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I would like to know a general formula which relates the field-of-view (FoV) to the size of an image and focal length.

The FoV formula for standard cameras is (see e.g. here)

formula

where f is the focal length. That formula can be derived from the relation formula which relates the distance between the image point and the principal point, r, and the angle between the principal axis and the incoming ray, formula, which reads:

formula

However, the above formula only works for standard cameras with perspective projection. Wide-angle fisheye cameras can have different relations formula, see for instance here different projections.

The only example in the web I found about FoV for fisheye cameras is in this page, where they say that the FoV is given by

formula

which would be consistent with the above reasoning assuming the equisolid angle projection from this paper.

Is it safe to assume that we can compute the FoV just by inverting the relation formula for each specific projection, as done with standard perspective projection? Is there any rigorous proof for that?

PabloVD
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  • What are you worried about? Is your question "For fisheye camera, is it possible that several different `theta` are projected to the same `r`?" ? – fana Aug 02 '23 at 01:11

1 Answers1

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Normally (at least, for fisheye), radial directional projection function describes one-to-one correspondence between r and theta (r increases as theta increases) for actual range of r.

Therefore, if the function is given as
r = F(theta),
you will be able to define the inverse:
theta = F-1(r)
for actual range of r.

If the above is not true for a camera, such camera will not be called as "fisheye", I think.

fana
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  • Thanks, that made it more clear. In general then, it is given by the inverse, as FoV = 2 F^(-1)(image_size/2). – PabloVD Aug 03 '23 at 08:42