7.3
Applications of Polynomial Filters
In the 1D case, polynomial filters have been successfully applied for modeling
nonlinear systems, quadratic detectors (Teager's operator, see Chapter
6), echo
cancellation, cancellation of nonlinear intersymbol interference, channel equal
ization in communications, nonlinear prediction, etc.
Applications to image processing are in enhancement (image sharpening,
edge
preserving smoothing, processing of document images), analysis (edge extraction,
texture discrimination), and communications (nonlinear prediction, nonlinear in
terpolation of image sequences). Overviews of some of these applications are
described next.
7.3.1
Contrast Enhancement
In the linear unsharp masking method, a fraction of the highpassfiltered version
v
(m,
n)
of the input image x(m, n) is used as a correction signal and added to
the original image, resulting in the enhanced image
y(m, n):
where
This method is very sensitive to noise due to the presence of the
highpass filter.
Polynomial unsharp
maslung techniques, in which a nonlinear filter is substituted
for the
highpass linear operator
in
the signal sharpening path, can solve ths prob
lem. Different polynomial functions can be used. In the Teagerbased operator,
details are amplified in bright regions, where the human visual system is less sensi
tive to luminance changes
(Weber's law), and reduced noise sensitivity is achieved
in dark areas. The correction signal in this case is
[Mitgl]
In the cubic unsharp maslung approach, the sharpening action is performed
only if opposite sides of the filtering mask are each deemed to correspond to a
different object
[Ram96a], thus avoiding noise amplification:
v
(m,
n)
=
[x(m

1,
n)

x(m
+
1,
n)12
x
[2x(m,n) x(m
1,n)
x(m+ 1,n)]
+
[x(m, n

1)

x(m, n
+
1)12
x
[2x(m,n) x(m,n
1)
x(m,n+
1)).
Figure 7.1 shows the results of the unsharp masking approaches to image contrast
enhancement. Figure
7.la is a portion of the original Lena test image; 7.lb was
CHAPTER
7:
POLYNOMIAL
AND
RATIONAL
OPERATORS
209
Figure
7.1:
(a) Original test image, and contrast enhanced versions obtained using unsharp
masking: (b) linear, (c) Teagerbased, and (d) cubic methods. (Reproduced with permission
from [Ram96a].
O
1996
SPIE.)
obtained using the linear method, 7.lc the Teagerbased method, and 7.ld the
cubic method.
Expressions of a slrmlar type can be used for
v
(m,
n).
For example, when the
data are noisy a more powerful edge sensor is needed and the Sobel operator can
be used [Ram96a].
7.3.2
Texture Segmentation
The segmentation of different types of textures present in
an
image can be per
formed based on local estimates of second and hlgherorder statistics [Mak94].
In particular, thlrdorder moments are the best features to use
in
noisy texture
discrimination.
A
pth order statistical moment estimator is a special polynomial
operator; for example, for
p
=
3
1
=

1
x(n)x(n
+
i)x(n
+
j),
M2
nEx
where
n
=
(nl
,
n2
),
i
=
(il
,
i2
)
,
and
j
=
(
jl
,
j2
)
.
The moments are evaluated within
an M
x
M
image block
X.
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