Structure of this Book
Chapters 2 through 8 provide foundations for digital geometry; they discuss grids, metrics, graphs, topology, and geometry and introduce concepts and methods used in digital geometry that are related to these subjects.
This book is organized as shown below.
Selected topics
Chapters 9–12: Straight Lines, Curves, Planes, Surfaces
Chapters 13–16: Hulls and Diagrams, Transformations (Geometrical, Morphological, Deformations)
Chapter 17: Other Properties and Relations
Chapters 9 through 13 discuss topics in digital geometry: digital “straightness” in Chapter 9; length and curvature of arcs and curves in Chapter 10; 3D straightness and planarity in Chapter 11; area and curvature of surfaces in Chapter 12; and hulls and diagrams in Chapter 13.
Chapter 14 discusses geometric operations on pictures; Chapter 15 discusses the application of operations of mathematic morphology to pictures; Chapter 16 discusses deformations of pictures; and Chapter 17 discusses picture properties and spatial relations.
Chapter 1 provides a general introduction and should be read first. Depending on the background of the reader, the different chapters may allow more or less independent reading. However, there are some obvious “clusters”, such as Chapters 4 and Chapter 5 (graph-theoretical models of pictures), Chapters 6 and 7 (basics of topology in the context of picture analysis), Chapters 8, 9, 11, 13, 14, and 17 (basics of geometry in the context of picture analysis), and Chapter 10 and 12 (performance evaluation of algorithms in digital geometry).
A third year undergraduate course about algorithms for digital pictures (in a program in electrical engineering, computer science or mathematics involving picture analysis or computer graphics) could focus on selected algorithms (see the List of Algorithms at the end of the book) and on the fundamentals that underlie these algorithms. The students would have the benefit of related mathematical topics and material for additional reading being provided in the same textbook. For example, the course could be structured as follows:
1. (1–2 lectures) Start with Section 1.1
2. (3–5 lectures) Follow this with Chapter 2, possibly shortening Section 2.3 and adding the example from Section 1.2.7 to the presentation of Section 2.4.
3. (3 lectures) Follow this with metrics and distance transforms (Chapter 3).
4. (2–3 lectures) Continue with the border tracing algorithm of Chapter 4 (with related property calculations; see, e.g., Section 8.1.6).
5. (2 lectures) Cover the frontier tracing algorithm of Chapter 5.
6. (2–3 lectures) Follow this with one or two DSS approximation algorithms (K1990 in Chapter 9, related to frontier tracing, or DR1995, related to border tracing of planar regions).
7. (3–8 lectures) Facilitate an extensive discussion about methods, algorithms, and performance evaluation for different arc length and curvature estimators (see Chapter 10).
8. (3–8 lectures) If time allows, algorithms for 3D region analysis could be added. This would include surface scanning from Section 8.4 (with related property calculations; see, e.g., Section 8.3.7), DPS approximation from Chapter 11, and surface area and curvature estimation with comparative performance evaluation from Chapter 12.
9. (3–6 lectures) Algorithms from Chapter 13 (hulls and diagrams; see also Section 1.2.9) or from Chapter 15 (morphologic operations) could also be added to the course.
Note that the exercises in this book are of varying complexity and should be selected carefully for such a course; however, all of the experimental exercises can be recommended for course work (assignments). The course could also cover other algorithms from the List of Algorithms (e.g., geometric transforms, which are not difficult to implement, or simple deformations, which require a good understanding of the “more challenging” concepts discussed in Chapter 16).
Graduate courses could focus on more specific topics clustered around selected sections in the book, such as the following:
(i) Picture Analysis and Topology (Chapter 2 as an introduction, then Chapters 4 through 7 and Chapter 16).
(ii) Multigrid Analysis of Property Estimators in Picture Analysis (basics from Chapter 2 and Chapter 3, including the example from Section 1.2.7, followed by multigrid subjects in Chapters 10, 12, and 17).
(iii) Combinatorial Picture Analysis (combinatorial subjects in Chapters 1 and 2 as an introduction, then Chapters 4 and 5, combinatorial subjects in Chapters 9 and 11, digital tomography in Chapter 14, and digital moments in Chapter 17).
(iv) The axiomatic approaches to different subdisciplines, especially the axiomatic theory of digital geometry in Chapter 14, could provide material for a graduate research seminar about Mathematical Fundamentals of Picture Analysis (see also the List of Axioms at the end of this book).
The extensive bibliography, with commented bibliography sections at the ends of the chapter, should also provide support for designing graduate student research seminars based on selected readings.