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2.1 Geometric primitives and transformations 29

Before we can intelligently analyze and manipulate images, we need to establish a vocabulary
for describing the geometry of a scene. We also need to understand the image formation
process that produced a particular image given a set of lighting conditions, scene geometry,
surface properties, and camera optics. In this chapter, we present a simplified model of such
an image formation process.
Section 2.1 introduces the basic geometric primitives used throughout the book (points,
lines, and planes) and the geometric transformations that project these 3D quantities into 2D
image features (Figure 2.1a). Section 2.2 describes how lighting, surface properties (Fig-
ure 2.1b), and camera optics (Figure 2.1c) interact in order to produce the color values that
fall onto the image sensor. Section 2.3 describes how continuous color images are turned into
discrete digital samples inside the image sensor (Figure 2.1d) and how to avoid (or at least
characterize) sampling deficiencies, such as aliasing.
The material covered in this chapter is but a brief summary of a very rich and deep set of
topics, traditionally covered in a number of separate fields. A more thorough introduction to
the geometry of points, lines, planes, and projections can be found in textbooks on multi-view
geometry (Hartley and Zisserman 2004; Faugeras and Luong 2001) and computer graphics
(Foley, van Dam, Feiner et al. 1995). The image formation (synthesis) process is traditionally
taught as part of a computer graphics curriculum (Foley, van Dam, Feiner et al. 1995; Glass-
ner 1995; Watt 1995; Shirley 2005) but it is also studied in physics-based computer vision
(Wolff, Shafer, and Healey 1992a). The behavior of camera lens systems is studied in optics
(M¨ oller 1988; Hecht 2001; Ray 2002). Two good books on color theory are (Wyszecki and
Stiles 2000; Healey and Shafer 1992), with (Livingstone 2008) providing a more fun and in-
formal introduction to the topic of color perception. Topics relating to sampling and aliasing
are covered in textbooks on signal and image processing (Crane 1997; J¨ ahne 1997; Oppen-
heim and Schafer 1996; Oppenheim, Schafer, and Buck 1999; Pratt 2007; Russ 2007; Burger
and Burge 2008; Gonzales and Woods 2008).
A note to students: If you have already studied computer graphics, you may want to
skim the material in Section 2.1, although the sections on projective depth and object-centered
projection near the end of Section 2.1.5 may be new to you. Similarly, physics students (as
well as computer graphics students) will mostly be familiar with Section 2.2. Finally, students
with a good background in image processing will already be familiar with sampling issues
(Section 2.3) as well as some of the material in Chapter 3.

2.1 Geometric primitives and transformations

In this section, we introduce the basic 2D and 3D primitives used in this textbook, namely
points, lines, and planes. We also describe how 3D features are projected into 2D features.
More detailed descriptions of these topics (along with a gentler and more intuitive introduc-
tion) can be found in textbooks on multiple-view geometry (Hartley and Zisserman 2004;
Faugeras and Luong 2001).

2.1.1 Geometric primitives

Geometric primitives form the basic building blocks used to describe three-dimensional shapes.
In this section, we introduce points, lines, and planes. Later sections of the book discuss