Fisheye lenses, different projections?

The Samyang 7.5mm f/3.5 fisheye lens is an interesting addition to the Micro Four Thirds lineup. While most third party Micro Four Thirds lenses so far have been existing manual lens designs given a new mount, this lens is designed for the Micro Four Thirds format from ground up.

How can I tell? A full frame fisheye lens has 180° diagonal field of view. Hence, the lens must match the sensor size exactly. If the lens was designed for APS-C, a larger format, then the corners would not correspond to 180° angle of view. A fisheye lens which does not project to 180° in the corners is pretty much useless. Then it is just a wide angle lens with a lot of geometric distortion.

Compared with the existing Lumix G 8mm f/3.5 fisheye lens, the Samyang lens is a much more traditional design, with a manual focus ring and aperture ring.

Some say the Samyang lens features a different projection type, the Stereographic projection. This is supposed to be less distorted than the Spherical projection traditionally associated with fisheye lenses. Let's look into how their projections differ. Here is a picture taken with both lenses, and also using the Olympus Zuiko Digital 9-18mm Four Thirds wide angle zoom at 9mm:

Samyang 7.5mm	Lumix G 8mm	Olympus 9-18mm @ 9mm

By superimposing both the fisheye images in one image, and doing edge detection, we can see how their projections compare:

And let's look at another example:

Samyang 7.5mm	Lumix G 8mm	Olympus 9-18mm @ 9mm

Superimposing the two gives:

(Click for larger images.)

Conclusion

So, what is the conclusion of all this? First of all, from the second example, we observe that both lenses have pretty much the same diagonal field of view. This experiment does not verify that the diagonal field of view is exactly 180°, as it should be, but at least both lenses have around the same maximum diagonal angle. To be more precise, the Samyang lens appears to render a slightly wider diagonal field of view. This might be due to the shorter focal length, 7.5mm versus 8mm.

Regarding the distortion, we can see that the Samyang lens renders objects which are inside the border of the image a little bit smaller. This means that it distorts the images somewhat less. So the rumor is true: The Samyang lens does give less "fisheye distortion".

But surely, the differences are pretty marginal. You're not likely to notice much difference, unless comparing head to head, as I do in this article. So if you're looking at the Samyang 7.5mm lens to avoid the fisheye distortion, you are going to be disappointed.

One thing to note is that if you plan to convert to rectilinear images in post processing, the Samyang lens has the potential for giving you the widest possible rectilinear images of the two fisheye lenses. This process is called defishing, and you can read about the topic here. It is probably not true that the Samyang lens features stereographic projection. It still has fisheye projection, but with slightly less distortion than the Lumix G 8mm f/3.5 fisheye lens.