7.3
Applications of Polynomial Filters
In the 1-D 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 highpass-filtered 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 Teager-based 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) Teager-based, and (d) cubic methods. (Reproduced with permission
from [Ram96a].
O
1996
SPIE.)
obtained using the linear method, 7.lc the Teager-based 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 hlgher-order statistics [Mak94].
In particular, thlrd-order 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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