From projective to affine

• Identifying the $\pi_\infty$ is equivalent to updating the projective structure to affine structure.
• An affine reconstruction is where the point coordinates $\{{\bf X^A}\}$ are known up to an affine transformation of their Euclidean values $\{{\bf X^E}\}$
  $${\bf X^A} = {\bf T_A}{\bf X^E} = \left[ \begin{array}{cc} {\bf A} & {\bf t_A} \\ {\bf0} & 1 \end{array} \right]$$
  $\{{\bf A},{\bf t_A}\}$ are unknown and same for all points.
• Choosing ${\bf a}$ corresponding to the `real' plane at infinity updates the projective reconstruction to affine.
• ${\bf T}$ then is the infinite homography, ${\bf H_\infty}$ , the homography through the plane at infinity.