Image Registration: Difference between revisions
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We want to find a rotation and translation from <math>(x,y)</math> to <math>(x',y')</math> such that <math>I_1(x,y) = I_2(x', y')</math>.<br> | We want to find a rotation and translation from <math>(x,y)</math> to <math>(x',y')</math> such that <math>I_1(x,y) = I_2(x', y')</math>.<br> | ||
This is represented as:<br> | This is represented as:<br> | ||
\[ | |||
\begin{align} | |||
x' &= a_1 x + a_2 y + a_3\\ | |||
y' &= a_4 x + a_5 y + a_6 | |||
\end{align} | |||
\] | |||
This can also be written as:<br> | This can also be written as:<br> | ||
<math> | <math> | ||
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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 https://home.cis.rit.edu/~cnspci/references/wolberg2000.pdf]. | 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 https://home.cis.rit.edu/~cnspci/references/wolberg2000.pdf]. | ||
The log-polar transformation is defined as follows:<br> | The log-polar transformation is defined as follows:<br> | ||
\[ | |||
\begin{align} | |||
r &= \sqrt{(x-x_c)^2 + (y-y_c)^2}\\ | |||
a &= \operatorname{arctan2}(y-y_c, x-x_c) | |||
\end{align} | |||
\] | |||
where <math>(x_c, y_c)</math> is the center of the image. | where <math>(x_c, y_c)</math> is the center of the image. | ||
Here a rotation in Cartesian coordinates <math>(x, y)</math> around the center \((x_c, y_c)\) corresponds to a shift in \(a\) in log-polar coordinates.<br> | |||
These translations can be found using [[Wikipedia: Cross-correlation]]. | |||
==References== | ==References== | ||
Revision as of 13:49, 15 May 2020
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.