Patent for Application Method and System for Adaptive Control of Real-Time Computer Graphics Rendering - Real-Time Rendering: Computer Graphics with Control Engineering - page 160

Given a condition where all the premise terms are used in every rule and a rule is
formed for each possible combination of premise elements, then we have rule set
with N i number of rules that can be expressed as:

n

∏

NNNN
i

=⋅

2 ...

⋅⋅ (12)

1 n

i = 1

Given the membership functions, the conversion of a crisp input value into its cor-
responding fuzzy value is known as fuzzification. The defuzzification of the resul-
tant fuzzy set from the inference system to a quantifiable value may be done using
the centroid (centre of gravity) method. The principle is to select the value in the
resultant fuzzy set such that it would lead to the smallest error on average given any
criterion. To determine y * the least square method can be used and the square of
the error is accompanied by the weightage of the grade of the membership µ B () .
Therefore, the defuzzified output, y * may be obtained by finding the solution to the
following equation.

( )

∫ µ

2

()

* *

y = argmin

*

y yydu − (13)

B

y

U

Differentiating with respect to y * and equating the derivative to zero yields

∫
∫

()

yydy µ

B

* =

y

Y

(14)

()

µ ydy

B

Y

which gives the value of the abscissa of the centre of gravity of the area below the
membership function µ B () .
The derivation of the membership functions is based on intuitive recognition of
the fundamental relationship between input and output of the rendering system. In the
context of the present invention, for example, there is an inverse relationship between
the frame rate and the total number of vertices used in the rendering process.
Figure B.7 indicates how this relationship may be developed in the form of a com-
bination of sigmoid functions for both the input and output variables. The diagram 701
illustrates the membership functions used for the input variable. There are two function
curves used for the linguistic value of the FPS error input variable. The function curve
at the left is to describe the extent of high and the one at the right is used to describe the
extent of low. In a similar manner, the diagram 702 shows the membership functions
for the output variable, which is the vertex count. The rule base of the fuzzy inference
rule set relating the input and output membership functions is shown in object 703.
In the same spirit as the closed-loop control feedback system shown in Figure B.2,
a fuzzy controller-based rendering system may be constructed using the aforemen-
tioned approach and using the derived fuzzy controller as the controller block in
Figure B.2.