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Next: Endending the two reference Up: Affine Structure Previous: Koenderink and Van Doorn

Extension to 2 reference plane

Figure 4: Affine Structure from Motion using two reference planes
\begin{figure}\centerline{\psfig{figure=2refplane-affine.ps}}\end{figure}

The above construction cannot be directly extended to the projective case because parallelism is not a projective invariant and we cannot compute ratios along parallel directions.

However, as a first step, we can extend the single reference plane method to a two reference plane method.

Let $P_j$, $j=1,2,3,4$ be four non-coplanar points in space and $p_j \leftrightarrow p_j'$ be their corresponding projections in the two views. The points $P_1,P_2,P_3$ and $P_2,P_3,P_4$ lie on twodifferent planes. Therefore, we can account for the motion of all points coplanar with either of these two planes.

Let $P$ be a points of interest not coplanar with either of the reference planes and let $\tilde{P}$ and $\hat{P}$ be its projections on to the two reference planes along the viewing direction in the first image. The points $P$, $\tilde{P}$ and $\hat{P}$ project on to $p'$, $\tilde{p}'$ and $\hat{p}'$ respectively, and an alternative affine structure invariant for $P$ can be computed as

\begin{displaymath}
\alpha_p = \frac{\mid P - \tilde{P} \mid}{\mid P - \hat{P} \mid} =
\frac{\mid p' - \tilde{p}' \mid}{\mid p' - \hat{p}' \mid}
\end{displaymath}

The main advantage in using the two reference plane construction is that $p'$, $\tilde{p}'$ and $\hat{p}'$ lie along one line in the second image.


next up previous
Next: Endending the two reference Up: Affine Structure Previous: Koenderink and Van Doorn
Subhashis Banerjee 2002-02-18