Fuzzy Methods in Nonlinear Signal Processing: Part I—Theory - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 57

applications, the observation sample located (spatially) in the center of the
observation window is particularly important in other applications. Thus, the
class of center affine filters is based on the concept of affinity to the central
observation sample.
Center affine filters are related to median affine filters through a simple
change in ordering. That is, rather than a generalization of the linear filter
based on affinity to the median, the center affine filter is a generalization
of the order-statistic weighted sum ( L ) filter based on affinity to the central
spatial sample. Accordingly, the output of the center affine filter is given by

i = 1 w ( i )

R

x

δ , ( i ) ( i )

CAFF[ x ] = , (2.37)

i = 1 | w ( i ) |

R
δ , ( i )

are the filter coefficients and R i, (δ) where the w ( i ) is the affinity of the i th order

statistic, x .

The center affine filter weights each order-statistic according to its affinity
to the central observation sample and according to its rank. The filter output,
therefore, is based mainly on those order statistics that are simultaneously
close to the central observation sample and preferable due to their rank order.
Note that, as opposed to the median affine filters, here the temporal weights
R δ , ( i )

, with respect to the central (spatial) sample reference point, x

( i ) δ

w ( i ) are constant. Like the
median affine filter, the center affine filter with nonnegative weights reduces
to its basic structures at the limits of the dispersion parameter

are time varying and the rank order weights

σ :

i = 1 w ( i )

x

( i )

lim
σ →

CAFF[ x ] = x and lim

σ →∞

CAFF[ x ] = . (2.38)

i = 1 | w ( i ) |

δ

0

Thus, the center affine filter reduces to the identity filter and the L -filter, with
coefficients w ( i ) , for σ → 0 and σ →∞ , respectively.

2.4.1.3 Optimization
To appropriately tune the performance of an affine filter, the filter coefficients
can be optimized. Two approaches to optimizing the parameters are pre-
sented. First, a simple suboptimal procedure that addresses only the affinity
spread function is presented. A more comprehensive optimization procedure
that addresses both the filter weights and affinity spread function is then
presented. This optimization is based on a stochastic adaptive procedure. In
both cases, the presented methods address the optimization of the median
affine filter. The methods can be applied to the design of center affine filters
by simply interchanging corresponding quantities.
A simple and intuitive design procedure can be derived from the fact that
the median affine filter behaves like a linear filter for to

a large initial value allows the use of the multitude of linear filter design
methods to find the

σ →∞ . Setting σ

w

i coefficients of the median affine filter. Holding the

w

i coefficients constant, the filter performance can, in general, be improved