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### Unclear how calibrateCamera estimates stdDeviations (perhaps wrong)

Hi, I'm researching camera calibration and uncertainty propagation. I'm trying to understand how calibrateCameraExtended estimates stdDeviationsIntrinsics and stdDeviationsExtrinsics. The docs say very little.

So I go straight to the source code. Check this lines of code where I understand the calculation is made. It starts with sigma2 (the "deviation of the noise") calculated as norm(allErrors, NORM_L2SQR) / (total - nparams_nz); which is just the formula for the unbiased estimator of the variance. Ok so far.

And then it calculates each s-element of the vector of standard deviations stdDevsM by

stdDevsM.at<double>(s) = std::sqrt(JtJinv.at<double>(j,j) * sigma2);

Where JtJinv is the pseudo-inverse of the jacobian calculated a few lines above from _JtJ which in turn comes from the LM solver invoked in previous lines.

First question: what exactly is _JtJ? I assume it must be the 1xN Jacobian of the projection error with respect to the parameters (there are N parameters). I've tried to trace the calculation of all the way to its origin, I got this far , but I'm not sure.

Second question: The moore-penrose of a 1xN matrix is a Nx1 matrix. So calling JtJinv.at<double>(j,j) with two indices j confuses me.

Third question: I couldn't make sense of the formula itself and I think It's wrong. The code has the comment

//see any papers about variance of the least squares estimator for
//detailed description of the variance estimation methods

But my understanding from simple uncertainty propagation for the case of uncorrelated parameters is that the Jacobian vector _JtJ, the variance of the projection error sigma2 and the vector of parameters standard deviations stdDevsM should follow (in matlab-like pseudocode):

sigma2 = dot(_JtJ.^2, stdDevsM.^2)

Which is ill-conditioned, the simplest solution would be

stdDevsM = sqrt(sigma2) .*_JtJ ./ norm(_JtJ.^2)

Is my reasoning correct? and, where does the calculation implemented in OpenCV come from?