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.
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
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
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.