equation is indicative of the fact that points and lines can be exchanged in many theories of projective geometry; such theories are termed dual theories. For example, the cross product of two lines, expressed in homogeneous coordinates, yields their intersecting point, and the cross-product of a pair of points gives the line between them.

Note that we can easily convert from homogeneous to inhomogeneous coordinates, simply by dividing through by the third element, thus [x1, x2, x3]T maps to

[ x1 , x2 ]T . A key point about homogeneous coordinates is that they allow the relevant

x3 x3

transformations in the imaging process to be represented as linear mappings, which of course are expressed as matrix-vector equations. However, although the mapping between homogeneous world coordinates of a point and homogeneous image coordinates is linear, the mapping from homogeneous to inhomogeneous coordinates is non-linear, due to the required division.

The use of homogeneous coordinates fits well with the relationship between image points and their associated back-projected rays into the scene space. Imagine a mathematical (virtual) image plane at a distance of one metric unit in front of the center of projection, as shown in Fig. 2.4. With the camera center, C, the homogeneous coordinates [x, y, 1]T define a 3D scene ray as [λx, λy, λ]T , where λ is the unknown distance (λ > 0) along the ray. Thus there is an intuitive link between the depth ambiguity associated with the 3D scene point and the equivalence of homogeneous coordinates up to an arbitrary non-zero scale factor.

Extending the idea of thinking of homogeneous image points as 3D rays, consider the cross product of two homogeneous points. This gives a direction that is the normal of the plane that contains the two rays. The line between the two image points is the intersection of this plane with the image plane. The dual of this is that the cross product of two lines in the image plane gives the intersection of their associated planes. This is a direction orthogonal to the normals of both of these planes and is the direction of the ray that defines the point of intersection of the two lines in the image plane. Note that any point with its third homogeneous element zero defines a ray parallel to the image plane and hence meets it at infinity. Such a point is termed a point at infinity and there is an infinite set of these points [x1, x2, 0]T that lie on the line at infinity [0, 0, 1]T ; Finally, note that the 3-tuple [0, 0, 0]T has no meaning and is undefined. For further reading on homogeneous coordinates and projective geometry, please see [21] and [12].

2.3.2 Perspective Projection Camera Model

We now return to the perspective projection (central projection) camera model and we note that it maps 3D world points in standard metric units into the pixel coordinates of an image sensor. It is convenient to think of this mapping as a cascade of three successive stages:

1.A 6 degree-of-freedom (DOF) rigid transformation consisting of a rotation, R (3 DOF), and translation, t (3 DOF), that maps points expressed in world coordinates to the same points expressed in camera centered coordinates.

We denote Pp as the 3 × 4 perspective projection matrix, defined by the value of f , in the above equation. If we consider an abstract image plane at f = 1, then points on this plane are termed normalized image coordinates4 and from Eq. (2.2), these are given by

2.3.2.3 Camera Modeling: Image Sampling

Typically, the image on the image plane is sampled by an image sensor, such as a CCD or CMOS device, at the locations defined by an array of pixels. The final part of camera modeling defines how that array is positioned on the [xc , yc ]T image plane, so that pixel coordinates can be generated. In general, pixels in an image sensor are not square and the number of pixels per unit distance varies between the xc and yc directions; we will call these scalings mx and my . Note that pixel positions have their origin at the corner of the sensor and so the position of the principal point (where the principal axis intersects the image plane) is modeled with pixel coordinates [x0, y0]T . Finally, many camera models also cater for any skew,5 s, so that the mapping into pixels is given by:

We denote Pc as the 3 × 3 projective matrix defined by the five parameters mx , my , s, x0 and y0 in the above equation.

2.3.2.4 Camera Modeling: Concatenating the Projective Mappings

We can concatenate the three stages described in the three previous subsections to give

λx = Pc Pp Pr X

or simply

where λ is non-zero and positive. We note the following points concerning the above equation

4We need to use a variety of image coordinate normalizations in this chapter. For simplicity, we will use the same subscript n, but it will be clear about how the normalization is achieved.

5Skew models a lack of orthogonality between the two image sensor sampling directions. For most imaging situations it is zero.