2.2. Notation and Problem Statement
We start with the notation used in this chapter.
2.2.1. Pinhole camera model
A 2D point is denoted by m = [u, v]T. A 3D point is denoted by M = [X, Y, Z]T. We use 
to denote the augmented vector by adding 1 as the last element: 
and 
. A camera is modeled by the usual pinhole (see Figure 2.1): The image of a 3D point M, denoted by m, is formed by an optical ray from M passing through the optical center C and intersecting the image plane. The three points M, m, and C are collinear. In Figure 2.1, for illustration purpose, the image plane is positioned between the scene point and the optical center, which is mathematically equivalent to the physical setup under which the image plane is in the other side with respect to the optical center. The relationship between the 3D point M and its image projection m is given by
Equation 2.1
Equation 2.2
Equation 2.3
Figure 2.1. Pinhole camera model.
where s is an arbitrary scale factor, (R, t), called the extrinsic parameters, is the rotation and translation that relates the world coordinate system to the camera coordinate system, and A is called the camera intrinsic matrix, with (u0, v0) the coordinates of the principal point, α and β the scale factors in image u and v axes, and γ the parameter describing the skew of the two image axes. The 3 × 4 matrix P is called the camera projection matrix, which mixes both intrinsic and extrinsic parameters. In Figure 2.1, the angle between the two image axes is denoted by θ, and we have γ = α cot θ. If the pixels are rectangular, then θ = 90° and γ = 0.
The task of camera calibration is to determine the parameters of the transformation between an object in 3D space and the 2D image observed by the camera from visual information (images). The transformation includes:
- Extrinsic parameters (sometimes called external parameters): orientation (rotation) and location (translation) of the camera (R, t);
- Intrinsic parameters (sometimes called internal parameters): characteristics of the camera (α, β, γ, u0, v0).
The rotation matrix, although consisting of nine elements, has only three degrees of freedom. The translation vector t obviously has 3 parameters. Therefore, there are six extrinsic parameters and five intrinsic parameters, totaling 11 parameters.
We use the abbreviation A–T for (A–1)T or (AT)–1.
2.2.2. Absolute conic
Now let us introduce the concept of the absolute conic. For more details, refer to [7, 15].
A point x in 3D space has projective coordinates 
= [x1, x2, x3, x4]T. The equation of the plane at infinity, Π∞, is x4 = 0. The absolute conic Ω is defined by a set of points satisfying the equation
Equation 2.4
Let x∞ = [x1, x2, x3]T be a point on the absolute conic (see Figure 2.2). By definition, we have 
. We also have 
and 
. This can be interpreted as a conic of purely imaginary points on Π∞. Indeed, let x = x1/x3 and y = x2/x3 be a point on the conic, then x2 + y2 = –1, which is an imaginary circle of radius 
.
Figure 2.2. Absolute conic and its image.
An important property of the absolute conic is its invariance to any rigid transformation. Let the rigid transformation be 
. Let x∞ be a point on Ω. By definition, its projective coordinates: 
with 
. The point after the rigid transformation is denoted by 
, and
Thus, 
is also on the plane at infinity. Furthermore, 
is on the same Ω because
The image of the absolute conic, denoted by ω, is also an imaginary conic and is determined only by the intrinsic parameters of the camera. This can be seen as follows. Consider the projection of a point x∞ on Ω, denoted by m∞, which is given by
It follows that
Therefore, the image of the absolute conic is an imaginary conic and is defined by A–T A–1. It does not depend on the extrinsic parameters of the camera.
If we can determine the image of the absolute conic, then we can solve the camera's intrinsic parameters, and the calibration is solved.
We show several ways in this chapter to determine ω, the image of the absolute conic.