How to make hyperbolic wallpaper

Given a repeating wallpaper pattern in the Euclidean plane, is there a good algorithm to turn it into a wallpaper pattern in the hyperbolic plane, as in the Circle Limit series of pictures by M.C. Escher? According to the recent paper The Smooth Power of the "Neandertal Method" by Aaron Montag, Tim Reinhardt, and Jürgen Richter-Gebert, the answer is “ye…