About the spherical controversy

(360° x 180°) Vs (360° x 360°)

 

Let us use two source images that were shot with a Canon EOS 5D and a Sigma 8mm f4.0 fisheye lens in two opposite directions i.e. the (yaw) angle between the image is 180°.

The angle of view is about or less than 183° in a circular fisheye lens projection:

While this operation is not usually required, let's now crop these two images so as to have an angle of view of exactly 180°:

OK, we are now ready for experimenting with these two images above. We may use PTGui for the rest of the demonstration.

 

 

First experiment (Equirectangular projection)

While this is also not mandatory, let's transform both images into an equirectangular projection with 180° (horizontal) x 180° (vertical) Field Of View (FOV):

By setting the yaw angle between the two images above as 180° we can now assemble the full panorama in equirectagular projection:

The panorama settings are :

(screen shot of PTGui panorama settings tab window)

 

 

Second experiment (Circular projection)

Let's transform both images into an circular projection with 180° (horizontal) x 180° (vertical) Field Of View (FOV) and let's set the yaw angle between the two images above as 180°:

By the way, another equivalent process could have been run by shifting the yaw setting position of both image in the Iimage Parameters window of PTGui:

360 X 360 single image with -90 deg yaw
360 X 360 single image with +90 deg yaw

 

We shall now assemble the panorama in circular projection:

The panorama settings are:

(screen shot of PTGui panorama settings tab window)

 

The project PTGui script that allowed to run the experiments above with the two Fisheye images.

How a 360° x 180° circular projected panorama looks like?

Discussion:

The equirectangular projection has many advantages:

BTW, Dr Helmut Dersh used to call the equirectangle perspective planar representation of a full spherical panorama a "PSphere".

In comparison the circular projection format for outputting the panorama is rarely used.

BTW, I was inspired by an old joke from Dr Helmut Dersh who told us that such an image could be produced by a perfect theoretical fisheye lens with 360° angle of view: I have humbly used this impossible lens concept to dodge reality and I have written "how to".

As a side note: nother (propriatory) panorama output projection format had been proposed for use in a commercial stitching program but it seems to have now be supplanted by all other open formats, including the above two image projections.

Conclusion:

IMHO, the FOV of a PSPhere (360° x 180°) and the FOV of a Circular image (360° x 360°) are equally correct and acceptable to define a spherical full "spherical" panorama that is remapped on a plane: these notions (and wordings) are equally purely conventional and should not be thought as mathematical concepts. Comparing and discussing about them out of their context has been the result of a huge misunderstanding between seasoned members of the community of panographers.

(360 x 180) and (360 x 360) should be always associated when needed (explicitly or implicitly) with their respective and different projection format to settle the never ending controversy...

Michel Thoby

1 January 2010; revised 2 jan 2009