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checkerboard depth location and parameter uncertainty

When I use Camera Calibration Toolbox for Matlab, I found checkerboard placement has some impact on the variance of the measured parameters. Specifically, if the checkerboard is placed relatively far away from the camera, the uncertainty of both focal length and principal axis increase. The placement is far away enough that the checkerboard only occupies maybe a quarter of the image.

My questions are how the toolbox compute the variances, why placing the checkerboard far away can increase the variances.

Also, even the checkerboard is placed relatively close to the camera, there is still depth variation due to the orientation of the plane and some small travel of the plane in depth direction. If the focus does not change, I would assume that the calibration works for all depth besides the calibration depth, since the physical property of the camera has not changed at all. Is that a safe assumption?

Thank you in advance.