Image Processing Reference
In-Depth Information
Appendix A: Review of Fourier
Analysis
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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
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