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I'm a high school student doing an essay on the applications of the Fourier transform on signal processing, but I haven't been able to find much information when applying the discrete fourier transform to an image and calculating it out by hand, and I'd like to check my understanding of the topic.

So from the research I have done, I understand that a black and white image can be represented with a matrix, with each pixel corresponding to a value from 0-255, depending on how black or white it is. Then you transform it row by row, and the new matrix you get you transform again, this time column by column, and then the result you get is the fourier transform of the image.

The thing I'm unsure about is whether you need to take the Fourier transform of the 1st Fourier transform's imaginary part as well as the real part. Since I read somewhere that preserving the phase of the transform is necessary if you want to inverse the transform. Also, are fourier transform representation of images the energy spectrum of the transform? Do I need to center the image somehow? And apply a logarithmic transform?

Essentially, what I'm trying to ask is that, how do I manually calculate the fourier transform representation of a small, 10x10 image, that would be identical to a program directly transforming the image to it's frequency domain.

Also, can anyone provide me with some detailed papers on the subject with manual calculations and explanations?

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  • $\begingroup$ "The thing I'm unsure about is whether you need to take the Fourier transform of the 1st Fourier transform's imaginary part as well as the real part." The answer is yes. The entire complex value of each "pixel" goes into the DFT. $\endgroup$ Sep 6 at 15:24
  • $\begingroup$ The DFT takes complex values in, and outputs complex values. When computing the DFT of a real-valued signal or image, we know that the imaginary component for each pixel is 0. All computations are done with complex numbers. $\endgroup$ Sep 7 at 1:37
  • $\begingroup$ “preserving the phase of the transform is necessary” Yes, this means don’t take the absolute value (magnitude) of the result. You’d often do this for display, but the inverse transform would be applied to the result of the DFT without that magnitude operation. Likewise with the log transformation. Useful for display, but not used in computations. $\endgroup$ Sep 7 at 1:40
  • $\begingroup$ Thanks for the answers. This cleared some stuff up. $\endgroup$
    – botman
    Sep 7 at 13:30
  • $\begingroup$ @robertbristow-johnson The Fourier transform of the image I manually computed has magnitude above 255, above the greyscale range. Do I just extend the range or round it down to 255? – $\endgroup$
    – botman
    Sep 7 at 14:21

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