New geodosic distance transforms for gray-scale images

In this paper, two new geodesic distance transforms for gray-scale images are presented. The first transform, the Distance Transform on Curved Space (DTOCS), performs the calculation with integer numbers. The second transform, the Weighted Distance Transform on Curved Space (WDTOCS), gives a weighted distance map with real numbers for an arbitrary gray-value image. Both transforms give a distance map in which the distance value of a single point corresponds to the length of the shortest discrete 8-path to the nearest background point. Both differ from the previously presented gray-level distance transforms by not weighting the distance values directly by the gray-values, but by gray-value differences.