Image Registration: Difference between revisions

Image Registration (view source)

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 ===Adaptive Log-Polar Transformation===
 The goal of Adaptive Polar Transform by Matungka ''et al.''<ref name="matungka2009adaptive"><cite class="journal">Rittavee Matungka, Yuan F. Zheng, and Robert L. Ewing (2009). ''Image Registration Using Adaptive Polar Transform'' DOI: [https://doi.org/10.1109/TIP.2009.2025010 10.1109/TIP.2009.2025010] URL: [https://home.cis.rit.edu/~cnspci/references/wolberg2000.pdf https://home.cis.rit.edu/~cnspci/references/wolberg2000.pdf]</cite></ref> is to address the non-uniform sampling of the log-polar transformation.
+Let the following
+* \(n_\rho\) be the number of samples (i.e. resolution) along the \(\rho\) axis
+* \(n_\theta\) be the number of samples along the \(\theta\) axis
+* \(r_i\) the radius size for pixel \(i=1,...,n_p\)
+* \(\theta = 0,...,n_\theta - 1\)
+To prevent undersampling along \(\rho\), we must have \(R_{n_\rho} - R_{n_\rho-1} \leq 1\).
+I.e. the outermost two circles must have \(\leq 1\) pixel difference.
+To prevent undersampling along \(\theta\), we must have at least \(2\pi R_{max}\) pixels along the \(\theta\) axis.
+This leads to the following equations:<br>
+\[
+\begin{align}
+R_i &= \exp(i \times \frac{\log R_{max}}{n_\rho}), \qquad R_{max} = R_{n_\rho}\\
+n_\rho &\geq \frac{\log R_{max}}{\log R_{max} - \log(R_{max}-1)}\\
+n_\theta &\geq 2\pi R_{max}
+\end{align}
+\]
 ==References==