This effect was generated in Filter Forge, a Photoshop plugin that allows you to create your own filters.
This filter converts an image from rectangular to polar coordinates, and vice versa, according to the option selected.
*Special thanks to byRo for his brilliant Spherify & Un-Spherify techniques which lend much of their logic to this effect.
spherify transform polar circular cartesian coordinates sphere spherize distort photoshop snippet math byro uber zev
This filter has earned its author one of our rewards. They are given to the authors of popular filters and include a free copy of Filter Forge.