Grey Weighted Polar Distance Transform for Outlining Circular and Approximately Circular Objects

We introduce the polar distance transform and the grey weighted polar distance transform for computation of minimum cost paths preferring circular shape, as well as give algorithms for implementations in a digital setting. An alternative to the polar distance transform is to transform the image to polar coordinates, and then apply a Cartesian distance transform. By using the polar distance transform, re-sampling of the image and interpolation of new pixel values are avoided. We also handle the case of grey weighted distance transform in a $5\times 5$ neighbourhood, which, to our knowledge, is new. Initial results of using the grey weighted polar distance transform to outline annual rings in images of log end faces are presented