This Univ. of California video of their robot folding five pieces of cloth is pretty amazing.  Clearly the robot has to inspect the item to determine its edges and length, then find the half-way points of each edge, and proceed to fold.

On the other hand, this video is running at 50X normal speed, which means it took over an hour and a half to complete the task.  At some point, I would just push it out of the way and say, "Oh never mind!   I'll do it!!"