Image Processing Reference
In-Depth Information
2.11
Determine the approximate number of multiplications required for the 2-D
convolution of f
(
n, m
)
and h
(
n, m
)
for the three cases given below. Assume
f
(
n, m
)
to be N
N and h
(
n, m
)
to be L
L.
a. Both f
(
n, m
)
and h
(
n, m
)
are nonseparable.
b. Both f
(
n, m
)
and h
(
n, m
)
are separable.
c.
f
(
n, m
)
is not separable but h
(
n, m
)
is separable.
2.12
Determine the convolution of f
x
(
n, m
)
with h
(
n, m
)
, where
2
4
3
5
1323
4710
3058
2026
4 2
26
x
(
n, m
) ¼
and
h
(
n, m
) ¼
2.13
An image of size 64
5 FIR LPF. Find
a. The number of real multiplications if the convolution is performed in the
pixel domain.
b. The number of real multiplications if the overlap
64 is to be
filtered using a 5
-
add technique is used
with an 8-point FFT algorithm.
2.14
The signal
f
(
x
)
its spectrum, F
(v)
is band limited so that
(i.e., Fourier
transform), is zero outside the interval
2
p
B
< v <
2
p
B.
a. Expand the
spectrum F
(v)
in a Fourier
series on the
interval
2
p
B
< v <
2
p
B, having the form
1
f
k
exp
j
k
2B
F
(v) ¼
k
¼1
b. Show that the coef
cients, f
k
, in this expansion are proportional to samples
of f
(
x
)
1
2B
unit of length. Using these values for f
k
in the
above equation, take the inverse Fourier transform of F
(v)
at the interval of
to give f
(
x
)
and
show that the result expresses the signal f
(
x
)
in terms of the sample values
f
(
2B
)
and an
interpolation function
of the form
''
''
2B
sin
2
p
Bx
2B
2
p
Bx
This last result is, then, actually a statement of the most popular form of the
sampling theorem, in that it expresses the band-limited signal f
(
x
)
in terms of
1
discrete samples taken at intervals of
2B
units of length.
2.15
Let P
(
x
) ¼
xe
0
:
5x
2
u
(
x
)
and let the number of quantizer levels be 8 (3 bit).
a. What are the decision and reconstruction levels?
b. Find the resulting MSE.
2.16
Design a 2-D 11
11 zero-phase FIR filter to approximate the desired fre-
quency response shown in Figure 2.68.
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