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?