With Tiny Planet Maker you can create nice Tiny Planet from 360° panoramic photographs or regular photographs. Just open the image from within the app or share it from another app (like gallery, or photo apps). Modify your Tiny Planets with sliders or with touch gestures. Note that the app itself does not take any images, instead it manipulates existing images. The tiny planet effect is also named small planet, globe photo or stereographic projection.
The app is open source: https://github.com/hollaus/TinyPlanetMaker