From projective to affine

Up to now it was assumed that these different strata could simply be overlaid onto each other, assuming that the plane at infinity is at its canonical position (i.e. ${\tt\Pi}_\infty = [0 \,0 \,0 \,1]^\top$). This is easy to achieve when starting from a Euclidean representation. Starting from a projective representation, however, the structure is only determined up to an arbitrary projective transformation. As was seen, these transformations do - in general - not leave the plane at infinity unchanged.

Therefore, in a specific projective representation, the plane at infinity can be anywhere. In this case upgrading the geometric structure from projective to affine implies that one first has to find the position of the plane at infinity in the particular projective representation under consideration.

This can be done when some affine properties of the scene are known. Since parallel lines or planes are intersecting in the plane at infinity, this gives constraints on the position of this plane. In Figure 2.1 a projective representation of a cube is given. Knowing this is a cube, three vanishing points can be identified. The plane at infinity is the plane containing these 3 vanishing points.

Figure 2.1: Projective (left) and affine (right) structures which are equivalent to a cube under their respective ambiguities. The vanishing points obtained from lines which are parallel in the affine stratum constrain the position of the plane at infinity in the projective representation. This can be used to upgrade the geometric structure from projective to affine.

Ratios of lengths along a line define the point at infinity of that line. In this case the points ${\tt M}_0$, ${\tt M}_1$, ${\tt M}_2$ and the cross-ratio $\{ {\tt M}_1, {\tt M}_2 ; {\tt M}_0, {\tt M}_\infty \}$ are known, therefore the point ${\tt M}_\infty$ can be computed.

Once the plane at infinity ${\tt\Pi }_\infty$ is known, one can upgrade the projective representation to an affine one by applying a transformation which brings the plane at infinity to its canonical position. Based on (2.9) this equation should therefore satisfy

$$\left[ \begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \end{array} \right]... ...}{c} 0 \\ 0 \\ 0 \\ 1 \end{array} \right] \sim {\tt\Pi}_\infty$$ (B24)

This determines the fourth row of ${\bf T}$. Since, at this level, the other elements are not constrained, the obvious choice for the transformation is the following

$${\bf T}_{PA} \sim \left[\begin{array}{cc} {\bf I}_{3 \times 3} & 0_3 \\ {\tt\pi}_\infty^\top & 1\end{array} \right]$$ (B25)

with ${\tt\pi}_\infty$ the first 3 elements of ${\tt\Pi }_\infty$ when the last element is scaled to 1. It is important to note, however, that every transformation of the form

$$\left[\begin{array}{cc} {\bf A} & 0_3 \\ {\tt\pi}_\infty^\top & 1\end{array} \right] \mbox{ with } \det {\bf A} \neq 0$$ (B26)

maps ${\tt\Pi }_\infty$ to $[0 \, 0\, 0\, 1]^\top$.