CHAPTER 12." NONLINEAR FILTERS BASED ON FUZZY MODELS

359

intersection connectives [Yag80]"

YI(Pl, P2, ..., Pn) = 1 - re_in 1, (1 - pi) p

i=1

1/p} .

(12.8)

Likewise, for an aggregation scheme ranging from maximum to unity (that is,

something more optimistic than the maximum), we can adopt the following class

of union aggregators:

: n{x

1/p} .

(12.9)

Finally, for an aggregation scheme ranging from minimum to maximum, we can

choose the generalized mean connective [Dyc84]:

lip

(12.10)

where Z in=l w i = 1. Indeed, this connective yields all values between minimum

and maximum by varying the parameter p between -co and + co.

Finally, we can resort to hybrid connectives to combine outputs of union and

intesection aggregators. The combination can be performed by using an additive

or a multiplicative model, respectively, as follows:

YH

= (1 --

u +

Y(Yu), (12.11)

YH = (YI) 1-y (yU) y.

(12.12)

The degree of compensation between the union and intersection components de-

pends on the value of the parameter u (0 < ), < 1).

12.3 Fuzzy Weighted Mean (FWM) Filters

These methods belong to the class of indirect approaches mentioned in Sec. 12.1.

FWM filters adopt fuzzy sets or fuzzy systems to evaluate the weights of a weighted

linear filter that, in turn, yields the output. To describe these and other nonlinear

techniques, we shall adopt a common mathematical notation. Suppose we deal

with digitized images having L gray levels. Let x(n) be the pixel luminance at

location n = [nx,

n2]

in the input image and let y(n) be the corresponding pixel

luminance in the output image (0 < x(n) < L - 1,0 < y(n) < L - 1). Let W(n) =

{xi(n); i = 0 .... ,N} be the set of pixel values which belong to a window around

x(n), where x0 = x(n). Finally, let Axi (n) = xi(n) - x (n) be the luminance

difference between the neighbor xi and the central pixel x.

360 FABRIZIO Russo

F

0

Luminance differences

Figure 12.2" A possible choice for

fuzzy

set F.

12.3.1 FWM Filters Based on Fuzzy Sets

These filters adopt weights that are based on the luminance values

Xo, x1,..., XN

in the window. The adoption of weights aims at preserving fine details and textures

during the smoothing process. The general structure of these filters is described

by the following relationship:

y(n) = ~'N=~

wi(n)xi(n).

(12.13)

N

Ei=o wi (n)

The simplest filter of this group adopts one fuzzy set only. The filter operation is

defined as follows:

y(n) =

EN=oI~F(~Xi)xi(n).

(12.14)

N

~'-i=0

[AF(AXi)

where F is a bell-shaped fuzzy set centered on zero. A possible choice for the

membership function/2F is shown in Fig. 12.2.

The fuzzy set shape aims at reducing the influence of pixels having large lu-

minance differences with respect to the central one. The filter is effective for im-

ages degraded by Gaussian or uniform noise, and a membership function defined

by two parameters suffices to obtain satisfactory results. Hybrid filters for mixed

Gaussian and impulse noise removal can be obtained by resorting to median-based

prefiltering to cancel outliers [Pen94, Pen95a,b,c].

Fuzzy cluster filters are another example of FWM filters because they typically

adopt a weighted mean structure. The output is usually obtained by means of an

iterative procedure:

. (k)

y(k+l)(n) = ~'iN1

wi

xi(n)

(k) ' (12.15)

where y(k+l) is the candidate output (cluster center) at step k + 1. Weights are

evaluated by means of fuzzy membership functions and usually depend on the

previous estimate of the cluster center [Dor96, Suc96].

An interesting class of fuzzy filters for color image processing is represented

by fuzzy vector directional filters [Pla95]. These filters process multichannel im-

age data by considering the weighted average of all of the vector-valued elements

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