Create MP4 as alternative to GIF

GIFs are an inefficient, low quality encoding. I'd like for the user to be able to optionally create an .mp4 file instead of a .gif. I tried to do this but couldn't figure out how to encode a series of bitmaps into an MP4. It's also important that it loops smoothly for viewers - I don't know if there's a flag in the file format to indicate looping, and how often video players respect that.

If anyone is willing to help with this, don't concern yourself with the UI for now. Just some code which makes an MP4 from bitmaps would be a good start.