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Image registration is recovering an affine transformation (rotation + translation) between two images.
Problem Statement
We are given two images \(\displaystyle I_1\) and \(\displaystyle I_2\).
Let \(\displaystyle (x,y)\) be uv coordinates within the image.
We want to find a rotation and translation from \(\displaystyle (x,y)\) to \(\displaystyle (x',y')\) such that \(\displaystyle I_1(x,y) = I_2(x', y')\).
This is represented as:
\(\displaystyle x' = a_1 x + a_2 y + a_3\)
\(\displaystyle y' = a_4 x + a_5 y + a_6\)
This can also be written as:
\(\displaystyle
\begin{pmatrix}
x' \\ y' \\ 1
\end{pmatrix}
=
\begin{pmatrix}
a_1 & a_2 & a_3\\
a_4 & a_5 & a_6\\
0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
x \\ y \\ 1
\end{pmatrix}
\)
References
