Image Registration
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:
\[
\begin{align}
x' &= a_1 x + a_2 y + a_3\\
y' &= a_4 x + a_5 y + a_6
\end{align}
\]
This can also be written as:
\(\displaystyle
\begin{pmatrix}
x' \\ y' \\ 1
\end{pmatrix}
=
\begin{pmatrix}
a_1 & a_2 & a_3\\
a_4 & a_5 & a_6\\
0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
x \\ y \\ 1
\end{pmatrix}
\)
Log-Polar Transformation
See Wolberg and Zokai<ref name="wolberg2000robust">George Wolberg, and Siavash Zokai. Robust Image Registration Using Log-Polar Transform URL:https://home.cis.rit.edu/~cnspci/references/wolberg2000.pdf.
The log-polar transformation is defined as follows:
\[
\begin{align}
r &= \sqrt{(x-x_c)^2 + (y-y_c)^2}\\
a &= \operatorname{arctan2}(y-y_c, x-x_c)
\end{align}
\]
where \(\displaystyle (x_c, y_c)\) is the center of the image.
Here a rotation in Cartesian coordinates \(\displaystyle (x, y)\) around the center \((x_c, y_c)\) corresponds to a shift in \(a\) in log-polar coordinates.
These translations can be found using Wikipedia: Cross-correlation.