Which of the following functions can be the covariance functions to a stationary process and which cannot?   A digital image consists of pixels, M(x,y), arranged in a large matrix. Each pixel takes a...

Which of the following functions can be the covariance functions to a stationary process and which cannot?

A digital image consists of pixels, M(x,y), arranged in a large matrix. Each pixel takes a real value, which determines its color. The image captured by a digital camera is often affected by noise if the light is dim. In order to reduce the effect of the noise we replace each pixel value of the image with the average of the original pixel value and the four closest pixel values:

Using statistical terminology, we might say that we estimate the “true” pixel value with the average of the outcomes of five (possibly correlated) random variables. Your task is to compute the variance of this estimate. We consider the image as a stochastic process:

where S(x,y) is a constant (the “true” image) and
  is correlated noise with covariance
  and δ is the ordinary euclidian distance between the pixel
  Using this model, compute the variance