7.4
Applications of Rational Filters
We mentioned in Sec.
7.2.3
that adaptation procedures can be used to determine
the coefficients of rational filters. However, this approach has been followed in the
literature only for simple
1-D
operators.
A
possible alternative for the design of
suitable rational filters for image processing to achieve a desired result is to em-
ploy some heuristic criteria and compose simple polynomials into a more complex
function. This section gives an overview of the applications of rational operators
in the fields of noise smoothing for image data (with various noise distributions),
of image interpolation, and of contrast enhancement.
7.4.1
Detail-Preserving Noise Smoothing
A
rational filter can be very effective in removing both short-tailed and medium-
tailed noise corrupting an image [Ram95, Ram96bI. For the sake of simplicity,
we refer to a one-dimensional operator, but its extension to two dimensions is
straightforward. The filter can be formulated as
where the detail-sensing function
SO
is defined as
and
S(x)
=
k[x(n
-
1)
-
x(n
+
1)12
for short-tailed and medium-tailed noise, respectively. In Eq.
(7.13),
A
is a suit-
ably chosen constant. The operator can be expressed in a form
sirmlar to that of
Eq.
(7.8).
In the latter case, for example, it becomes
x(n
-
1)
+
x(n
+
1)
+
(A
-
2)x(n)
+
kx(n)[x(n
-
1)
-
x(n
+
1)12.
y(n)
=
k[x(n
-
1)
-
x(n
+
1)12
+A
9
(7.14)
a constant and a quadratic term are recognizable in the denominator, while the
numerator consists of the sum of a linear function and a cubic function of the
input signal.
We can examine the behavior of these filters for different positive values of
the parameter
k:
If
k
is very large,
y(n)
2:
x(n)
and the filter has no effect,
while if
k
1:
0
and a suitable value is chosen for
A,
the rational filter becomes a
simple linear
lowpass filter; for intermediate values of
k,
the output of the sensor
S(x)
modulates the response of the filter. Hence, the rational filter can act as an
edge-preserving smoother conjugating the noise attenuation capability of a linear
lowpass filter and the sensitivity to hlgh frequency details of the edge sensor.
Practical tests show that the value of
k
is not critical; moreover, it is not a function
of the processed image but rather of the amount of present noise.

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