Design and Implementation - Segmentation of Hand Bone for Bone Age Assessment - page 88

variables of any observation, fuzzy concept is useful to describe the vague nature
of reality and to comprehend the natural linguistic semantic of human; however,
when we want to take a subsequent action, crisp logic is useful to make a binary
decision in accordance to the interpreted fuzzy information. This resembles the
decision making process of a human expert.
This process of transforming fuzzy output set into crisp output set is termed
as 'defuzzification'. The concept is similar to rounding up from decimal numeri-
cal number into integer numerical number in mathematics. Analogously, in the
context of fuzziness, this process can be viewed geometrically as rounding up the
range value of the output membership function to the nearest vertex of the cor-
responding membership function. There are five common fuzzifiers: smallest of
max, largest of max, centroid of area, bisector of area, and mean of max. In the
proposed application, centroid of area method is adopted to find the centroid of the
aggregated output membership function, c ∗ , defined mathematically as follows:

c ∗ = ∫ µ A ( c ) . cdc
∫ µ A ( c ) dc

(3.61)

Decide to continue if
( Quadruple division | NDE , NCL , HDT , CER )
> 1 − ( Quadrupl e division | NDE , NCL , HDT , CER )
otherwise decide to terminate

(3.62)

For better understanding, the idea is illustrated with an arbitrary numerical
example. Let the NDE = 0.2, NCL = 0.4, HDT = 0.6, CER = 0.8. The resultant
computed centroid is shown in Fig. 3.19 which amounts to 0.292. In this particular
case, according to classification rule, the logical decision is to be terminated as in
the decision rule below. This is the quadruple division is controlled automatically
using fuzzy logic by incorporating human expert knowledge (Fig. 3.20 ).

Fig. 3.19 Overall input-output relationship using max-min composition and centroid defuzzi-
fier. The red line in the aggregated output in bottom right denotes the centroid position of the
aggregated output membership area