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Overview of the different strata

The properties of the different strata are briefly summarized in Table 2.1 . The different geometric strata are presented. The number of degrees of freedom, transformations and the specific invariants are given for each stratum. Figure 2.5 gives an example of an object which is equivalent to a cube under the different geometric ambiguities. Note from the figure that for purposes of visualization at least a metric level should be reached (i.e. is perceived as a cube).

Table 2.1: Number of degrees of freedom, transformations and invariants corresponding to the different geometric strata (the coefficients $r_{ij}$ form orthonormal matrices)
ambiguity DOF transformation invariants
projective 15 $ {\bf T}_P=\left[ \begin{array}{cccl} p_{11} & p_{12} & p_{13} & p_{14} \\
p_{...
...2} & p_{33} & p_{34} \\
p_{41} & p_{42} & p_{43} & p_{44} \end{array} \right] $
cross-ratio
affine 12 $ {\bf T}_A=\left[ \begin{array}{cccl} a_{11} & a_{12} & a_{13} & a_{14} \\
a_{...
...24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\
0 & 0 & 0 & 1
\end{array} \right] $
relative distances
along direction
parallelism
plane at infinity
metric 7 $ {\bf T}_M=\left[ \begin{array}{cccl}
\sigma r_{11} & \sigma r_{12} & \sigma r...
...1} & \sigma r_{32} & \sigma r_{33} & t_z \\
0 & 0 & 0 & 1
\end{array} \right] $
relative distances
angles
absolute conic
Euclidean 6 $ {\bf T}_E=\left[ \begin{array}{cccc} r_{11} & r_{12} & r_{13} & t_x \\
r_{21}...
...} & t_y \\ r_{31} & r_{32} & r_{33} & t_z \\ 0 & 0 & 0 & 1
\end{array} \right] $
absolute distances


Figure 2.5: Shapes which are equivalent to a cube for the different geometric ambiguities
\begin{figure}\centerline{
\psfig{figure=geometry/figures/Ambiguities.ps, width=8cm}}\end{figure}


next up previous contents
Next: Conclusion Up: The stratification of 3D Previous: Euclidean stratum   Contents
Marc Pollefeys 2002-11-22