Three view geometry

Considering three views it is, of course, possible to group them in pairs and to get the two view relationships introduced in the last section. Using these pairwise epipolar relations, the projection of a point in the third image can be predicted from the coordinates in the first two images. This is illustrated in Figure 3.7. The point in the third image is determined as the intersection of the two epipolar lines. This computation, however, is not always very well conditioned. When the point is located in the trifocal plane (i.e. the plane going through the three centers of projection), it is completely undetermined.

Fortunately, there are additional constraints between the images of a point in three views. When the centers of projection are not coinciding, a point can always be reconstructed from two views. This point then projects to a unique point in the third image, as can be seen in Figure 3.7, even when this point is located in the trifocal plane.

Figure 3.7: Relation between the image of a point in three views. The epipolar lines of points ${\tt m}$ and ${\tt m}'$ could be used to obtain ${\tt m}''$. This does, however, not exhaust all the relations between the three images. For a point located in the trifocal plane (i.e. the plane defined by ${\tt C}, {\tt C}'$ and ${\tt C}''$) this would not give a unique solution, although the 3D point could still be obtained from its image in the first two views and then be projected to ${\tt m}''$. Therefore, one can conclude that in the three view case not all the information is described by the epipolar geometry. These additional relationships are described by the trifocal tensor.

For two views, no constraint is available to restrict the position of corresponding lines. Indeed, back-projecting a line forms a plane, the intersection of two planes always results in a line. Therefore, no constraint can be obtained from this. But, having three views, the image of the line in the third view can be predicted from its location in the first two images, as can be seen in Figure 3.8.

Figure 3.8: Relation between the image of a line in three images. While in the two view case no constraints are available for lines, in the three view case it is also possible to predict the position of a line in a third image from its projection in the other two. This transfer is also described by the trifocal tensor.

Similar to what was derived for two views, there are multi linear relationships relating the positions of points and/or lines in three images [181]. The coefficients of these multi linear relationships can be organized in a tensor which describes the relationships between points [176] and lines [63] or any combination thereof [65]. Several researchers have worked on methods to compute the trifocal tensor (e.g. see [193,194]).

The trifocal tensor ${\bf T}$ is a $3 \times 3 \times 3$ tensor. It contains 27 parameters, only 18 of which are independent due to additional nonlinear constraints. The trilinear relationship for a point is given by the following equation^C1:

$$m_i(m'_j m''_k T_{i33} - m''_k T_{ij3} - m'_j T_{i3k} + T_{ijk})=0$$ (C31)

Any triplet of corresponding points should satisfy this constraint.

A similar constraint applies for lines. Any triplet of corresponding lines should satisfy:

$$l_i \sim l'_j l''_k T_{ijk}$$