I wanted to create a "little planet" video from a 360 video clip, but I didn't want to spend $100 upgrading my video editing software. It was pretty trivial to create a "plane" in Blender to remap the geometry. Result is pretty successful.
Got sample input video from
https://www.mettle.com/360vr-master-series-free-360-downloads-page/
(which redirects to Vimeo downloads)
Experiment result is at:
https://vimeo.com/298199363/088a04c5f1
(embed isn't working today)
Probably could be done much more elegantly with custom transform like
https://cgcookie.com/course/working-with-custom-transform-nodes-in-cycles
After quite a bit of head-scratching, I did get a custom Cycles material to do the remapping too. I haven’t compared processing times.