The next natural step from the stereo formalism and the fundamental matrix is a multi-camera situation (i.e. 3 or more projections). The trifocal tensor approach is such an extension and maintains a similar projective geometry spirit. This model has been proposed and developed by Sashua [46], Hartley [23] and Faugeras [19] among others. Figure 5 represents the imaging scenario.

Here, the trifocal plane is formed by the three optic centers ** COP_{1}**,

To map points, one merely considers intersections of mapped lines. The
tensor
can be considered as a
**3 x 3 x 3** cube
operator (i.e. defined by 27 scalars in total). It can also be
represented as the concatenation of three **3 x 3**
matrices: ** G_{1}**,

If a set of corresponded points are known in each of the 3 images,
the tensor can be estimated in a similar way as the fundamental
matrix. For instance, one can perform a least-squares linear
computation to recover the 27 parameters [23]
[46]. However, the trifocal tensor's 27 scalar
parameters are not all independent unknowns. Not every
**3 x 3
x 3** cube is a tensor. It too has constraints (like the
fundamental matrix) and really has only 18 degrees of freedom. The
above linear methods for recovering the tensor do not impose the
constraints and can therefore produce invalid tensors.

By making an appeal to Grassmann-Cayley algebra, Faugeras gracefully derives the algebraic constraints on trifocal tensors which can be viewed as higher order (4th degree) polynomials on the parameters [19]. The 9 constraints are folded into a nonlinear optimization scheme which recovers the 18 remaining degrees of freedom of the tensor from image correspondences.