258 CHAPTER 8. APPLICATIONS
solution to the changes required in the colorant concentrations. Final analysis is
based upon a visual assessment of the color quality that is both time-consuming
and prone to errors.
In a fuzzy logic-based approach, a qualitative description of the partial dif-
ferential equations, governing the change in tristimulus values due to changes
in colorant concentration, is coded in the form of a fuzzy associative memory
(FAM). Table 8.3 depicts the FAM table for one of the tristimulus coefficients.
In this table, one of the two inputs to the FAM is the change in tristimulus
value∆X=X 0 −X,whereX 0 is the reference andXis the tristimulus value
of the sample, and the other input is the rate of change in tristimulus value with
respect to the colorant concentration, namely,∆X/C 1. The output of the FAM
is the change in colorant∆C 1. Note that the behavior of the other tristimulus
values, namely,∆Yand∆Zbear similar relationships to colorantsC 2 andC 3 ,
and consequently the association of subsets in each case will be identical. As
such, there are a total of nine FAMs that need to be generated.
Table 8.3. Fuzzy associative memory
←− ∆X −→
LN SN ZE SP LP
↑ LN SP SP ZE SN SN LN= Large Negative
∆X/C 1 SN LP SP ZE SN SN SN= Small Negative
↓ ZE LN LN ZE LP LP ZE=Zero
SP SN SN ZE SP LP SP=SmallPositive
LP SN SN ZE SP SP LP= Large PositiveIn the preceding table, the fuzzy sets representing the input and output
may be defined in terms of triangular, trapezoidal, Gaussian, or any other suit-
able membership functions. Of particular significance is the column of entries
pertaining to the change in tristimulus value∆X=ZEsubset. Note that in
a real-time dynamical process, one needs to consider the appropriate control
actions to compensate for the degree of variation in the ìvelocityî term even
when the ìpositionî term is part of the zero error subset. This is necessary
to prevent a deadband in the output control function and to maintain a stable
response. However, in the color matching problem when the tristimulus values
areinthezerosubset,becausetheprocessisnotareal-timesystem,therate
of change in tristimulus value with respect to a colorant has no effect on the
change in colorant. This is so because the process is static and a rate of change
in tristimulus value with respect to a colorant cannot be associated. This would
imply that no change in colorant is required when the deviation between the
tristimulus value of the sample and that of the standard is part of the zero error
subset. Figure 8.7 illustrates the subsets in the form of triangular membership
functions.