Ok, the GIS software is more difficult than I had expected.
This is one of the poles - flat picture, I don't know which projection. How do I turn it into a flat small band at the southern top of the map? I tried G.Projector, but couldn't load it in pole position.
http://www.jhendor.de/tutorials/cc2-ft.pdf describes how Ralf Schemann over at ProFantasy used a little toy I wrote a long time ago to do the sort of thing that you're asking. You would be particularly interested in the "Converting the image to an equirectangular projection" step, I'm guessing.
I haven't had a pressing need to use one of the projection tools out there, so I can't help you much with those, I'm afraid.
This image shows what the UI for ReprojectImage would look like to unproject a stereographic image
And this is a 4096x2048 export from those settings.
Other projections are pretty straightforward.
ReprojectImage is an ancient tool, but it sort of works for this kind of thing.
Excellent! Thank you, waldronate, that's exactly what I needed.
Another proof of the excellent support and help in this forum. I took the freedom to donate a few bucks to the site.