Mapping the value of a sample in a 2-D DFT to cycles/pixel

Question

If I have an image and its 2-D DFT of that image, what is the mapping between the value of the DFT at (u,v), and the frequency in the spatial domain in the x and y components, in cycles/pixel? I want to be able to say "There is a peak in the DFT at (u,v), which corresponds to this frequency". I don't think the answer is u cycles/pixel, since the values shift with different image sizes, even if the image content remains the same.

Answer 1

Well the Nyquist frequency is 0.5 cycles / pixel and the range of frequencies can be considered to range from -0.5 to +0.5 cycles / pixel. So if the dimension in a given axis is N pixels then each pixel is 1 / N cycles per pixel.

Depending on your particular DFT/FFT implementation you may have zero frequency in the middle of the image (M/2, N/2) or in the corner (0, 0). For the former case the spatial frequency at u, v will be:

(M/2 - u) / M, (N/2 - v) / N