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Endending the two reference plane method to projective

Figure 5: Affine Structure from Motion using two reference planes
\begin{figure}\centerline{\psfig{figure=2refplane-proj.ps}}\end{figure}

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

\begin{displaymath}
\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}
\end{displaymath}

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