# FLAME-VQA: A Fuzzy Logic-Based Model for High Frame Rate Video Quality Assessment
## 4 Fuzzy Logic in User Experience Assessment
### The Model Development Process
#### 4.2.2 A Set of Fuzzy Rules and Defuzzification to the Scalar Result  
In the preceding subsection, we delved into the procedure of transforming the crisp values of input and output parameters into fuzzy variables, a pivotal step in constructing the model's inference system. Yet, to generate a measurable output, i.e., the quantified level of video quality rating in numerical terms (referred to as MOS), the fuzzy values must undergo a process of defuzzification (as illustrated in Figure 5). This entails establishing a set of fuzzy rules, choosing between a conjunctive or disjunctive rule system, and implementing a defuzzification technique to compute the output. Several widely used defuzzification methods include the max membership principle, centroid method, weighted average method, and center of sums, among others.  
The model's inference system relies on a set of 36 fuzzy rules, which can be found in Table A1 of Appendix A. The rules were defined using a hybrid method that combines multiple approaches. As domain experts, the authors formed an initial set of linguistic rules which were then iteratively fine-tuned using rule induction from the data. We used previously defined fuzzy cluster boundaries and tested each combination of input parameters to see the desired outcome, i.e., MOS value from the ground truth. We then compared the output of our model with the sought-after outcome and, when necessary, tweaked the rules to achieve error reduction. The connections between the linguistic values of the input and output variables are established using IF, AND, and THEN logical operators. The model was implemented using the widely used Mamdani inference system, where the output \(y^{k}\) consists of a set of \(r\) propositions:  
\[\begin{split}\text{IF }x_{1}\text{ is }A_{1}^{k}\text{ AND }x_{2}\text{ is }A_{2}^{k}\text{ AND }x_{3}\text{ is }A_{3}^{k}\text{ AND }x_{4}\text{ is }A_{4}^{k}\text{ THEN }y^{k}\text{ is }B^{k}\text{ with }W_{k}\\ \text{ for }k=1,\;2,\;\ldots,\;r.\end{split} \tag{7}\]  
\(x_{1}\), \(x_{2}\), \(x_{3}\) and \(x_{4}\) from Equation (7) are the inputs, \(A_{1}^{k}\), \(A_{2}^{k}\), \(A_{3}^{k}\) and \(A_{4}^{k}\) are the fuzzy sets representing the \(k\)-th input quadruplets and \(B^{k}\) is the fuzzy set representing the \(k\)-th output. While experimenting with the FIS of the model, we found that adding certainty grade (i.e., rule weights \(W_{k}\)) [48] to the rules improves the accuracy and helps in capturing the ambiguity of the subjective ratings. The model adopts the widely used disjunctive system of rules. Hence, the output \(y\) is represented by the fuzzy union of all individual rule contributions \(y^{i}\), where \(i=1,\;2,\ldots r\), and \(r\) is the number of IF-THEN propositions, as follows:  
\[y=y^{1}\cup y^{2}\cup\ldots\cup y^{r} \tag{8}\]  
Figure 5: FIS block diagram of the FLAME-VQA model.  
Finally, for defuzzification purposes, we used the centroid method. The method calculates the center of gravity, or centroid, of the fuzzy set's membership function. This centroid represents the typical or representative value of the fuzzy number. It is calculated by finding the weighted average of the universe of discourse using the membership degrees as weights. The method can be described with Equation (9), where \(y^{*}\) is the defuzzified value and \(u(y)\) is the curve describing the fuzzy union derived from Equation (8). Note that Equations (7)-(9) are derived from [49].  
\[y^{*}=\frac{\int u(y)\cdot y\,dy}{\int u(y)\,dy} \tag{9}\]