In 1D projective geometry, I want to compute the 2x2 Homography matrix $H$ (in homogeneous coordinates), given 3 pairs of corresponding points.
i.e. I want to find H such that: $$\left(\begin{array}{cc} h_{11} & h_{12}\\ h_{21} & h_{22}\end{array}\right)\left(\begin{array}{ccc} 0 & a & a+b\\ 1 & 1 & 1 \end{array}\right) = \left(\begin{array}{ccc} 0 &a' &a'+b'\\ 1 & 1 & 1 \end{array}\right).$$
However, I've got 6 equations here and only 3 unknowns. (dof(H) = 4 elements less one for scaling = 3).
I thought about 3 scaling factors that would add up to 6 unknowns, s.t. we would have a unique solution. But how exactly do I insert the scaling factors into the matrices and how can I compute H then?
Do you have a clue?