Distinguished planes and ${\bf F}$

• Let ${\bf T^*}$ be a homography between two images through a plane ${\bf\pi}$ . Thus we have (projectively)
  $${\bf x'} = {\bf T^*}{\bf x}$$

• Given ${\bf F}$ we know ${\bf e}$ and ${\bf e'}$ .
• Now if ${\bf F} = [{\bf e'}]_{\times}{\bf T^*}$ is a valid decomposition, then so is ${\bf F} = [{\bf e'}]_{\times}{\bf T}$ , where
  $${\bf T} = {\bf T^*} + {\bf e'}{\bf a}^T$$
  where ${\bf a}$ is an arbitrary 3 vector.
• Thus, the most general projection matrices consistent with ${\bf F}$ are
  $${\bf P}= {\bf [I \mid 0]}$$    and $${\bf P'} = [{\bf T} + {\bf e'}{\bf a} \mid b{\bf e'}]$$
  This is four parameter family, $b$ is an arbitrary scalar.
• The matrix ${\bf T} = {\bf T^*} + {\bf e'}{\bf a}^T$ is a projective transformation of the plane $\pi$ represented by the four vector $(0,0,0,1)^T$ , since for points on this plane ${\b X_4} = 0$ , and
  $${\bf x} = {\bf P} \left[ \begin{array}{c} X\\ Y\\ Z\\ 0 \end{arra... ... 0 \end{array} \right] = \left[ \begin{array}{c} X\\ Y\\ Z \end{array} \right]$$
  $${\bf x'} = {\bf P'} \left[ \begin{array}{c} X\\ Y\\ Z\\ 0 \end{ar... ...ht] = {\bf T} \left[ \begin{array}{c} X\\ Y\\ Z \end{array} \right] = {\bf Tx}$$

• Thus, the choice of ${\bf a}$ corresponds to which actual plane in 3-space is $\pi_\infty$ .