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.

David Marshall 1994-1997