PERSPECTIVE CAMERAS

Figure 9.1(a) simplifies the imaging geometry of conventional perspective cameras: an in-coming ray of light through the center of the lens is focused on the receptive surface, producing an upside-down and right-and-left reversed image. The symmetry axis of the lens is called the optical axis. Figure 9.1(b) shows its abstraction, using the coordinate system with the origin O at the lens center and the z-axis along the optical axis. The receptive surface is also called the image plane. Its intersection with the optical axis is called the principal point , and the distance from the lens center O is generally known as the focal length. It is not necessarily the same as the optical focal length of the lens itself, but both coincide for scenes infinitely far apart (for near scenes, the exact focus point is determined by solving what is called the lens equation). Let θ be the incidence angle, i.e., the angle between the optical axis and the ray of light passing through the lens center O. As shown in Fig. 9.1(b), the ray focuses on the image plane at distance d from the principal point given by

d = f tan θ, (9.1)

where f is the focal length. The mapping from the outside scene to the image plane defined in this way is called perspective projection. In the perspective projection model of Fig. 7.2 in Chapter 7, the image plane is placed before the lens center, but the geometric relationships are the same.

145

f d

θ O x

z

(a) (b)

FIGURE 9.1 (a) Camera imaging geometry. (b) Perspective projection.

z

O x

P

p θ

z

O x

P

p θ

(a) (b)

FIGURE 9.2 Spherical camera models. (a) Central image sphere for omnidirectional cameras. (b) Perspective image sphere for perspective cameras.

Since all points on an incoming ray are projected to the same point, a camera is essen-tially a device to record incoming rays. Hence, wherever the image plane is placed, or even if it is a nonplanar shape, the recorded information is the same. From this point of view, the simplest mathematical model is to consider a sphere surrounding the lens center O and regard the (color or intensity) value of the ray as recorded at its intersection with the sphere (Fig. 9.2(a)). We call such a sphere the image sphere.

We should note that for the usual camera only those rays incoming from the front are recorded, while for this spherical camera model incoming rays from all directions are recorded. This means that Fig. 9.2(a) is a mathematical idealization of omnidirectional (or catadioptric) cameras. For recording only the front rays, we place the sphere so that it passes through the lens center O (Fig. 9.2(b)). Hereafter, we call the sphere of Fig. 9.2(a) the central image sphere and the sphere of Fig. 9.2(b) the perspective image sphere.

Let f be the radius of the image sphere in Fig. 9.2(b). If we consider an image plane that passes through the center of the sphere and is orthogonal to the optical axis, the correspondence between the image sphere and the image plane is 1 to 1 and given by stereographic projection, as shown in Fig. 9.3(a); a point p on the image sphere is mapped to the intersection p^′ of the image plane z = f with the line passing through p and the origin O (֒→ Fig. 4.2 in Chapter 4 and Fig 8.2 in Chapter 8). This stereographic projection is actually an inversion with respect to a sphere surrounding the origin O with radius√

2f (the dotted circle in Fig. 9.3), which we call the inversion sphere. If a point p on the image sphere is inverted with respect to this sphere, it is mapped, by definition, to a point p^′ on

Perspective cameras  147

O x

z

p p’

z=f

θ

θ/2

O x

z

p p’

z=2f

θ

(a) (b)

FIGURE 9.3 Stereographic projection of a sphere onto a plane. (a) Perspective camera model. (b) Fisheye lens camera model.

A

B C

α

β γ

a b

c S

FIGURE 9.4 Spherical triangle ABC on a unit sphere.

the line Op such that |Op^′| = 2f²/|Op|. We can see that this point is on the image plane by the following reasoning.

The inversion sphere, the image sphere, and the image plane share a circle C of radius f as their intersection. The circle C is on the inversion sphere, so it is unchanged by the inversion. Since a sphere is inverted to a sphere and since the origin O is inverted to infinity, a sphere that passes through O is inverted to a sphere of an infinite radius, i.e., a plane, that contains the circle C. Hence, the inversion of the image sphere coincides with the image plane. Thus, the stereographic projection results in an inversion (֒→ Exercise 9.1).

Traditional World 9.1 (Spherical trigonometry) As is well known, the shortest path that connects two points on a sphere is the great circle (= circle that has the same radius as the sphere) passing through them. A spherical triangle is obtained by connecting three points on a sphere by great circles, and the study of spherical triangles is known as spherical trigonometry. Let α, β, and γ be the interior angles (= the angles made by the tangents to the great circles) at vertex A, B, and C, respectively, of a spherical triangle on a unit sphere.

Let a, b, and c be the lengths (= the angles made by the vectors from the sphere center) of the sides opposite to A, B, and C, respectively (Fig. 9.4). The following relationships are well known:

sin α

sin a =sin β

sin b =sin γ

sin c, (9.2)

cos a = cos b cos c + sin b sin c cos α, cos α = − cos β cos γ + sin β sin γ cos a. (9.3)

These correspond to the law of sines, sin α

a = sin β

b =sin γ

c , (9.4)

and the law of cosines,

a²= b²+ c²− 2bc cos α, (9.5)

of a planar triangle, where α, β, and γ are the interior angles of the vertices and a, b, and c are the lengths of their opposite sides, respectively. Hence, Eqs. (9.2) and (9.3) are called the law of sines and the law of cosines, respectively, of a spherical triangle. As is intuitively evident, the sum of the interior angles of a spherical triangle is larger than π. It is known that the area S of this spherical triangle is given by

S = α + β + γ − π. (9.6)

If the sphere has radius r, the area S is magnified r² times.