How is WSNR (weighted signal to noise ratio) defined?
There might be several versions, yet the most probable is as follows. An SNR is an energy-dependent measure. In the time or spatial domain, it is agnostic to location: all samples have the same "weight" (uniform weighting). As an energetic measure, it can be computed in the Fourier domain as well (or any orthogonal transformation). 

Then, from the sensory system, we know that in hearing, or vision, not all frequencies have the same perceptual weight: a human hearing a loud 35 kHz tone won't probably not notice it, while a bat or a dog will notice. Hence, the relative importance of a sound should be weighted by the perceptual sensitivity of the subject (bat, dog, human). This yields a different weighting across different frequencies. One shall not care about a 35 kHz tone for a human ear, and give it a little weight, compared to the [1-2] kHz band. Same happens for vision. Such weightings are used for instance in JPEG compression.

So instead of computing, in the frequency domain some:

$$-10 \log \sum_k \frac{\left(\hat{s_k}-\hat{n_k}\right)^2}{\hat{s_k}^2}$$
with $k$ an index of frequency bins, you use a predefined weighting for each $k$:
$$-10 \log \sum_k w_k\frac{\left(\hat{s_k}-\hat{n_k}\right)^2}{\hat{s_k}^2}\,,$$
where  $w_k$ give less weight to less important frequencies. In sound, you can find A-weighting curves (and B-, C-, D- and Z-). 

In images, a Matlab code function ratio = wsnr(orig, dith, nfreq) gives references to ancient papers:


J. Mannos and D. Sakrison, "The effects of a visual fidelity criterion on the encoding of images", IEEE Trans. Inf. Theory, July 1974.
T. Mitsa and K. Varkur, "Evaluation of contrast sensitivity functions for the formulation of quality measures incorporated in halftoning algorithms", IEEE Int. Conference on Acoustics, Speech and Signal Processing, 1993