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### Relation between projection matrices and image homographies

The homographies that will be discussed here are collineations from . A homography describes the transformation from one plane to another. A number of special cases are of interest, since the image is also a plane. The projection of points of a plane into an image can be described through a homography . The matrix representation of this homography is dependent on the choice of the projective basis in the plane.

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,

 (C13)

Now, if the camera projection matrix is , then the projection of onto the image is
 (C14)

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:

 (C15)

where is the rotation matrix that describes the relative orientation from the camera with respect top the one.

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

 (C16)

where is the projection of the center of projection of the first camera (in this case, ) in image . This point is called the epipole, for reasons which will become clear in Section 3.2.1.

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

 (C17)

This equation gives an important relationship between the homographies for all possible planes. Homographies can only differ by a term . This means that in the projective case the homographies for the plane at infinity are known up to 3 common parameters (i.e. the coefficients of in the projective space).

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.

Next: Deviations from the camera Up: The projection matrix Previous: The projection matrix   Contents
Marc Pollefeys 2002-11-22