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