Image Processing Reference
In-Depth Information
ω
y
ω
s
y
ω
y
0
ω
x
-ω
s
x
ω
s
x
ω
x
0
-ω
s
y
FIGURE 2.9
Fourier transform of the sampled signal.
v
s
x
v
x0
v
x0
v
s
y
v
y0
v
y0
!
v
s
x
2
v
x0
f
s
x
2f
x0
v
y0
!
(
2
:
38
)
v
s
y
2
f
s
y
2f
y0
Thus we obtain the Nyquist theorem for sampling,
1
2f
x0
D
x
(
2
:
39
)
1
2f
y0
D
y
2.4.1 T
WO
-D
IMENSIONAL
S
AMPLING
T
HEOREM
Suppose a 2-D function f
a
(
x, y
)
, that is, the Fourier
transform of the function is zero for all frequency pairs outside the support shown in
Figure 2.6. Then the function f
a
(
x, y
)
is band limited to
(v
x0
,
v
y0
)
is completely determined by samples of the
units apart. This means that the signal has to
D
y
) ¼
v
x
0
,
v
y
0
function spaced
(D
x,
be sampled at the minimum rate of
samples per unit length in the x- and
y-directions. This minimum required rate is called the Nyquist rate or the Nyquist
frequency. Consequently, if a certain sinusoid contained in the original signal needs
to be accurately reproduced after sampling, it has to be sampled at least twice at each
cycle. Thus, any sampling rate will capture all the information contained in frequen-
cies below one-half of the sampling rate. If a signal is undersampled (i.e., sampled at
a rate less than the Nyquist rate), aliasing will occur. Aliasing causes the high
frequencies to appear as low frequencies in the sampled signal. To avoid aliasing,
(
2f
x0
,2f
y0
)
Search WWH ::
Custom Search