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linear self-calibration

The first step consists of normalizing the projection matrices. The following normalization is proposed:

\begin{displaymath}
{\bf P}_N = {\bf K}_N^{-1} {\bf P} \mbox{ with }
{\bf K}_N=...
...0 &\frac{w}{2} \\ & w+h &\frac{h}{2}\\ & & 1\end{array}\right]
\end{displaymath} (F12)

where $w$ and $h$ are the width, resp. height of the image. After the normalization the focal length should be of the order of unity and the principal point should be close to the origin. The above normalization would scale a focal length of a 60mm lens to 1 and thus focal lengths in the range of 20mm to 180mm would end up in the range $[1/3,3]$. The aspect ratio is typically also around 1 and the skew can be assumed 0 for all practical purposes. Making these a priori knowledge more explicit and estimating reasonable standard deviations one could for example get $f \approx rf \approx 1 \pm 3$, $u \approx v \approx 0 \pm 0.1$, $r \approx 1 \pm 0.1$ and $s=0$. It is now interesting to investigate the impact of this knowledge on $\omega^*$:
\begin{displaymath}
\omega^* \sim {\bf KK}^\top =
\left[ \begin{array}{ccc} f^2+...
... 1 \pm 9 & \pm 0.1 \\ \pm 0.1 & \pm 0.1 & 1 \end{array}\right]
\end{displaymath} (F13)

and $\omega_{22}^*/\omega^*_{11} \approx 1 \pm 0.2$. The constraints on the left-hand side of Equation (6.7) should also be verified on the right-hand side (up to scale). The uncertainty can be take into account by weighting the equations.
\begin{displaymath}
\begin{array}{rcl}
\frac{1}{9 \nu} \left( P_1 \Omega^* {P_1...
...u} \left( P_2 \Omega^* {P_3}^\top \right)& = &0 \\
\end{array}\end{displaymath} (F14)

with $P_i$ the $i$th row of ${\bf P}$ and $\nu$ a scale factor that is initially set to 1 and later on to ${P_3} \tilde{\Omega}^* {P_3}^\top$ with $\tilde{\Omega}^*$ the result of the previous iteration. Since $\Omega ^*$ is a symmetric $4 \times 4$ matrix it is parametrized through 10 coefficients. An estimate of the dual absolute quadric $\Omega ^*$ can be obtained by solving the above set of equations for all views through linear least-squares. The rank-3 constraint should be imposed by forcing the smallest singular value to zero. This scheme can be iterated until the $\nu$ factors converge, but in practice this is often not necessary. Experimental validation has shown that this approach yields much better results than the original approach described in [114,105]. This is mostly due to the fact that constraining all parameters (even with a small weight) allows to avoid most of the problems due to critical motion sequences [144,63] (especially the specific additional case for the linear algorithm [99]).



Subsections
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Marc Pollefeys 2002-11-22