Page 144 -

3.4 Fourier transforms 123

3.4.2 Two-dimensional Fourier transforms

The formulas and insights we have developed for one-dimensional signals and their trans-
forms translate directly to two-dimensional images. Here, instead of just specifying a hor-
izontal or vertical frequency ω x or ω y , we can create an oriented sinusoid of frequency
(ω x ,ω y ),
s(x, y) = sin(ω x x + ω y y). (3.62)
The corresponding two-dimensional Fourier transforms are then

∞ ∞

H(ω x ,ω y )= h(x, y)e −j(ω x x+ω y y) dx dy, (3.63)
− −
and in the discrete domain,

M−1 N−1
1 −j2π k x x+k y y
H(k x ,k y )= h(x, y)e MN , (3.64)
MN
x=0 y=0
where M and N are the width and height of the image.
All of the Fourier transform properties from Table 3.1 carry over to two dimensions if
we replace the scalar variables x, ω, x 0 and a with their 2D vector counterparts x =(x, y),
ω =(ω x ,ω y ), x 0 =(x 0 ,y 0 ), and a =(a x ,a y ), and use vector inner products instead of
multiplications.

3.4.3 Wiener filtering
While the Fourier transform is a useful tool for analyzing the frequency characteristics of a
filter kernel or image, it can also be used to analyze the frequency spectrum of a whole class
of images.
A simple model for images is to assume that they are random noise fields whose expected
magnitude at each frequency is given by this power spectrum P s (ω x ,ω y ), i.e.,

2
[S(ω x ,ω y )] = P s (ω x ,ω y ), (3.65)
where the angle brackets ·x denote the expected (mean) value of a random variable. 9 To
generate such an image, we simply create a random Gaussian noise image S(ω x ,ω y ) where
10
each “pixel” is a zero-mean Gaussian of variance P s (ω x ,ω y ) and then take its inverse FFT.
The observation that signal spectra capture a first-order description of spatial statistics
is widely used in signal and image processing. In particular, assuming that an image is a
sample from a correlated Gaussian random noise field combined with a statistical model of
the measurement process yields an optimum restoration filter known as the Wiener filter. 11
To derive the Wiener filter, we analyze each frequency component of a signal’s Fourier
transform independently. The noisy image formation process can be written as
o(x, y)= s(x, y)+ n(x, y), (3.66)

9
The notation E[·] is also commonly used.
10
We set the DC (i.e., constant) component at S(0, 0) to the mean grey level. See Algorithm C.1 in Appendix C.2
for code to generate Gaussian noise.
11 Wiener is pronounced “veener” since, in German, the “w” is pronounced “v”. Remember that next time you
order “Wiener schnitzel”.