I am not in this field but I just remember from my studies that variance is the most important parameter for noise. Mean can always be subtracted mathematically to remove its undesirable effect but variance is the parasitical effect of noise (which varies unpredictably) in our signal (image or communication signal). They are different technique to evaluate this variance such as the maximum likelihood technique.

I am not sure if this can help you.

If you would calculate the eigenvalues of the MRI image you will observe that there are a few very large ones and the others are very small. The assumption mostly used is that the significant eigenvalues correspond to the useful data and the remaining smaller ones are contributed by noise. This you may have observed in several papers which retain the components that explain a certain percentage of the variance (e.g. 90 %) are retained and the other are discarded as a pre-filtering step.

In this case I think that the authors are trying to establish  a new criteria to do so. The mean as you know is effected by the outliers (in this case the high eigenvalues) hence the median is used. This will then allow then to establish a more fair criteria of component rejection.

Hope this is relevant

Hi. The PCA makes a reduction of dimension based on eigenvalues with a variance conservation. It is assumed that the eigenvectors produced by the method are associated with two axis: one with the signal and noise and the other by the noise only. Thus, the variance in the second coordinate system is an approximation of the noise value and behavior on the original image with the assumption that the original image will have a minimum variance in the second axis. Good luck!