# 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.

• This is amazing. What is the required knowledge to study image processing ? – user230452 Feb 26 '16 at 2:15
• There is no reason, a priori, to assume that the removed components are indeed noise. – copper.hat Feb 26 '16 at 2:16
• they estimate the power spectral density of the noise, and a filter to remove it, in the 2D discrete Fourier transform domain, probably the DFT of the auto-correlation. alternatively, it could be an autoregressive model/LPC coefficient estimation, again from the auto-correlation of the image – reuns Feb 26 '16 at 4:53
• It is unusual for an optical image to have spikes in the frequency domain so far away from DC, so I'm not as dismissive as others that there are no applicable approaches. In general, you want to know the process of how the image was formed and use that to tailor your approach. For instance, you may know that you have a bad sensor that creates these spikes for every image, and then this approach would work just fine. – AnonSubmitter85 Feb 26 '16 at 14:46
• @copper.hat I'm not so sure. How often does a properly formed optical image have spikes that far away from DC? – AnonSubmitter85 Feb 26 '16 at 14:50

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.

• Spikes in the spectrum at this location, regardless of whether they form a circular group or are perfect sinusoids, are in all likelihood not part of an accurate optical image. – AnonSubmitter85 Feb 26 '16 at 14:49
• @leonbloy Indeed, there's no magic. I AM, however, not ashamed to use the word, as it perfectly conveys my enthusiasm for this example (and to hopefully stir up some enthusiasm from any young readers that come across this post)! Since this example has sent me down a path to understand the topic further, and when it's methods are appropriate, I thought I'd try to pass the feeling along to other enthusiastic learners here on the messageboards... I seem to have succeeded with user230452 above :D – Lewkrr Feb 29 '16 at 23:17