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Image Registration: Difference between revisions
→Log-Polar Transformation
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The log-polar transformation is defined as follows:<br> | The log-polar transformation is defined as follows:<br> | ||
\ | \( | ||
\begin{align} | \begin{align} | ||
r &= \sqrt{(x-x_c)^2 + (y-y_c)^2}\\ | r &= \sqrt{(x-x_c)^2 + (y-y_c)^2}\\ | ||
a &= \operatorname{arctan2}(y-y_c, x-x_c) | a &= \operatorname{arctan2}(y-y_c, x-x_c) | ||
\end{align} | \end{align} | ||
\ | \)<br> | ||
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> | 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]]. | These translations can be found using [[Wikipedia: Cross-correlation]]. | ||
A scale change (i.e. enlarge or stretch) is a shift in log-space:<br> | |||
\( \lambda x \mapsto \log(\lambda x) = \log(\lambda) + \log(x) \) | |||
==References== | ==References== | ||