Within my field a recent study suggested to use the symmetric properties of certain image datasets to improve signal to noise ratio (SNR). I will spare you the details, but in the end one can get a symmetric dataset (as described below) with improved SNR. However the way that one typically computes SNR is to compare the "amount of signal" (sum of the pixels in the central region, more or less), to the overall standard deviation (std).
The properties of the symmetrized data are such that if one computes the std over the entire image (in a standard way), std is indeed lower than in the original image, leading to the improved SNR.
However I am bit skeptical regarding the fact that one should compute the std in the same way as in a "normal image". Indeed, here we have an image with identical pixels according to certain symmetries (around central vertical axis, horizontal vertical axis, both diagonals, and [$\pi/2$] rotations). Many pixels are hence correlated, and I am not sure that simply computing the std across the whole map is a fair assessment of the SNR.
The question is: how do I compute the standard deviation from such a dataset? Should I take into account the fact that many pixels are not independent (correlated)? How?
example symmetric data set (don't mind the central black circle): https://i.stack.imgur.com/TzlAS.png