I am reading a paper about High Dynamic Range (HDR) reconstruction. One of the core problems there is how to reconstruct pixel irradiance from multiple observations. Typically, weighted averaging is used; a number of weighting schemes were presented in the literature. In the mentioned paper they evaluate different weighting schemes by comparing their achieved signal-to-noise ratios. I do not understand how exactly this comparison is done.

The outlined approach is as follows. There is a pre-recorded set of images. First, using a reference method, "ground truth" reconstruction $\mu_X$ and its "optimal" variance $\sigma_{\mu_X}^{2(opt)}$ are computed. Then, for every scheme, weighted mean $\hat{\mu}_X$ and its variance $\sigma_{\hat{\mu}_X}^{2(w)}$ are calculated. Finally, they go on to say:

From $\sigma_{\hat{\mu}_X}^{(w)}$, the ground truth HDR image $\mu_X$, and the bias $(\mu_X - \hat{\mu}_X)$, we compute the signal-to-noise ratio achieved by each weighting.

Unfortunately, no formula is given. I know that in imaging, SNR is defined as $\frac{\mu}{\sigma}$. But how exactly they incorporate bias there?

In principle, one can just compute $\mathrm{SNR}^{(w)}=\frac{\hat{\mu}_X}{\sigma_{\hat{\mu}_X}^{(w)}}$ for each scheme and compare them. However, it does not feel right, because it does not take into account how close the computed mean is to the ground truth.

So my question is: which formula they use? Or, more generally, how to compare SNR of different weighting schemes, given that ground truth signal is known.



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