Fuzzy Methods in Nonlinear Signal Processing: Part I—Theory - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 56

{x(n)} x(n-N) x(n-N+1) x(n)

-1
z

-1
z

-1

z

R 1 ,(δ) R R N ,(δ)

w w w N

1 2 ,(δ) 2

^

d (n)

Σ Σ

Median

FIGURE 2.8
Structure of the (unnormalized) median affine filter.

emphasizes the linear properties of the filter, while a small value puts more
weight on its median properties. Of special interest are the limiting cases. For
σ →∞

, the affinity function is constant on its entire domain. The estimator,
therefore, weights all observations strictly according to their natural order,
i.e.,

i = 1 w

i x i
i = 1 w

=

lim
σ →∞

MAFF[ x ] (2.35)

i

and the median affine estimator reduces to a normalized linear filter. In con-
trast, for σ → 0 the affinity function shrinks to an impulse at MED ( x ) . Thus,

the constant weights w i are disregarded and the estimate is equal to the me-

dian, i.e.,

lim
σ →

MAFF[ x ] = MED ( x ). (2.36)

0

In addition to the limiting cases, the median affine filter includes important
filters, such as the MTM filter, 12 as subclasses.
The median affine filter also possesses several desirable properties, 11 in-
cluding: (1) data-adaptiveness, (2) translation invariance, (3) the ability to
suppress impulses, and (4) the ability to preserve signal trends and discon-
tinuities. These properties lend understanding to the filtering process and
help explain the improved performance achieved through the incorporation
of sample affinity into the filter structure.

2.4.1.2 Center Affine Filters
A second important subclass of affine filters is the class of center affine fil-
ters. While the median serves as an acceptable reference point for certain