Fuzzy Methods in Nonlinear Signal Processing: Part I—Theory - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 63

The optimization of the fuzzy median follows directly from the above by
simply setting j 2. Thus, in the fuzzy median case there is

only a single parameter to optimize. In the case of an FWM filter, the spatial
weights and membership function spread parameter must be optimized. The
optimization of the WM filter spatial weights is similarly carried out under the
MAE criteria utilizing a gradient-based algorithm. 17 , 18

= δ = ( N + 1 )/

Such an optimization

yields the following weight update expression:

d

w ( + ) = w ( ) + µ ω e ( ) (w ( )) (w ( )) ( ) − ( ) .

n 1 n n sgn n sgn[sgn n x j n n ] (2.51)

j j j j

As in the previous case, this optimization can be combined with the iterative
membership function optimization expression and applied in an alternating
fashion. Although this does not guarantee globally optimal results, the pro-
cedure is simple, computationally efficient, and has yielded good results.

2.5 Extensions to Multivariate Data

The fuzzy relations and resulting filtering algorithms can be readily extended
to the multivariate data case. Indeed, multivariate signals arise naturally
in many applications, such as color image processing, velocity estimation,
three-dimensional surface moving, etc. Such signals can be processed in a
component-wise fashion, although this approach fails to exploit correlations
between components. Therefore, the more appropriate approach is to operate
directly on the multivariate data, taking advantage of the natural correlations
between signal components.
The fuzzy SR relations, upon which all of the discussed methods are
founded, are based on two relations: (1) spatial-rank ordering and (2) fuzzy
affinity relations between observation samples. The spatial, or temporal, or-
dering of the samples is defined naturally by the observation window con-
figuration and movement. While no universally accepted concept of rank
ordering exists for multivariate data, numerous ordering approaches have
been defined, e.g., sum distance ordering. Thus, to extend the fuzzy SR re-
lations concepts and filtering extensions to multivariate data, one need only
select the appropriate rank ordering procedure and define a fuzzy relation
between vector-valued samples.
Since fuzzy relations are a function of the distance between points, care
must be taken in defining the appropriate multivariate distance metric D ( a , b )

M . The selection of the vector difference metric is
application dependent. For example, if the directions that a , b

between two vectors, a , b ∈

M represent
are the main features of concern, then the angle between a and b is a good
difference metric. 19 Conversely, if the distance between vectors is the feature
of concern, then the L p norm 20 is the appropriate metric. Numerous other
metrics, of course, are also possible, depending on the problem at hand.

∈