Comparison of conversion methods
Picture straight from the camera (Olympus OM-D EM-5 and fisheye Samyang 7.5/f3.5, with diagonal field of view 180 degrees).
Rectilinear projection. Picture was intentionally downsized to show extremely stretched corners.
This is the greatest rectangle, possible to cut out from previous image. Areas near left and right borders look unnatural ...
... so they are often cut off. Result image is cropped to 3:2 proportions and has horizontal field of view about 121 degrees. This image corresponds to focal length about 10mm on full frame camera.
We can apply cylindrical projection instead of rectilinear. Only vertical lines are straightened. Radial lines are also straight - this kind of cylindrical projection is often named "Panini" in honor of the Italian painter, who showed in this way church interiors. Horizontal field of view (135 degrees) and vertical field of view (101 degrees) are preserved. Horizontal lines remain bent ...
... so we can additionaly straighten horizontal lines, but there is always a compromise - one line is straight, others are bent.