Why not 3.4 or 4.x?

@asmorkalov backport to 4.x:#22996

Thank you for the invitation! I still think that the existing formula is correct. The proposed formula for sigma2 is only asymptotically correct (that is, when the number of points is huge).

As a "proof", the existing formula seems to be equivalent to the equation (1) of this paper:https://arxiv.org/abs/2107.13484

The issue isn't really specific to camera calibration. Here is a general explanation of why the number of parameters needs to be included in the denominator:https://stats.stackexchange.com/a/64495

I find the Cramer-Rao lower bound a bit confusing and unrelated to this discussion.

@lamm45 Thanks for your contribution!

Not sure if I've really got that explanation on StackExchange (while I've found another reference to it in the Multiple View Geometry in Computer Vision book, see section 5.1.3 at page 134).

Anyway, if we believe in it, then the best solution would be to forbid the calibration whentotal < nparams_nz since all the calibration makes no sense.

(Maybe this condition is incorrect since each point gives at least 2 equations - one per coordinate - and the prohibition bound should be twice bigger here)

And to leave stddevs empty whentotal == nparams_nz.

What do you think?

my memories about this part of statistics are very fuzzy, but if I recall correctly the formula you propose is sort of biased estimate of the standard deviation. I would keep the original formula and, as you suggest, avoided cases when total is less than nparams_nz.

(Maybe this condition is incorrect since each point gives at least 2 equations - one per coordinate - and the prohibition bound should be twice bigger here)

I think you are right, the error estimator should use2*total. And not just for the prohibition bound, but for the actual error computation as well. So basically all estimates have been wrong by up to a factor of two, which is quite significant.

I've finally understood that denominator correction. While it's more related to ordinary least squares estimation variance, I also found a more general explanation of it in MVG book (see the reference in the PR description).
This explanation also contain a note that is strictly related to the calibration problem: when fitting model in the projection of N points, a dimensionality of residuals should be 2*N, not N.
At the same time, I think that error computation (not stddev) is OK since this is how a reprojection error should work: an average across all the points.
So basically the only place wheretotal should be fixed is stddev estimation.

I should have been more precise, but when I said error estimator or computation, I meant the estimator or computation for thestandard error. In this context, standard error is essentially the same as stddev, but I prefer the former.

@asmorkalov Backport to 3.4:#23025

@alalek Backport to 4.x is aligned with 5.x code, please check:#22996

Merged through#23189