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Unclear how calibrateCamera estimates stdDeviations (perhaps wrong) [closed]

asked 2019-10-15 20:26:26 -0500

JamesBaxter gravatar image

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?

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Closed for the following reason the question is answered, right answer was accepted by JamesBaxter
close date 2019-10-16 13:54:30.952053

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answered 2019-10-16 03:23:44 -0500

Eduardo gravatar image

updated 2019-10-16 03:30:42 -0500

Quick answer:

  1. J is the Jacobian that should be of size MxN with M the number of samples and N the number of estimated parameters; JtJ is simply J^T x J that should be of size NxN
  2. See 1
  3. I believe the following applied to non-linear least-squares/non-linear minimization should have the necessary information
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Perfecto, thank you! I spent a long time trying to trace the meaning of JtJ and couldn't figure it out. Also I was confusing the Jacobian wrt the optimisation function (projection error) with the Jacobian of the projection function.

JamesBaxter gravatar imageJamesBaxter ( 2019-10-16 13:52:32 -0500 )edit

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Asked: 2019-10-15 20:26:26 -0500

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Last updated: Oct 16 '19