I just learned about the 1-dimensional discrete Fourier transform which got introduced to me by means of circulant matrices (in the context of convolutions) and noticing that they all have the same eigenvectors. Now I want to learn about the 2D Fourier transform but I couldn't find an explanation so far that satisfied me (most of them solely state the formula without deriving it), which is why I'm asking whether someone knows a good source where the 2D discrete Fourier transform gets properly derived/developed.
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$\begingroup$ You could try to review a book about Optics, like "Fourier Optics" by J. Goodman.. the continuous version of the 2D Fourier Transform is important in Optics since the Far Field profile of a light beam is the 2D Fourier Transform of the incoming beam, as is also the profile that happens in the focal plane of a aspherical lens, so you could in practice to calculate and apply filters in the Fourier Plane doing calculations at speedlight because of this, also taking cross-correlations. In the same book, I believe is constructed the discrete case from the continuous case, to use it in simulations. $\endgroup$– JoakoNov 7, 2021 at 21:51
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$\begingroup$ One can consider 2D Fourier transform as a sequence of two 1-dimensional discrete Fourier transforms: applied to the first variable and then to the second. The properties follow immediately. Not much new in the multidimensional case. $\endgroup$– AndrewNov 8, 2021 at 5:38
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2 Answers
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I think any image processing course will introduce the 2-D DFT. It is hard to recommend anything in particular not knowing what you want to do with it, but:
If you are interested in image processing oriented introduction, let me recommend this: https://www.robots.ox.ac.uk/~az/lectures/ia/lect2.pdf
If you want something a bit broader, I would start with this : https://see.stanford.edu/materials/lsoftaee261/chap8.pdf
If you want something more theoretical and mathematical, .... then the relevant wikipedia pages are a good place to start in my opinion.
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$\begingroup$ I have already found all the sources you mentioned (before asking this question I clearly did some research), but none of them really derive the 2D DFT but rather define it. Hence, I seek an explanation that introduces this topic by maybe approximating 2D functions with the respective 2D trig. functions. $\endgroup$ Nov 4, 2021 at 15:50
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$\begingroup$ I see. Sorry I could not be more helpful. Can you please pinpoint what is missing from e.g. the Wikipedia page en.wikipedia.org/wiki/Fourier_transform ? $\endgroup$ Nov 4, 2021 at 16:08
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$\begingroup$ Yes, I might go through all of the stated sources, I just hoped that someone on this platform might have a good recommendation. $\endgroup$ Nov 4, 2021 at 18:10
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I can recommend "Fouerier Analysis and its applications" by Gerald B. Folland. If you first want to take a look into the book (or can't afford it) you can find a photocopy of it as one of the first google entries when searching the title.
