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If I have understood correctly, camera intrinsic parameters and pose estimated with 7x4 chessboard corners give a lower reprojection error than using 7x4 + 4x5x4 points?

If you have correctly updated the 3D objects points to take into account the new 4 Aruco corners, my impression would be that more points would theoretically be better in the ideal case.

Since you have observed the inverse, I agree with @swebb_denver The discrepancy comes most likely from the accuracy of the detected 2D corners. Could be that in one case cornerSubPix is used for Aruco but maybe line intersection is used for chessboard? Or maybe the refinement algorithm works better with corners from intersection compared to corners from a square shape? By the way, it looks like the chessboard squares overlap a little. Maybe the figure or the script generator have to be fixed (see this PR)?

To validate or not this hypothesis, I would generate the camera intrinsic parameters and some realistic camera poses. This way, you can use simulated data and try to see the influence of a small noise in the 2D corner locations with the estimated intrinsics and camera pose. Then, multiple experiments could be done:

1. accuracy when using only chessboard corners with some noise
2. accuracy when using chessboard corners + Aruco corners with the same noise applied
3. accuracy when using chessboard corners + Aruco corners with a bigger noise for the Aruco corners

I would expect:

• RMS_2 < RMS_3
• RMS_1 < RMS_3?
• and normally RMS_2 <= RMS_1

If RMS_2 > RMS_1, in my opinion that would mean that the geometric configuration of the 3D object points influences on the accuracy of the estimation. And, or the repartition of the 2D corners in the whole image.

This is indeed the case but here we should have more or less the same configuration, a regular grid with a rectangular shape, for both cases.

There is a paper on this topic, /cc @racuna :

Insights into the robustness of control point configurations for homography and planar pose estimation, Raul Acuna, Volker Willert

The abstract:

In this paper, we investigate the influence of the spatial configuration of a number of n≥4 control points on the accuracy and robustness of space resection methods, e.g. used by a fiducial marker for pose estimation. We find robust configurations of control points by minimizing the first order perturbed solution of the DLT algorithm which is equivalent to minimizing the condition number of the data matrix. An empirical statistical evaluation is presented verifying that these optimized control point configurations not only increase the performance of the DLT homography estimation but also improve the performance of planar pose estimation methods like IPPE and EPnP, including the iterative minimization of the reprojection error which is the most accurate algorithm. We provide the characteristics of stable control point configurations for real-world noisy camera data that are practically independent on the camera pose and form certain symmetric patterns dependent on the number of points. Finally, we present a comparison of optimized configuration versus the number of control points.