I know we have some members which are experts in map projections and I need some support/help/info for a project I'm working on.

The goal is to create a world map. I decided for a map in a 2:1 ratio. And now the difficult part: I have two images of the poles as seen from space. These I want to get on the map in the north and south.

What is the best way to get a relatively good result for this? Which (free) software is able to do this?

I'm thankful for any suggestions.