Gaussian weighted distance between pixel amplitudes: motivations and sources? In several image processing papers, and for instance in Random Walks for Image Segmentation (2006) by Leo Grady, changes in image intensities are turned into edge weights. In some fields, it is called Geographically Weighted Regression. This applies to graphs too, and has become popular for instance in non-local means denoising algorithms. A popular weighting is Gaussian. Quoting Grady (Equation 1, page 1772):


  In this work, we have preferred (for empirical reasons) the typical
  Gaussian weighting function given by: 
  $$ w_{ij} =  \exp{\left(-\beta(g_i-g_j)^2\right)} $$ 
  where $g_i$ indicates the
  image intensity at pixel $i$


My questions are:


what are the earliest/most authoritative references for such a typical Gaussian weighting?
since Gaussian models are not considered the most appropriate for structured image modelling, what are the non-trivial "empirical" benefits of such a weighting, with respect to, eg, a near-Gaussian bisquare function?
$$w_{ij} = (1-\beta(g_i-g_j)^2)^2$$
Maybe not an exact answer, but I'll to give a direction.

What you are using is essentially an RBF-Kernel. First, it has a ready interpretation as a similarity measure (satisfies Mercer's conditions). In essence, other such kernels can be used. For instance, an inner product (linear kernel) would define an angular distance between pixels/features. For a deeper understanding, I would dive in the literature of kernels.

Having said that, the simplest form of such relations are called inverse distance weighting (Grady could have used this as well), where any identity like $1/d(g_i, g_j)^p$ can be used. These methods also cover a wide spectrum.