Fourier transform of a 2D image, and noise cancelation I'm a statistics grad student, and I just started getting into Digital-Image-Processing (an analogy for processing super-large contingency tables).  In the book "Digital Image Processing" by Gonzalez and Woods, I was reading chapter 2, page 94 of the third edition, and found the following image that is MAGIC to me...

They begin with a noisy image, identify some 'hot points' surrounding the center of the Fourier transform, then simply remove them, reverse the transform, and POW!  Magically the image is crystal clear!!!
Does anyone have any insite into this?  Are there any statistical approaches that can be used to test a hypothesis that these 'hot points' are in fact interference?  Are there other interesting approaches to analyzing these transforms statistically?? 
If any of the StackExchange users a familiar with this topic and can suggest some articles/books that I might read, I would be very grateful. 
 A: Don't believe in magic. In practice image restoration (same as audio restoration) very rarely gives subjectively impressive results (specially by such simple methods). 
This example is very artificial. Starting from a sharp an "natural" photograph, we have added a synthesized noise (or "interference") consisting of perfect 2D sinusoids of different frequencies $\omega_1,\omega_2$, with the property $\omega_1^2+\omega_2^2=\alpha$ (constant). Because of this fortunate (and artificial) property, we can filter them by using a 2D notch filter having that circular form. Nothing magic.
My advice: many (not all) 2D algorithms (specially linear filters, Fourier processing, etc) are generalizations of 1D algorithms. So, start by understanding traditional 1D signal processing. 
In this case, imagine you can a "natural" audio signal, which happens to be corrupted by a pure sinusoidal noise. By appyling a notch filter, you can suppress it (almost) perfectly. Impressive? Perhaps no so much (we cannot "see" that). Anyway, natural audio signals are not typically corrupted by pure sinusoids.
