Theoretical analysis on random errors for different shape functions in digital image correlation

In 2002, it is found that second-order shape functions exhibit roughly twice random error as first-order shape functions (Schreier et al. Exp Mech, 42(3) 303-310). Or equivalently, the variances of second-order shape function 4 times that of first order shape function.

I'm interested in the phenomenon of the ratio being 4. Since there have been theoretical formulae for random error (Wang et al. Strain, 45 (2) (2009), 160-178, Wang et al. Exp Mech 55 (9) (2015) 1717-1727), I did some calculation. Nevertheless, I found that the ratio approaches 3.5 rather than 4, as this source code will indicates.

You can recheck my finding by running file "CompareHessianAndEnsenbleAverage.m" with MATLAB. There are 10 speckle images can be chosen by modifying the "img_id". I suggest to use Pattern05 because that pattern seems more statistical stable.

I built a model to explain why the ratio is 3.5 instead of 4. I think the deep reason is uniformity and isotropy of speckle pattern. I wrote a paper and it has been published in Optics and Lasers in Engineering.

I should emphasize that this is not an applied research, because the difference between 3.5 and 4 is negligible in practice. This research is just an answer to my question.