Image Registration: Difference between revisions

Image Registration (view source)

199 bytes added , 15 May 2020

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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>
 This is represented as:<br>
-<math> x' = a_1 x + a_2 y + a_3</math><br>
+\[
-<math> y' = a_4 x + a_5 y + a_6</math><br>
+\begin{align}
+x' &= a_1 x + a_2 y + a_3\\
+y' &= a_4 x + a_5 y + a_6
+\end{align}
+\]
 This can also be written as:<br>
 <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].
-\(
-\DeclareMathOperator{\atantwo}{arctan2}
-\)
 The log-polar transformation is defined as follows:<br>
-<math>r = \sqrt{(x-x_c)^2 + (y-y_c)^2}</math><br>
+\[
-<math>a = \atantwo(y-y_c, x-x_c)</math><br>
+\begin{align}
+r &= \sqrt{(x-x_c)^2 + (y-y_c)^2}\\
+a &= \operatorname{arctan2}(y-y_c, x-x_c)
+\end{align}
+\]
 where <math>(x_c, y_c)</math> is the center of the image.
+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==