Image Processing Reference
In-Depth Information
13
World Geometry by Direction in
N
Dimensions
The origins of the concept ”perspective” as it came to be used in the field of computer
vision can be traced to two key fields, photogrammetry and geometry. Although the
problems and the concepts studied in the latter since the early 1800s accounts for
a large portion of contemporary computer vision, these fields have a long history
that covers several schools of thought and art. Many of these concepts date back at
least to the Renaissaince or even earlier. In 1480, Leonardo da Vinci formulated a
definition of the perspective images [56] that was very close to our contemporary
understanding of it. Scientists continued the work of da Vinci on projections and
geometry. Albrecht Duerer created an instrument that could be used to create a true
perspective drawing in 1525 [93]. Girard Desargues contributed to the foundation of
projective geometry, Traite de la section perspective (1636). Other significant contri-
butions were by Johan Heinrich Lambert, with the treatise “Perspectiva Liber” (The
Free Perspective, 1759), and the establishment of the relationship between projec-
tive geometry and photogrammetry (1883), by R. Sturms and G. Haick [56]. In the
subsequent sections, we will outline the basic principles of measurements of world
geometry from photographs, to help us understand the structure of a world scene.
13.1 Camera Coordinates and Intrinsic Parameters
We assume that we have a perspective camera. This can be imagined as a box with
a very small hole through which the light rays hit the image plane behind. For this
reason the perspective camera is also called the
pinhole camera
. It is an ideal
camera
obscura
of the kind that was carried on horses or on the back of the artist in the
medieval age. The pinhole camera is illustrated in Fig. 13.1 with the image plane
shown in green.
First, we make some notational precisons. Points are represented by capital let-
ters such as
P, O, Q
. Since the geometry of points and vectors between them in the
ordinary Euclidean space
E
3
is discussed in this chapter, an alternative notation will
be used for vector representation. Consequently, to represent a vector between the
