Definition of contrast-to-noise ratio?

Question

I have been looking at definitions of the contrast-to-noise ratio. As indicated in numerous sources (e.g. here (page 12)), the contrast to noise ratio between two signals $A$ and $B$ is: $$ CNR=\frac{S_A-S_B}{\sigma}=|SNR_A-SNR_B|$$ Where $SNR_i$ denotes the signal-to-noise ratio of the $i$th signal, $S_i$ is the $i$th signal and $\sigma$ the standard deviation due to noise.

The problem with this is that the standard deviation $\sigma$ may be different for the two signals $A$ and $B$. In such a case what is the typical value used for $\sigma$? (A source would be helpful)

Answer 1

This link, http://users.fmrib.ox.ac.uk/~robson/internal/SignaltoNoise.html talks about the difference, but it only has a prescription for what to use for sigma for MRI or fMRI images for equipment depending on number of coils. No analysis and not too useful for general cases.

I think is issue is more than medical imaging, any image recognition need will have this issue. It's not a hard topic, you may need to get the sigmas in regions of interest, and then use the right one for each S. This link will have more, but not sure the material is free http://www.dspguide.com/ch25/3.htm