Question: Each pixel value in 500 × 500 pixel image I is an independent normally distributed random variable with zero mean and standard deviation one. I is convolved with the 2k + 1 x 2k + 1 kernel G.
What is the covariance of pixel values in the result? (There are two ways to do this; on a case-by-case basis - e.g. at points that are greater than 2k + 1 apart in either the x or y direction, the values are clearly independent - or in one fell swoop. Don't worry about the pixel values at the boundary.)