.”

The first chapter, on digital topology, discusses adjacency and connectedness in the discrete spaces Z2 and Z3, mentions the Jordan curve theorem (but without proof), and covers topological thinning (where an object is shrunk to a smaller object that retains only the topological structure of the original) and skeletonization. There are many algorithms for these in both two and three dimensions, but few of them are discussed. The safe algorithm given in Section 1.4.1.1 uses a subfield approach in which, at any stage, only a certain subset of foreground pixels is marked for deletion. This would have been a good place to introduce subfield or subiteration approaches to thinning. Note that Section 1.5 is headed “Euler Characteristics,” which is wrong.

The diagrams, certainly in the first chapter, are not as useful as they could be. For example, Figures 1.9 and 1.10 contain two diagrams each, both very small, and with the important part embedded in a larger grid. In chapter 2, many diagrams are copied from earlier work by the authors; an opportunity to redesign those diagrams using new software was missed.

I found chapter 2 (on distance functions) to be messy and confusing. An example is Definition 2.16, where a distance function is given without any prior justification. Note that there are several metrics known for discrete grids, including lattice-based grids such as the hexagonal grid (in 2D), and the face-centered cubic grid (in 3D). Not using the theory of geometric lattices means that the authors must not only adopt a somewhat ad hoc approach, but that they are also unable to generalize their results to different grids. Another curious error is on page 48, Lemma 2.6, where a condition of positive definiteness is that “2{n-1}p ≤ 2{n-1}.” The authors are too fond of obstructive symbols; instead of using a description based mainly on words, they use only symbols, such as in Definition 2.24 for a “wave front set.” I could probably work out what they mean here, but surely a simple word-based description (with an accompanying diagram or example) would be better for the reader. Note also in this chapter that Figure 2.12 is completely unreadable. We also find definitions that are somewhat nonstandard: for example, a neighborhood of a point p is defined as the points that are adjacent to p “in some sense.” Surely it would be simpler to first formally define adjacency, then neighborhood, then connectedness (and show this to be an equivalence relation), and finally a connected component as the connected equivalence class of a point.

As a passing comment, the index is not complete: I couldn’t, for example, find “adjacency,” surely one of the most fundamental definitions.

Chapters 3 through 5 consider straight lines, straightness in general, and the polygonal approximation of curves. These are important topics, but all too often material is introduced without sufficient reason. For example, there is considerable discussion of continued fractions, but nothing of the number theory that underpins their theory; hence, it is not made clear why continued fractions are needed. Note in Section 3.4.1 that a plane is defined by the equation z = ax+by+c; this is not a useful equation (how could you express the plane y = 0 in this form?). It should of course be ax+by+cz=d with a, b, c not all zero.

The text is also made harder to read by the authors’ use of abbreviations: “If there is a USF of an n-DPS P, then...” Given that digital geometry and topology is somewhat of a niche area, the authors do not make the readers’ job easier by expecting them to keep up with abbreviations. They occur seldom enough that they could be written out in full each time. In such places, the text gives the impression that the authors are not really writing for clarity, but more to display what they know.

Chapter 6 explores the medial axis transform, which can be defined as the union of the centers of all circles of maximum size that fit within the image. This is a venerable algorithm, and in many ways has been superseded by other methods. What this chapter (and possibly the rest of the book) needs would be some computer programs so that the relative efficiencies of different methods could be compared. The medial axis was introduced originally for shape analysis, but that can be done by topological or morphological methods (the latter are not discussed in this text.)

The final chapter investigates methods of approximating a surface by voxels (a “voxel” being the 3D analog of a pixel). This is a fascinating topic, and the authors cover a lot of ground in this chapter, with numerous examples; however, it is let down, like the rest of the book, by confusing terminology, notation, and descriptions.

In spite of my remarks, there is a lot in this book to like: it covers a great deal of material, and with some careful editing and better attention paid to the presentation, it would be a valuable and instructive addition to the literature. I hope to see a second edition--this book deserves one, and the material deserves plenty of readers.

Reviewer:  Alasdair McAndrew Review #: CR141804 (1402-0123)
1) Klette, R.; Rosenfeld, A. Digital geometry: geometric methods for digital picture analysis. Elsevier, San Francisco, CA, 2004.
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