As an image is obtained by perspective projection, the relation between points
belonging to a plane in 3D space and their projections
in the image is mathematically expressed by a homography
. The matrix of this homography is found as follows. If the plane is given by
and the point
of is represented as
, then
belongs to if and only if
. Hence,

Consequently, .

Note that for the specific plane the homographies are simply given by .

It is also possible to define homographies which describe the transfer from one image to the other for points and other geometric entities located on a specific plane. The notation will be used to describe such a homography from view to for a plane . These homographies can be obtained through the following relation and are independent to reparameterizations of the plane (and thus also to a change of basis in ).

In the metric and Euclidean case,
and the plane at infinity is
. In this case, the homographies for the plane at infinity can thus be written as:

In the projective and affine case, one can assume that
(since in this case is unknown). In that case, the homographies
for all planes; and thus,
. Therefore can be factorized as

Note that this equation can be used to obtain and from , but that due to the unknown relative scale factors can, in general, not be obtained from and . Observe also that, in the affine case (where ), this yields .

Combining equations (3.14) and (3.16), one obtains

Equation (3.16) also leads to an interesting interpretation of the camera projection matrix:

(C18) | |||

(C19) | |||

(C20) |

In other words, a point can thus be parameterized as being on the line through the optical center of the first camera (i.e. ) and a point in the reference plane . This interpretation is illustrated in Figure 3.4.