Biomedical Engineering Reference
In-Depth Information
where
fðm; nÞ¼
X
M
1
N
1
X
Fðu; vÞe
2p
jð
u
M
þ
v
N
Þ
;
d
ðwÞ¼
1
w ¼
0
;
0 otherwise
:
u¼
0
v¼
0
m ¼
0
;
1
;
2
;
.
; M
1
;
n ¼
0
;
1
;
2
;
.
; N
1
:
The forward or inverse Fourier transform of an
N N
image, computed directly with the preceding defini-
tions, requires a number of complex multiplications
and additions proportional to
N
2
.Bydecomposingthe
expressions and eliminating redundancies, the fast
Fourier transform (FFT) algorithm reduces the
number of operations to the order of
N
log
2
N
[5]
.
The computational advantage of the FFT is significant
and increases with increasing
N.
When
N ¼
64 the
number of operations are reduced by an order of
magnitude and when
N ¼
1024, by two orders of
magnitude.
A useful image enhancement operation is convolution
using local operators, also known as kernels. Considering
a kernel
w
(
k, l
) to be an array of (2
K þ
1
2
L þ
1)
coefficients where the point (
k, l
)
¼
(0,0) is the center of
the kernel, convolution of the image with the kernel is
defined by:
gðm; nÞ¼wðk; lÞ
*
fðm; nÞ
¼
X
K
X
L
wðk; lÞ
$
fðm k; n lÞ;
k¼K
l ¼L
where
g
(
m, n
) is the outcome of the convolution or output
image. To convolve an image with a kernel, the kernel is
centered on an image pixel (
m, n
), the point-by-point
products of the kernel coefficients and corresponding
image pixels are obtained, and the subsequent summation
of these products is used as the pixel value of the output
image at (
m, n
). The complete output image
g
(
m, n
)is
obtained by repeating the same operation on all pixels of
the original image
[4,5,13]
. A convolution kernel can be
applied to an image in order to effect a specific enhance-
ment operation or change in the image characteristics.
This typically results in desirable attributes being ampli-
fied and undesirable attributes being suppressed. The
specific values of the kernel coefficients depend on the
different types of enhancement that may be desired.
Attention is needed at the boundaries of the image
where parts of the kernel extend beyond the input image.
One approach is to simply use the portion of the kernel
that overlaps the input image. This approach can, however,
lead to artifacts at the boundaries of the output image. In
this chapterwe have chosen to simply not apply the filter in
parts of the input image where the kernel extends beyond
the image. As a result, the output images are typically
smaller than the input image by the size of the kernel.
The Fourier transform
F
(
u, v
) of an image
f
(
m
,
n
)is
defined as
6.3.3 Pixel operations
In this section we present methods of image enhance-
ment that depend only upon the pixel gray level and do
not take into account the pixel neighborhood or whole-
image characteristics.
6.3.3.1 Compensation for nonlinear
characteristics of display or print media
Digital images are generally displayed on cathode ray
tube (CRT) type display systems or printed using some
type of photographic emulsion. Most display mecha-
nisms have nonlinear intensity characteristics that result
in a nonlinear intensity profile of the image when it is
observed on the display. This effect can be described
succinctly by the equation
eðm; nÞ¼Cðfðm; nÞÞ;
where
f
(
m, n
) is the acquired intensity image,
e
(
m, n
)
represents the actual intensity output by the display
system, and
C
() is a nonlinear display system operator. In
order to correct for the nonlinear characteristics of the
display, a transform that is the inverse of the display's
nonlinearity must be applied
[14, 16]
.
MN
X
X
M
1
N
1
Fðu; vÞ¼
1
fðm; nÞe
2p
jð
u
M
þ
v
N
Þ
;
m¼
0
n¼
0
gðm; nÞ¼Tðeðm; nÞÞ
y
C
1
ðCðfðm; nÞÞÞ
u ¼
0
;
1
;
2
;
.
; M
1
;
v ¼
0
;
1
;
2
;
.
; N
1
;
gðm; nÞ
y
fðm; nÞ;
where
u
and
v
are the spatial frequency parameters. The
Fourier transform provides the spectral representation of
an image, which can be modified to enhance desired
properties. A spatial-domain image can be obtained from
a spectral-domain image with the inverse Fourier trans-
form given by
where
T
( ) is a nonlinear operator which is approximately
equal to
C
1
( ), the inverse of the display system opera-
tor, and
g
(
m
,
n
) is the output image.
Determination of the characteristics of the non-
linearity could be difficult in practice. In general,
if
