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Additive noise is primarily a local process

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.



Todd Veldhuizen
Fri Jan 16 15:16:31 EST 1998