Noise suppression by smoothing
1. Prove that it is not possible to improve the signal-to-noise ratio for a arbitrary single wave number with a linear smoothing filter H. (Hint: write the image G as a sum of the signal part S and the noise part N.)
2. Assume white noise (equally distributed over all wave numbers), but a spectrum of the signal that is only equally distributed up to half of the maximum wave number. Is it now possible to improve the signal-to-noise ratio integrated over all wave numbers? What is the shape of the transfer function that optimizes the signal-to-noise ratio?