Epipolar geometry: calibrated case

• There are two camera coordinate systems related by ${\bf R, T}$
  $${\bf X'} = {\bf RX} + {\bf T}$$

• Taking the vector product with ${\bf T}$ followed by the scalar product with ${\bf X'}$
  $${\bf X'} \cdotp ({\bf T} \times {\bf RX}) = 0$$
  which expresses that vectors ${\bf OX}$ , ${\bf O'X'}$ and ${\bf OO'}$ are coplanar.
• This can be written as
  $${\bf X'}^T{\bf E}{\bf X} = 0$$
  where
  $${\bf E} = \left[ {\bf T} \right]_{\times} {\bf R}$$
  is the Essential matrix.
• Image points and rays in Euclidean 3-space are related by:
  $$\left[ \begin{array}{c} x \\ y \\ 1 \end{array} \right] = {\bf C} \left[ \begin{array}{c} X \\ Y \\ Z \end{array} \right]$$    and $$\left[ \begin{array}{c} x' \\ y' \\ 1 \end{array} \right] = {\bf C'} \left[ \begin{array}{c} X' \\ Y' \\ Z' \end{array} \right]$$

• Hence, we have
  $${\bf x'}^{T} {\bf C'}^{-T}{\bf E}{\bf C}^{-1}{\bf x} = 0$$

• Thus, the relation between the essential and fundamental matrix is:
  $${\bf F} = {\bf C'}^{-T}{\bf E}{\bf C}^{-1}$$