next up previous
Next: Algebra of the two Up: Affine Structure Previous: Extension to 2 reference

Endending the two reference plane method to projective

Figure 5: Affine Structure from Motion using two reference planes

Since lines are projective invariants, the two reference plane method can easily be extended to deal with the projective case.

Consider the above figure. The left epipole is denoted as $V_l$ and the ray $PV_1$ projects on to the line $p'V_l$ which is an epipolar line.

The points $p'$, $\tilde{p}'$, $\hat{p}'$ and $V_l$ are collinear and projectively related to $P$, $\tilde{P}$, $\hat{P}$ and $V_1$, and therefore have the same cross-ratio. The projective structure invariant can thus be defined as

\alpha_p = \frac{\mid P - \tilde{P} \mid}{\mid P - \hat{P} \...
\frac{\mid V_l - \hat{p}' \mid}{\mid V_l - \tilde{p}' \mid}

Note that when the center of projections are at infinity the epipole $V_l$ also becomes a point at infinity and the structure invariant $\alpha_p$ reduces to the affine structure invariant.

Subhashis Banerjee 2002-02-18