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coupled self-calibration

In some cases different (sub)sequences have been recorded with the same camera and the same settings. In this case it is possible to combine the equations derived above into a larger system of equations. This can for example be used to solve the problem of a dominant plane separating a sequence in two subsequences (see Section 5.4).

When choosing ${\bf P}=[{\bf I}\vert{\bf0}]$ for one of the projection matrices it can be seen from Equation (6.11) that ${\bf\Omega}^*$ can be written as:

\begin{displaymath}
{\bf\Omega}^* = \left[\begin{array}{cc}
{\bf KK}^\top & {\bf a}\\ {\bf a}^\top & b
\end{array}\right]
\end{displaymath} (F15)

Now the set of equations (6.14) can thus be written as:
\begin{displaymath}
\left[\begin{array}{cc}{\bf C} &{\bf D}\end{array}\right]
\l...
...egin{array}{c}{\bf a} \\ b\end{array}\right]\end{array}\right]
\end{displaymath} (F16)

where ${\bf k}$ is a vector containing 6 coefficients representing the matrix ${\bf KK}^\top$, ${\bf a}$ is a 3-vector and $b$ a scalar and ${\bf C}$ and ${\bf D}$ are matrices containing the coefficients of the equations. Note that this can be done independently for every 3D subsequence.

If the sequence is recorded with constant intrinsics, the vector ${\bf k}$ will be common to all subsequences and one obtains the following coupled self-calibration equations:

\begin{displaymath}
\left[\begin{array}{ccccc}{\bf C}_1& {\bf D}_1& {\bf0} & \cd...
...array}{c}{\bf a}_n \\ b_n\end{array}\right]
\end{array}\right]
\end{displaymath} (F17)

As will be seen in the experiments this approach is very successful. The most important feature is that through the coupling it allows to get good results even for the shorter subsequences. For each subsequence a transformation to upgrade the reconstruction from projective to metric can be obtained from the constraint ${\bf T}_i{\bf\Omega}_i^*{\bf T}_i^\top=\mbox{diag} (1,1,1,0)$ (through Cholesky factorization).


next up previous contents
Next: non-linear self-calibration refinement Up: linear self-calibration Previous: linear self-calibration   Contents
Marc Pollefeys 2002-11-22