Image Processing Reference
In-Depth Information
Appendix A: Review of Fourier
Analysis
bACkgRound to the FouRIeR tRAnSFoRM
The Fourier transform in 1-D, with the notation for k -space, is written as
+∞
Fk
()
=
fxe
()
ik x
x
(
) d
x
(A.1)
x
−∞
The Fourier transform in 2-D k -space is
+∞
+∞
+∞
+∞
∫∫
∫∫
ik xky
x
(
+
) dd
Fk k
(, )
=
f xye
( ,)
x
y
=
f xy
(, )
d yye
ik x
x
(
) d
x
(A.2)
y
xy
−∞
−∞
−∞
−∞
For the 1-D example, we can write this as
+∞
Fk
()
=
fx
( (
cos(
kx
)
+
i
sin(
kxx
) d
)
(A.3)
x
x
x
−∞
This can be interpreted as the multiplication of f ( x ) by a cosine and a sine
function followed by integration over all of x . Another description of this is
a “projection” of a cosine or sine onto f ( x ). This operation provides a measure
of the relative weight of each specific (sine and cosine) spatial frequency that
is required to reconstruct f ( x ) from this particular basis or expansion set of
functions representing f ( x ), that is, a basis set that happens to be sines and
cosines. The same terminology applies if we were to represent f ( x ) using a dif-
ferent basis or set of functions such as those discussed in the chapter on the
PDFT (Chapter 7 and Appendix C). F ( k x ) is a complex number and the Fourier
coefficient for the spatial frequency k x .
The case when k x = 0, this gives F
+∞ d which is a measure of
the area under f ( x ) or, in terms of the Fourier transform, is the zero frequency
coefficient which is a measure of the DC level in f ( x ) . The inverse Fourier
transform is written as fx
()
0 = −∞
f xx
()
+∞
()
=
Fk e
()
ik x
(
)
d
k
=
FF
1
{}
. Clearly, we have
x
x
x
−∞
f = F −1 F { f }.
Some properties of the Fourier transform are:
Linearity: F { Af + Bg } = AF + BG
Similarity or scaling:
F { f ( ax )} = [ F ( k x / a )]/| a |
189
Search WWH ::

Custom Search