Nonlinear Techniques for Color Image Processing - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 464

where α( F i , F j ) is the directional (angular) distance defined in Equation 12.37

and distance ρ( F i , F j ) could be calculated using the Minkowski L p norm. The

parameter regulates the influence of angle and distance components.

As for any other ranked-order filter, an ordering of the B i implies the same
ordering of the corresponding vectors F i . Thus, DDF defines the F ( 0 )

ς

vector as

its output: F DDF = F 0 . For ς = 0weobtain the VMF and for ς = 1 the BVDF.

The DDF is defined for 5 and its usefulness stems from the fact that it

combines both the criteria used in BVDF and VMF. 100 , 104

ς = 0 .

12.4.1.10 Hybrid Directional Filter
Another efficient rank-ordered operation called hybrid directional filter (HDF)
has been proposed. 95 This filter operates independently on the direction and
magnitude of the color vectors and then combines them to produce a final
output. This hybrid filter, which can be viewed as a nonlinear combination
of the VMF and BVDF filters, produces an output according to the following
rule:

F VMF

if F VMF = F BVDF

F ∗ =

, (12.40)

|| F VMF ||

F BVDF otherwise

|| F BVDF ||

where F BVDF is the output of the BVDF filter, F VMF is the output of the VMF
and ||·|| denotes the vector norm.

12.4.2 Fuzzy Adaptive Filters

The performance of the different nonlinear filters based on order statistics de-
pends heavily on the problem under consideration. The types of noise that are
present in an image considerably affect the filter performance. To overcome
difficulties associated with the uncertainty that comes with the data, adaptive
designs based on local statistics have been introduced. 105 - 109 , 134 Such filters
utilize data-dependent coefficients to adapt to local image characteristics. The
weights of the adaptive filters are determined by fuzzy transformations based
on features from local data. The general form of the fuzzy adaptive filters is
given as a nonlinear transformation of a weighted average of the input vectors
inside the processing window:

f f

,

n n n

F ∗ =

0 w i F i

=

0 w i F i 0 w i

(12.41)

i = i = i =

where f is a nonlinear function that operates over the weighted average

of the input set. The relationship between the pixel under consideration and
each pixel in the window should be reflected in the decision for the filter's
weights. In the adaptive design, the weights provide the degree to which
an input vector contributes to the output of the filter. They are determined

( · )