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Image Registration: Difference between revisions
→Adaptive Log-Polar Transformation
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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. | 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 | {{ hidden | Motivation | | ||
Let the following: | |||
* \(n_\rho\) be the number of samples (i.e. resolution) along the \(\rho\) axis | * \(n_\rho\) be the number of samples (i.e. resolution) along the \(\rho\) axis | ||
* \(n_\theta\) be the number of samples along the \(\theta\) axis | * \(n_\theta\) be the number of samples along the \(\theta\) axis | ||
* \(r_i\) the radius size for pixel \(i=1,..., | * \(r_i\) the radius size for pixel \(i=1,...,n_\rho\) | ||
* \(\theta = 0,...,n_\theta - 1\) | * \(\theta = 0,...,n_\theta - 1\) | ||
To prevent undersampling along \(\rho\), we must have \(R_{n_\rho} - R_{n_\rho-1} \leq 1\). | 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. | I.e. the outermost two circles must have \(\leq 1\) pixel difference.<br> | ||
To prevent undersampling along \(\theta\), we must have at least \(2\pi R_{max}\) pixels along the \(\theta\) axis. | To prevent undersampling along \(\theta\), we must have at least \(2\pi R_{max}\) pixels along the \(\theta\) axis. | ||
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n_\rho &\geq \frac{\log R_{max}}{\log R_{max} - \log(R_{max}-1)}\\ | n_\rho &\geq \frac{\log R_{max}}{\log R_{max} - \log(R_{max}-1)}\\ | ||
n_\theta &\geq 2\pi R_{max} | n_\theta &\geq 2\pi R_{max} | ||
\end{align} | |||
\] | |||
However, using these would lead to oversampling of the fovea region wasting computation resources. | |||
}} | |||
Let the following: | |||
* \(n_r\) be the number of samples (i.e. resolution) along the \(r\) axis | |||
* \(n_\theta\) be the number of samples along the \(\theta\) axis | |||
Given an image of size \(2 R_{max} \times 2 R_{max}\), let (R_i\) the radius size for pixel \(i=1,...,n_\rho\). | |||
The main idea is that the number of pixels \(n_\theta\) should be adaptive to the radius. | |||
\[ | |||
\begin{align} | |||
n_r &= R_{max}\\ | |||
n_{\theta_i} &= R_i | |||
\end{align} | \end{align} | ||
\] | \] | ||
==References== | ==References== | ||