The new geometry compared with the two-view case is the ability to transfer from two views to a third: given a point correspondence over two views the position of the point in the third view is determined; and similarly, given a line correspondence over two views the position of the line in the third view is determined. This transfer property is of great benefit when establishing correspondences over multiple views. [1]

Like the fundamental matrix, it can be computed from image correspondences alone, without requiring knowledge of the motion or calibration.

\[T_i = a_ib_4^T − a_4b_iT\]

Point–point–point correspondence. Given point \(\mathbf{x}\) in view 1, \(\mathbf{x}^\prime\) in view 2, and \(\mathbf{x}^{\prime\prime}\) in view 3, …

\[[\mathbf{x}^\prime]_{\times} \Big(\sum\limits_i x^i \mathbf{T}_i\Big) [ \mathbf{x}^{\prime\prime}]_{\times} = \mathbf{0}_{3 \times 3}\]

Let \(\mathbf{x} = (x_1, y_1, 1)\), \(\mathbf{x}^\prime = (x_2, y_2, 1)\), and \(\mathbf{x}^{\prime\prime} = (x_3, y_3, 1)\).

\[\begin{bmatrix} 0 & -1 & y_2 \\ 1 & 0 & -x_2 \\ -y_2 & x_2 & 0 \end{bmatrix} \Big(\sum\limits_i x^i \mathbf{T}_i\Big) \begin{bmatrix} 0 & -1 & y_3 \\ 1 & 0 & -x_3 \\ -y_3 & x_3 & 0 \end{bmatrix}\] \[M(\mathbf{x}) = \Big(\sum\limits_i x^i \mathbf{T}_i\Big)\]

References

[1] Hartley Zisserman. Multiple View Geometry.

[2] Amnon Shashua. Algebraic Functions For Recognition. TPAMI, 1995. PDF.