Image Registration: Difference between revisions
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==Log-Polar Transformation== | ==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 https://home.cis.rit.edu/~cnspci/references/wolberg2000.pdf]. | |||
\( | |||
\DeclareMathOperator{\atantwo}{arctan2} | |||
\) | |||
The log-polar transformation is defined as follows:<br> | |||
<math>r = \sqrt{(x-x_c)^2 + (y-y_c)^2}</math><br> | |||
<math>a = \atantwo(y-y_c, x-x_c)</math><br> | |||
where <math>(x_c, y_c)</math> is the center of the image. | |||
==References== | ==References== | ||
Revision as of 12:55, 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:
\(\displaystyle x' = a_1 x + a_2 y + a_3\)
\(\displaystyle y' = a_4 x + a_5 y + a_6\)
This can also be written as:
\(\displaystyle
\begin{pmatrix}
x' \\ y' \\ 1
\end{pmatrix}
=
\begin{pmatrix}
a_1 & a_2 & a_3\\
a_4 & a_5 & a_6\\
0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
x \\ y \\ 1
\end{pmatrix}
\)
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.
\(
\DeclareMathOperator{\atantwo}{arctan2}
\)
The log-polar transformation is defined as follows:
\(\displaystyle r = \sqrt{(x-x_c)^2 + (y-y_c)^2}\)
\(\displaystyle a = \atantwo(y-y_c, x-x_c)\)
where \(\displaystyle (x_c, y_c)\) is the center of the image.