Consider an image degraded by zero-mean, Additive White Gaussian Noise
(AWGN) with variance . The power spectrum of the noise is:
The noise has the same power ( ) everywhere in the frequency
domain. One might conclude from this that all spatial frequencies are
affected equally by the noise, but this is misleading.
Figure 1.12: Frequency-domain view of additive noise
Figure 1.12 illustrates the noise power spectrum. Low frequencies
lie (roughly) inside the circle (
is the
wavelength of the frequency component, in pixels). The amount of
noise power inside this circle can be approximated by comparing its
area to the area of the transform. The circle area is
. The domain of the transform
is
, which has area 1. So the fraction of noise
which lies at wavelengths above 10 pixels is only 0.0314, or about
3%. Similarly, only 12.6% of the noise lies at wavelengths above
5 pixels.
The signal strength is typically very strong for wavelengths below
10 pixels, so the effect of the low-frequency noise on the signal-to-noise
ratio is minor.
From these arguments, one can conclude that additive white noise is primarily a local phenomenon. This implies that the use of a local filter to remove additive noise is reasonable.