Firstly, let's say that in order to smooth an image, I convolve it with a Gaussian function having standard deviation $\sigma_x$ and $\sigma_y$. I am now interested in knowing if there exist methods for estimating $\sigma_x$ and $\sigma_y$ from the smoothed image?
Secondly, to potentially make things more complicated, let's say the image pixels instead are drawn from a zero mean unit variance normal distribution (top image), which is then convolved as above (bottom image), could the standard deviations be estimated from the smoothed image?
Here is the MATLAB code used to generate the images. What I would like to achieve is to estimate the standard deviations, which in this case were $\sigma_x=2$ and $\sigma_y=2$.
img = randn(128,128); figure(1); imagesc(img); colormap(gray); axis off; title('Noise image') nimg = imgaussfilt(img, [2 2]); figure(2); imagesc(nimg); colormap(gray); axis off; title('Smoothed noise image')