2.1  Projective geometry


Projective geometry serves as a mathematical framework for 3D multi-view imaging and 3D computer graphics. It is used to model the image-formation process, generate synthetic images, or reconstruct 3D objects from multiple images. To model lines, planes or points in a 3D space, usually the Euclidean geometry is employed. However, a disadvantage of the Euclidean geometry is that points at infinity cannot be modeled and are considered as a special case. This special case can be illustrated by using a perspective drawing of two parallel lines. In perspective, two parallel lines, e.g., a railroad track, meet at infinity at the vanishing point (see Figure 2.1). However, the intersection of the parallel lines at infinity is not easily modeled by the Euclidean geometry. A second disadvantage of Euclidean geometry is that projecting a 3D point onto an image plane requires a perspective scaling operation. As the scale factor is a parameter, the perspective scaling requires a division that becomes a non-linear operation. Therefore, the use of the Euclidean geometry should be avoided.

PIC
Figure 2.1 Two parallel rails intersect in the image plane at the vanishing point.


Projective geometry constitutes an attractive framework to circumvent the above disadvantages of the Euclidean geometry. In Euclidean space, a point defined in three dimensions is represented by a 3-element vector (X,Y,Z)T . In the projective space, the same point is described using a 4-element vector (X1,X2,X3,X4)T such that

X  = X1∕X4,   Y = X2 ∕X4,  Z = X3 ∕X4,           (2.1)
where X4≠0. Usually, the coordinates (X,Y,Z)T and (X1,X2,X3,X4)T are called inhomogeneous coordinates and homogeneous coordinates, respectively.

As a generalization, the mapping from a point in the n-dimensional Euclidean space to an (n + 1)-dimensional projective space can be written as

              T                        T
(◟X1,X2,◝..◜.,Xn-)◞ →  (λ◟X1,-λX2,◝..◜.,λXn,-λ)◞,
  Euclidean space         projective space
(2.2)

where λ≠0 corresponds to a free scaling parameter. This free scaling parameter λ is often called the homogeneous scaling factor. Using the presented projective geometry framework, we now introduce the pinhole camera model.