I know there's a benefit of knowing the Fourier Transform for image processing, but is there a benefit to know Fourier series, or could you just skip them? Would you recommend skipping Fourier series or would that make it too abstract? What's your opinion on this?

  • 2
    $\begingroup$ If you need to process an image of a chessboard, Fourier series might be more useful to know than Fourier Transforms. But yes, you can skip Fourier series as useless knowledge if you already know Fourier Transforms and also believe in impulses (also known as delta functions). $\endgroup$ May 2 '13 at 21:39
  • $\begingroup$ They're kind of the same thing $\endgroup$
    – endolith
    May 3 '13 at 13:36
  • $\begingroup$ @endolith Really? I guess I have different definition of 'the same' than you.... $\endgroup$
    – user8005
    May 3 '13 at 13:37
  • $\begingroup$ Perhaps "the same" is too strong. Look at the table that @endolith linked. Personally I found it easiest to learn by starting in the upper-right hand corner and then moving to the left or down in the table by taking limits (either dividing time more and more finely, or by extending the summation/integration bounds to infinity.) $\endgroup$ May 3 '13 at 16:58
  • $\begingroup$ @user8005: Yes, they are kind of the same: blogs.mathworks.com/steve/2010/03/15/the-dft-and-the-dtft $\endgroup$
    – endolith
    May 3 '13 at 17:38

In image processing, you are always dealing with discrete data, so the transform isn't really a Fourier transform, it is a discrete Fourier transform (DFT with a sum rather than an integral). The DFT can be understood as providing the coefficients of a Fourier series from which a periodic version of you're original image can be reconstructed. The "checker board" image is always implied in the inverse of the DFT which would construct an infinite checker board of squares made up of your original image.

Normally the area outside the boundaries of you're original image is just ignored (treated as undefined), but if for some reason that area must be considered when reconstructing from a DFT (you transformed a sub-section of an image for example) that area will be filled with a periodic replication of of the transformed portion of you're image.

I would say it is worth the little bit of time it takes to understand Fourier series before moving on to Fourier transforms so you understand the underlying assumptions and mechanics of the transform.


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