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The image of the absolute conic

One of the most important concepts for self-calibration is the Absolute Conic (AC) and its projection in the images (IAC) F2. Since it is invariant under Euclidean transformations (see Section 2.2.3), its relative position to a moving camera is constant. For constant intrinsic camera parameters its image will therefore also be constant. This is similar to someone who has the impression that the moon is following him when driving on a straight road. Note that the AC is more general, because it is not only invariant to translations but also to arbitrary rotations.

It can be seen as a calibration object which is naturally present in all the scenes. Once the AC is localized, it can be used to upgrade the reconstruction to metric. It is, however, not always so simple to find the AC in the reconstructed space. In some cases it is not possible to make the difference between the true AC and other candidates. This problem will be discussed in the Section 6.2.5.

In practice the simplest way to represent the AC is through the Dual Absolute Quadric (DAQ). In this case both the AC and its supporting plane, the plane at infinity, are expressed through one geometric entity. The relationship between the AC and the IAC is easily obtained using the projection equation for the DAQ:

\begin{displaymath}
\omega_i^* \sim {\bf P}_i \Omega^* {\bf P}_i^\top \enspace .
\end{displaymath} (F7)

with $\omega_i^*$ representing the dual of the IAC, $\Omega ^*$ the DAQ and ${\bf P}_i$ the projection matrix for view $i$. Figure 6.1 illustrates these concepts.
Figure 6.1: The absolute conic (located in the plane at infinity) and its projection in the images
\begin{figure}\centerline{\psfig{figure=selfcal/AbsoluteConic.ps, width=7.6cm}}\end{figure}
For a Euclidean representation of the world the camera projection matrices can be factorized as: ${\bf P}_i = {\bf K}_i {\bf R}_i^\top [ {\bf I} \,\vert\, \mbox{-}{\tt t}_i ] $ (with ${\bf K}_i$ an upper triangular matrix containing the intrinsic camera parameters, ${\bf R}_i^\top$ representing the orientation and ${\tt t}_i$ the position) and the DAQ can be written as $\Omega^* = diag(1,1,1,0)$. Substituting this in Equation (6.7), one obtains:
\begin{displaymath}
\omega_i^* \sim {\bf K}_i {\bf K}_i^\top \enspace
\end{displaymath} (F8)

This equation is very useful because it immediately relates the intrinsic camera parameters to the DIAC.

In the case of a projective representation of the world the DAQ will not be at its standard position, but will have the following form: $\Omega^*={\bf T}\Omega^*_{M} {\bf T}^{\top}$ with ${\bf T}$ being the transformation from the metric to the projective representation. But, since the images were obtained in a Euclidean world, the images $\omega^*_i$ still satisfy Equation (6.8). If $\Omega ^*$ is retrieved, it is possible to upgrade the geometry from projective to metric.

The IAC can also be transferred from one image to another through the homography of its supporting plane (i.e. the plane at infinity):

\begin{displaymath}
\omega_j \sim {{\bf H}^\infty_{ij}}^{-\top} \omega_i {{\bf H...
...}^\infty_{ij} \omega_i^* {{\bf H}^\infty_{ij}}^\top \enspace .
\end{displaymath} (F9)

It is also possible to restrict this constraint to the epipolar geometry. In this case one obtains the Kruppa equations [75] (see Figure 6.2):

Figure 6.2: The Kruppa equations impose that the image of the absolute conic satisfies the epipolar constraint. In both images the epipolar lines corresponding to the two planes through ${\tt C}_i$ and ${\tt C}_j$ tangent to $\Omega $ must be tangent to the images $\omega _i$ and $\omega _j$.
\begin{figure}\centerline{\psfig{figure=selfcal/Kruppa.ps, width=7.6cm}}\end{figure}

\begin{displaymath}[{\tt e}_{ij}]^\top_\times {\bf KK}^\top [{\tt e}_{ij}]_\times
\sim {\bf F}_{ij} {\bf KK}^\top {\bf F}_{ij}^\top
\end{displaymath} (F10)

with ${\bf F}_{ij}$ the fundamental matrix for views $i$ and $j$ and ${\tt e}_{ij}$ the corresponding epipole. In this case only 2 (in stead of 5) independent equations can be obtained [176]. In fact restricting the self-calibration constraints to the epipolar geometry is equivalent to the elimination of the position of infinity from the equations. The result is that some artificial degeneracies are created (see [142]).


next up previous contents
Next: Self-calibration methods Up: Self-calibration Previous: Geometric interpretation of constraints   Contents
Marc Pollefeys 2002-11-22