Spatially-variant morphological operations on binary images based on the polar distance transform

Binary mathematical morphology can be computed by thresholding a distance transform, provided that the distance transform is a metric. Here we show that the polar distance transform is a metric and use it for morphological operations. The polar distance transform varies with the spatial coordinates of the image, resulting in spatially-variant morphology. In this distance transform each pixel is related to an image origin. We prefer angular propagation over radial, thus we construct structuring elements that are elongated in the angular direction, which is useful when circular segments are handled. We show an example where segments of annual rings on a log end face are connected using mathematical morphology based on the polar distance transform.