Graphics Reference
In-Depth Information
and its Fourier transform on a “frequency” domain. Comparing equations (21.11) and
(21.16) we see that if f(x) were expressed as a Fourier series, then F(u) picks up the
coefficient of an appropriate periodic term of the series and intuitively expresses the
contribution of that term to the values of f. The Fourier transform and its inverse take
us back and forth between these domains.
Moving on to functions of two variables, assuming that integrals over unbounded
regions have been defined, we have
If f, G:
R
2
Definition.
Æ
R
, then
(
)
(
)
=
Ú
(
)
-
2
p
i
ux
+
vy
F u v
,
f x y e
,
dxdy
(21.19)
2
R
is called the
Fourier transform of f(x,y)
and
(
)
(
)
=
Ú
(
)
2
p
i
ux
+
vy
g x y
,
G u v e
,
dudv
(21.20)
2
R
is called the
inverse Fourier transform of G(u,v)
. Like in the one-dimensional case, the
functions F and g will be denoted by FT(f) and FT
-1
(G), respectively.
Sufficient conditions for a function f(x,y) to have a Fourier or inverse Fourier
transform are similar to the one variable case. The case of “separable” functions
simplifies the analysis somewhat. A function f(x,y) is called
separable
if
(
)
=
() ()
12
fxy
,
f xf y
for some functions f
i
(x). In that case, the integral in equation (21.19) can be separated
into two one-dimensional integrations and the proofs in that case can be used to show
that the integral in equation (21.19) exists.
1
2
The box function B(x,y):
B (x,y) = 0, |x|, |y|>
,
1
2
= 1, |x|, |y|£
.
The function sinc(x,y):
sinc(x,y) = sinc(x) sinc(y)
Figure 21.7 shows the Fourier transform of the function B(x,y). It is just sinc(u,v).
Figure 21.7.
The Fourier transform of the box
function B(x,y).
