Would this GIMP script be useful? To use it, you need to have your shape as the only non-alpha portion of the GIMP image. This script converts the image to a square, does a rectangular-to-polar conversion, and does a linear gradient down each column along the non-transparent portions of the drawing before going back to rectangular coordinates. It can also do contours, if you so desire. The script will show up in the Filters/SambrookJM menu under GIMP.
It's not perfect, but it could be a good starting point. It's not a fast script at all, so any ideas on how to speed things up would be greatly appreciated. I'd also like to know how you delete the selection-mask channel, since the ways I've tried haven't worked.
Hope this helps out somewhat...