From affine to metric

• Suppose that
  $${\bf P_E} = {\bf C}\left[ {\bf I} \mid {\bf0} \right]$$    and $${\bf P'_E} = {\bf C}\left[ {\bf R} \mid {\bf t} \right]$$
  i.e., we assume that the two camera positions have fixed internals.
• Suppose, also, that we have recovered affine structure $\{{\bf X^A}\}$ and a pair of projection matrices
  $${\bf P} = \left[ {\bf I} \mid {\bf0} \right]$$    and $${\bf P'} = \left[ {\bf T} \mid {\bf e'} \right]$$

• Then we know that ${\bf T} = {\bf H_\infty}$ .
• We also know that
  $${\bf P_E} = {\bf P}{\bf H}^{-1}$$    and $${\bf P'_E} = {\bf P'}{\bf H}^{-1}$$
  where,
  $${\bf H}^{-1} = \left[ \begin{array}{cc} {\bf C}^{-1} & {\bf0} \\ {\bf0} & k \end{array} \right]$$
  is the affine transformation that upgrades the affine reconstruction to Euclidean.
• The above gives us
  $${\bf T} = {\bf H_\infty} = {\bf CRC}^{-1}$$

• Rearranging and using ${\bf RR}^T = {\bf I}$ , we have
  $${\bf K} = {\bf T}^{-1} {\bf K} {\bf T}^T$$
  where
  $${\bf K} = {\bf CC}^T$$

• Thus, the internal parameter matrix can be computed from the infinite homography using Cholesky decomposition.