Nonlinear Image Processing

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.