for a small project I need to compute the similarity transformation matrix which transforms 2d coordinates from one image (left image) in another image (right image).
I know that the right image is generated from the left image by applying a similarity transformation (rotation, translation).
Further I know that the following relation between a point from the left image (u,v) and a point from the right image (u',v') (a match) holds:
$$
\begin{align}
\begin{bmatrix}
u' \\
v'
\end{bmatrix} = \begin{bmatrix}
a & -b & c \\
b & a & d
\end{bmatrix} \cdot \begin{bmatrix}
u \\
v \\
1
\end{bmatrix}
\end{align}
$$
We have now 4 unknowns => we need to collect 4 points (2 matches) and get the following equations:
$$
u_0' = au_0 – bv_0 + c\\
v_0' = bu_0 + av_0 + d\\
u_1' = au_1 – bv_1 + c\\
v_1' = bu_1 + av_1 + d
$$
My question is now how to bring this in a form where the SVD can be applied to find the best similarity matrix.
Thank you!
edit
Attention: The matches might be noisy – in other words, not all matches follow exactly the same transformation.
Best Answer
The system of equations is already in matrix form ($4\times4$).
There is no benefit using an SVD, Gaussian elimination is good enough, as the solution is exact.