In the context of image processing (and computer vision), the concept of convolution comes up a lot. Convolution is quite related to the concept of Fourier transform and DFT. In the context of image processing, we are mainly interested in DFTs, given that signals (images) are discrete and finite. In this context, I have heard the expression "periodic signal", which reminds me of a sine or a cosine waves, which have a period of $2\pi$, that is, every $2\pi$ the output of these functions repeats. Now, in the case of images, it is not clear to me the meaning of a periodic signal. What would be a periodic image? How is this concept related to DFT and convolution?


You can think of the periodic signals like sine and cosine are basis for building other signal. See this figure for a 1-D example.

For images, sine and cosine extend to 2-D patterns as something like in here. As in 1-D signal, any images can be represented as a weighted sum of these basis images. And roughly speaking, DFT are just weights in the weighted sum above.

Convolution connects with DFT by the nice convolution theorem. Namely, if you have two images (one of them is typically a filter, but you can consider any filter as an image patch as well) A & B, F(A * B)=F(A)F(B). That is, the Fourier transform of the convolved image of A and B is equal to the product of the Fourier transform of image A and the Fourier transform of image B.

  • $\begingroup$ Ok, I know the convolution theorem. But I am more interested in the "periodic" part of "periodic signal" and its related to DFT and convolution, that is, do DFTs and convolution assume some kind of periodicity in their formulations? $\endgroup$ – nbro Apr 4 at 8:44
  • $\begingroup$ You may want to also check out this dsp.stackexchange.com/questions/16586/…. For "classic" signal processing, we always assume signal extend to infinity. So essentially you can interpret that DFT is roughly the DFS (discrete fourier series) of the infinite size image created by repeatedly tiling your original image. So that infinite image is indeed periodic. $\endgroup$ – user559678 Apr 4 at 14:09
  • $\begingroup$ If you are happy with interpretation of finite size signal/image, there is no periodicity besides the part that the basis in DFT (sine and cosine) are "periodic" when extending them to infinity. Note that the low-frequency basis are actually not so periodic within the coverage of the original image. And their periods are not the same. That is why they can add up to represent any image. In terms of convolution, I don't think there is any role of periodicity there. But symmetry plays a role. If one of the images is symmetric, then convolution is the same as correlation (i.e., matching). $\endgroup$ – user559678 Apr 4 at 14:21

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