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Next: Extension to 2 reference Up: Affine Structure Previous: Change of basis

Koenderink and Van Doorn

Figure 3: Affine Structure from Motion
\begin{figure}\centerline{\psfig{figure=afs.ps}}\end{figure}

Instead of choosing ${\bf E_{3}} = {\bf X_3 - X_0}$, KVD choose ${\bf E_{k}} = {\bf k}$ (i.e. the direction of viewing in the first frame). Since ${\bf e_k} = {\bf ME_k} = {\bf0}$, the projection of ${\bf E_{k}}$ in the first image is degenerate reducing it to a single point. Thus, only two basis vectors are chosen in the first image

\begin{displaymath}
{\bf\Delta x_i} = \alpha_i {\bf e_1} + \beta_i {\bf e_2}
\end{displaymath}

In the second image, the third axis vector is no longer degenerate, given by ${\bf e'_k} = {\bf ME'_k} = {\bf MAE_k}$. ${\bf e'_k}$ is actually an epipolar line. If we use ${\bf e'_1}$ and ${\bf e'_2}$ to predict the position where each point would appear in image 2, as if they lay on plane $\{{\bf E_{1}},{\bf E_{2}}\}$, we get

\begin{displaymath}
{\bf\hat{x}'_i} = {\bf x'_0} + \alpha_i {\bf e'_1} + \beta_i {\bf e'_2}
\end{displaymath}

the disparity between the predicted position and the observed position

\begin{displaymath}
{\bf x'_i} = {\bf x'_0} + \alpha_i {\bf e'_1} + \beta_i {\bf e'_2} + \gamma_i {\bf e'_k}
\end{displaymath}

is solely due to the $\gamma_i$ component

\begin{displaymath}
{\bf x'_i} - {\bf\hat{x}'_i} = \gamma_i {\bf e'_k}
\end{displaymath}

Geometrically, according to the notations of the figure, the projections of a fourth point $P$ and an arbitrary fifth point $Q$ form two similar trapezoids $P\tilde{P}p'\tilde{p}'$ and $Q\tilde{Q}q'\tilde{q}'$. From similarity of trapezoids we have

\begin{displaymath}
\gamma_q = \frac{\mid Q - \tilde{Q} \mid}{\mid P - \tilde{P}...
...=
\frac{\mid q' - \tilde{q}' \mid}{\mid p' - \tilde{p}' \mid}
\end{displaymath}

By assuming that $q$ and $q'$ are cooresponding points, we obtain the affine structure invariant $\gamma_q$.


next up previous
Next: Extension to 2 reference Up: Affine Structure Previous: Change of basis
Subhashis Banerjee 2002-02-18