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