Officially known as Stereographic Projection, here are three simple steps to create your Little Planet Images through Photoshop.
Go to Image > Image Size and make sure Constrain Proportions is turned off. Also create the images into a square by having the height the same size as the with.
Invert photograph by going to Image > Image Rotation > 180°. If you wish to have the image circle inward, leave as is and do not rotate.
To create the image into a circle, choose Filter > Polar Coordinates. If the filters doesn’t seem to be working, got to step 4 and return to step 3 when complete. After applying this filter, editing is required to get rid of the line.
Another way to get rid of the line is to create a panorama and flipping the image (duplicated). When distorted, it will appear seamless.
Match Height Size to Width Size to Create Square
Invert Photo and Distort
Inverted Planet without Inversion
The reason most of the filters are not highlighted is because the plugins only operate in 8-Bits/Channel. You can change this by going to Image > Mode > 8-Bits/Channel.
Here are some of my images.
Buenos Aires, Argentina at Night
Thunder and Lightning over Mississauga, Ontario
Sunset in Mississauga, Ontario
Sunset in Leamington, Ontario
Sturgeon Bay Provincial Park at Night
Beagle Channel, Patagonia, South America