Introduction

What are vanishing points?

With the assumption of perfect projection, e.g. with a pin-hole camera, a set of parallel lines in the scene is projected onto a set of lines in the image that meet in a common point. This point of intersection is called the vanishing point. A vanishing point can be a finite (real) point or an infinite (ideal) point on the image plane. Vanishing points which lie on the same plane in the scene define a line in the image, the so-called the vanishing line. Figure 1 shows the three vanishing points and vanishing lines of a cube, where a vanishing point at infinity is depicted by a direction on the image plane. How vanishing points are analytically defined in the 3D projective space $P^3$ and in the projective space$P^2$ of the image plane is explained in (Vanishing points by Stan Birchfield) and [7]. In this tutorial a finite point $(x,y)^T$ on the image plane is expressed in homogeneous coordinates as $(x,y,1)^T$ and points at infinity in homogeneous coordinates as $(x',y',0)^T$.

Figure 1: The three vanishing points and vanishing lines of a cube.

Why are vanishing points important?

A man-made environment has two characteristic properties: Many lines in the scene are parallel and various edges in the scene are orthogonal. In an indoor environment this is true for e.g. shelves, doors, windows and corridor boundaries. In an outdoor environment e.g. streets, buildings and pavements satisfy this assumption. This means that vanishing points provide strong cues for inferring information about the 3D structure of a scene. If e.g. the camera geometry is known, each vanishing point corresponds to an orientation in the scene and vice versa.

Therefore, the understanding and interpretation of a man-made environment can be significantly simplified by the detection of vanishing points. For instance, this has been done in the field of navigation of robots and autonomous vehicles [15], in the field of object reconstruction [5,14] or for the calibration of cameras [4,9,16]. In [14] is shown that three orthogonal vanishing points of an object, e.g a building, can be used for a direct euclidean reconstruction on the basis of multiple views of the object.