Computing the epipolar lines

 Given a point m that has coordinates (u₁,v₁) in the first retinal plane $\Pi$,it is known that its correspondence and the fundamental matrix are related by:

$$\left(u \quad v \quad 1 \right) \, \mbox{\bf F} \left(\begin{array} {c} u_1 \\ v_1\\ 1 \end{array}\right) = 0.$$

This equation can be expanded to

$$(F_{11} u_1 + F_{12} v_1 + F_{13}) \, u + (F_{21} u_1 + F_{22} v_1 + F_{23}) \, v + (F_{31} u_1 + F_{32} v_1 + F_{33}) = 0,$$

where u₁ and v₁ are known entities, u and v are variables. This is the equation of a line in the second retinal plane. It is the epipolar line on which m' must lie.

Given a point m that has coordinates (u₁,v₁) in the first retinal plane $\Pi$,to compute the epipolar line l$_{\bf m}$ that contains m, we must first compute the epipole e $\in \Pi$.

e $\equiv$ intersection point of all epipolar lines in $\Pi$.

• The epipole e in $\Pi$ $\equiv$ projection of C' onto $\Pi$ $\equiv$ null vector of F.
• The epipole $\mbox{\bf e}'$ in $\Pi'$ $\equiv$ projection of C onto $\Pi'$ $\equiv$ null vector of F$^\top$.

The coordinates of l$_{\bf m}$ can then be computed as the cross-product of e and m.