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I often see examples in textbooks where someone constructs a 2D Fourier spectrum of an image by fixing two (symmetric) pixels to a given value, and computes the inverse DFT to get a nice waveform image. I want to know how these images are created. It seems to me that they pick one pixel that would give them the desired spacing and orientation, fill in the symmetric opposite pixel with the conjugate value, and then somehow fill in the DC component.

This last part is what's confusing me. I want to answer the more general question: if you're given the non-DC frequency components of the Fourier spectrum of an image, how do you compute the DC component? I would be happy with an algorithmic solution, as I'm working on signals in Python with the scipy/numpy suite.

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Not at all. By definition, the components are orthogonal.

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  • $\begingroup$ Then what do they use to decide how to set the DC component? Does it just control the overall brightness of the image? $\endgroup$ – JeremyKun Dec 27 '13 at 0:40
  • $\begingroup$ That's how you could say it, yes. $\endgroup$ – user7358 Dec 27 '13 at 0:59
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    $\begingroup$ The point is that image data is non-negative. So the control of the DC component allows to make sure that the resulting signal is non-negative. For your scenario the DC component should be twice the magnitude of each of the two bins you use for synthesis. For more complicated setups the solution is not that simple however. $\endgroup$ – Jazzmaniac Dec 27 '13 at 13:12
  • $\begingroup$ Most of the time, though, when plotting images, the image library takes care of normalizing all of the values involved, so nonnegativity isn't an issue. $\endgroup$ – JeremyKun Dec 29 '13 at 20:23

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