Biomedical Engineering Reference
In-Depth Information
such that
where
W
(
u, v
) and
F
(
u, v
) are the Fourier transforms of
the kernel and the image, respectively. Therefore, en-
hancement can be achieved directly in the frequency
domain by multiplying
F
(
u, v
), pixel-by-pixel, by an ap-
propriate
W
(
u, v
) and forming the enhanced image with
the inverse Fourier transform of the product. Noise
suppression or image smoothing can be obtained by
eliminating the high-frequency components of
F
(
u, v
),
while edge enhancement can be achieved by eliminating
its low-frequency components. Since the spectral
filtering process depends on a selection of frequency
parameters as high or low, each pair (
u, v
) is quantified
with a measure of distance from the origin of the
frequency plane,
Dðu; vÞ¼
Efgðm; nÞg ¼ fðm; nÞ
and
s
d
s
g
¼
p ;
where
E
{
} is the expected value operator, s
g
is
the standard deviation of
g
(
m, n
), and s
d
is that of the
noise. Noise suppression is more effective for larger
values of
Q
.
$
6.3.5.2 Change enhancement by image
subtraction
q
u
2
þ v
2
;
Image subtraction is generally performed between two
images that have significant similarities between them.
The purpose of image subtraction is to enhance the dif-
ferences between two images (1). Images that are not
captured under the same or very similar conditions may
need to be registered
[17]
. This may be the case if the
images have been acquired at different times or under
different settings. The output imagemay have a very small
dynamic range and may need to be rescaled to the avail-
able display range. Given two images
f
1
(
m, n
) and
f
2
(
m, n
),
the rescaled output image
g
(
m,n
) is obtained with
bðm; nÞ¼f
1
ðm; nÞf
2
ðm; nÞ
which can be compared to a threshold
D
T
to determine if
(
u, v
) is high or low. The simplest approach to image
smoothing is the ideal low-pass filter
W
L
(
u, v
), defined to
be 1 when
D
(
u, v
)
D
T
and 0 otherwise. Similarly, the
ideal high-pass filter
W
H
(
u, v
) can be defined to be 1
when
D
(
u, v
)
D
T
and 0 otherwise. However, these
filters are not typically used in practice, because images
that they produce generally have spurious structures that
appear as intensity ripples, known as ringing
[5]
. The
inverse Fourier transform of the rectangular window
W
L
(
u, v
)or
W
H
(
u, v
) has oscillations, and its convolution
with the spatial-domain image produces the ringing.
Because ringing is associated with the abrupt 1 to 0 dis-
continuity of the ideal filters, a filter that imparts
a smooth transition between the desired frequencies and
the attenuated ones is used to avoid ringing. The com-
monly used Butterworth low-pass and high-pass filters
are defined respectively as
bðm; nÞ
min
fbðm; nÞg
max
fbðm; nÞg
min
fbðm; nÞg
gðm; nÞ¼f
max
$
where
f
max
is the maximum gray level value available,
b
(
m,n
) is the unstretched difference image, and
min{
b
(
m,n
)} and max{
b
(
m,n
)} are the minimal and
maximal values in
b
(
m,n
), respectively.
1
1
þ c½Dðu; vÞ=D
T
2
n
B
L
ðu; vÞ¼
6.3.6 Frequency domain
techniques
and
1
1
þ c½D
T
=Dðu; vÞ
2
n
;
B
H
ðu; vÞ¼
Linear filters used for enhancement can also be imple-
mented in the frequency domain by modifying the
Fourier transform of the original image and taking the
inverse Fourier transform. When an image
g
(
m, n
)is
obtained by convolving an original image
f
(
m, n
) with
a kernel
w
(
m, n
),
where
c
is a coefficient that adjusts the position of the
transition and
n
determines its steepness. If
c ¼
1, these
two functions take the value 0.5 when
D
(
u, v
)
¼ D
T
.
Another common choice for
c
is
p
1, which yields
0.707 (
3 dB) at the cutoff
D
T
.
The most common
choice of
n
is 1; higher values yield steeper transitions.
The threshold
D
T
is generally set by considering the
power of the image that will be contained in the pre-
served frequencies. The set
S
of frequency parameters
(
u, v
) that belong to the preserved region, i.e.,
D
(
u, v
)
D
T
for low-pass and
D
(
u, v
)
D
T
for high-pass,
gðm; nÞ¼wðm; nÞ
*
fðm; nÞ;
the convolution theorem states that
G
(
u, v
), the Fourier
transform of
g
(
m, n
) is given by
Gðu; vÞ¼Wðu; vÞFðu; vÞ;
