# What are the parameters of a calibrated spherical camera?

I'm aware if we have a pinhole camera model several parameters describe the specific camera (such as aspect-ratio, focal length, principal point, distortion parameters etc).

There're standard procedures to estimate such parameters that one can follow.

I was wondering however if I replaced the pinhole camera with a spherical camera (such as 360 camera for example) what parameters describe such camera?

I would suppose radius is an example of parameter, are there any other relevant parameters in such case?

Is there any reference you can point out?

...if we have a pinhole camera model several parameters describe the specific camera (such as aspect-ratio, focal length, principal point, distortion parameters etc).

"Distortion parameters" does not sound like a typical pinhole camera model. A pinhole camera does not have a "finite apperture" or way of focusing light other than a tiny little opening which all rays of light (that will go on to produce the image) go through. You can take lens distortions into account by adding them to the pinhole model as additional operations but that then is not the standard pinhole camera model.

The pinhole camera model is a very simple transformation of a point from a global 3D coordinate system to a local 2D coordinate system. It does nothing more than that. It maps the angle at which a point appears within its field of view to a point on its focusing plane. Another useful way to be thinking about lenses is in terms of mapping points in front of them to points behind them.

The pinhole camera model is the simplest of those transformations and requires:

1. The position of the camera in global coordinates (Where is it?).
2. The orientation of the camera (Where does it point to?)
3. The "Focal Length"
4. The sensor size
5. The principal point

The focal length in particular is exactly the "thing" that creates this simple map. The shorter the focal length, the wider is the field of view within which points get mapped to a plane behind the pinhole and vice versa.

But notice here, rays in front of the pinhole enter it at some angle and leave exactly behind the pihnole at the same angle.

I was wondering however if I replaced the pinhole camera with a spherical camera (such as 360 camera for example) what parameters describe such camera?

A pinhole 360 camera is a pinhole camera with 360 degrees field of view. Exactly the same set of parameters but now practically points can enter the pinhole from anywhere around it and depart at the same angle behind it. Notice here that in this view the "focal length" is still valid as a "ratio" of lengths.

But in reality, there are no 360 degree "pinholes". Instead, the wide field of view is acquired in some other way. That might be a mirror for example or multiple cameras with various lenses.

In this case, it is even more useful to be thinking in terms of rays entering the lens and rays departing from the lens (or optical system in general). The general model in this case is established around a transfer matrix that maps "entry points" to "exit points" and basically can model any sort of configuration you like.

In terms of references, Geyer and Daniilidis:A unifying theory for central panoramic systems and practical implications is probably still relevant and since then, Daniilidis has also put together a volume of relevant work, the contents of which you might find useful, even if it is just for more relevant terminology to get you closer to what you are dealing with.

Hope this helps.

• Is the book any good? Also does the book cover the "360 pinhole camera" model? Apr 2, 2019 at 9:43
• Sorry also you mention "a pinhole 360 camera is a pinhole camera with 360 degree field of view". The image plane isn't a plane however you have a sphere which is a different manifold. Apr 2, 2019 at 10:02
• @user8469759 I do not have the book, the paper was very useful however but as a reference it was becoming old by now. Looking around, I spotted the book which brings together more work on the topic and this is why I suggested it. On the second point: I was thinking more in terms of mapping angles to image pixel locations. Are you looking at particular application?
– A_A
Apr 2, 2019 at 10:27
• Think at classic structure from motion, but change the pinhole camera with a 360. Just want to know how to estimate the parameters in these case, if there's the need. In classic reconstruction you would estimate the intrinsics and the pose. It's clear (to me at least) that in the case of spherical camera a matrix would still describe the camera pose, as far as the intrinsics goes I'm not sure what's the actual equivalent. And I'd really like to understand what's the image model in this case. Apr 2, 2019 at 10:49
• @user8469759 There is a lot of literature on "Structure from motion with panoramic cameras" (?) and the recommended book also has a section on it and calibration specifically. With 360s you still determine relative location and orientation. Think of it like recovering the camera up vector which might have an offset compared to another view.
– A_A
Apr 2, 2019 at 11:21