Fourier Transforms and Convolutions

One major reason that Fourier transforms are so important in image processing is the convolution theorem which states that

If f(x) and g(x) are two functions with Fourier transforms F(u) and G(u), then the Fourier transform of the convolution f(x)*g(x) is simply the product of the Fourier transforms of the two functions, F(u) G(u).
Thus in principle we can undo a convolution. e.g. to compensate for a less than ideal image capture system:

• Take the Fourier transform of the imperfect image,
• Take the Fourier transform of the function describing the effect of the system,
• Divide the former by the latter to obtain the Fourier transform of the ideal image.
• Inverse Fourier transform to recover the ideal image.

This process is sometimes referred to as deconvolution.
See Handouts/Books for other useful Fourier Transform Properties.