Generalized Fuzzy Linguistic Bicubic BSpline Surface Model for Uncertain Fuzzy Linguistic Data
Abstract
:1. Introduction
2. Introduction to BSpline Function
3. Fuzzy Set Approach to BSpline
4. Literature Review
5. Preliminaries
6. Linguistic Variables
7. Fuzzy Linguistic Point Relation
8. Fuzzy Linguistic Bicubic BSpline Surface Model
Algorithm 1 Fuzzy linguistic bicubic Bspline surface modeling 
Step 1: Define all fuzzy data points. Let ${\tilde{P}}_{i,j}=\left\{{\tilde{P}}_{\left(0,0\right)},{\tilde{P}}_{\left(1,0\right)},{\tilde{P}}_{\left(2,0\right)},\dots ,{\tilde{P}}_{n,m}\right\}\in X\times Y\mathrm{where}{\tilde{P}}_{0}=\left\{{x}_{0},{y}_{0},{\tilde{z}}_{0}\right\},{\tilde{P}}_{1}=\left\{{x}_{1},{y}_{1},\tilde{{z}_{1}}\right\},{\tilde{P}}_{2}=\left\{{x}_{2},{y}_{2},{\tilde{z}}_{2}\right\},\dots ,{\tilde{P}}_{n}=\left\{{x}_{n},{y}_{n},{\tilde{z}}_{n}\right\}$. 
Step 2: Define all fuzzy point relations where ${\mu}_{L}:L\to I=\left[0,1\right],{\mu}_{M}:M\to I=\left[0,1\right]\mathrm{and}{\mu}_{H}:H\to I=\left[0,1\right]$. 
Step 3: Define all fuzzy control points where
$${\tilde{P}}_{i,j}=\left[\begin{array}{cccc}\left({x}_{0},{y}_{0}\right),{\mu}_{{\tilde{P}}_{(0,0)}}\left({x}_{0},{y}_{0}\right)& \left({x}_{0},{y}_{1}\right),{\mu}_{{\tilde{P}}_{(1,0)}}\left({x}_{0},{y}_{1}\right)& \cdots & \left({x}_{0},{y}_{m}\right),{\mu}_{{\tilde{P}}_{(0,m)}}\left({x}_{0},{y}_{m}\right)\\ \left({x}_{1},{y}_{0}\right),{\mu}_{{\tilde{P}}_{(1,0)}}\left({x}_{1},{y}_{0}\right)& \left({x}_{1},{y}_{1}\right),{\mu}_{{\tilde{P}}_{(1,1)}}\left({x}_{1},{y}_{1}\right)& \cdots & \left({x}_{1},{y}_{m}\right),{\mu}_{{\tilde{P}}_{(1,m)}}\left({x}_{1},{y}_{m}\right)\\ \vdots & \vdots & \ddots & \vdots \\ \left({x}_{n},{y}_{0}\right),{\mu}_{{\tilde{P}}_{(n.0)}}\left({x}_{n},{y}_{0}\right)& \left({x}_{n},{y}_{1}\right),{\mu}_{{\tilde{P}}_{(n.1)}}\left({x}_{n},{y}_{1}\right)& \cdots & \left({x}_{n},{y}_{m}\right),{\mu}_{{\tilde{P}}_{(n.m)}}\left({x}_{n},{y}_{m}\right)\end{array}\right]$$

Step 4: Define all linguistic functions $\left[{L}_{i,},{M}_{i},{H}_{i}\right]$ to form fuzzy linguistic control points. 
Step 5: Develop fuzzy linguistic bicubic Bspline surface model using fuzzy linguistic control points blended with Bspline basis function. 
9. Numerical Example and Visualization
10. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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(x_{n}_{,} y_{n})  y_{0}  y_{1}  y_{2}  y_{3}  y_{4}  y_{5}  y_{6}  y_{7}  y_{8}  y_{9} 

x_{0}  $\left(0,0\right),1,00$  $\left(0,1\right),0.84$  $\left(0,2\right),0.84$  $\left(0,3\right),1.00$  $\left(0,4\right),1.00$  $\left(0,5\right),1.00$  $\left(0,6\right),1.00$  $\left(0,7\right),0.84$  $\left(0,8\right),0.84$  $\left(0,9\right),1.00$ 
x_{1}  $\left(1,0\right),1.00$  $\left(1,1\right),0.84$  $\left(1,2\right),0.84$  $\left(1,3\right),1.00$  $\left(1,4\right),1.00$  $\left(1,5\right),1.00$  $\left(1,6\right),1.00$  $\left(1,7\right),0.84$  $\left(1,8\right),0.84$  $\left(1,9\right),1.00$ 
x_{2}  $\left(2,0\right),0.84$  $\left(2,1\right),0.67$  $\left(2,2\right),0.67$  $\left(2,3\right),0.84$  $\left(2,4\right),0.84$  $\left(2,5\right),0.84$  $\left(2,6\right),0.84$  $\left(2,7\right),0.67$  $\left(2,8\right),0.67$  $\left(2,9\right),0.84$ 
x_{3}  $\left(3,0\right),0.84$  $\left(3,1\right),0.67$  $\left(3,2\right),0.67$  $\left(3,3\right),0.84$  $\left(3,4\right),0.84$  $\left(3,5\right),0.84$  $\left(3,6\right),0.84$  $\left(3,7\right),0.67$  $\left(3,8\right),0.67$  $\left(3,9\right),0.84$ 
x_{4}  $\left(4,0\right),1.00$  $\left(4,1\right),0.84$  $\left(4,2\right),0.84$  $\left(4,3\right),1.00$  $\left(4,4\right),1.00$  $\left(4,5\right),1.00$  $\left(4,6\right),1.00$  $\left(4,7\right),0.84$  $\left(4,8\right),0.84$  $\left(4,9\right),1.00$ 
x_{5}  $\left(5,0\right),1.00$  $\left(5,1\right),0.84$  $\left(5,2\right),0.84$  $\left(5,3\right),1.00$  $\left(5,4\right),1.00$  $\left(5,5\right),1.00$  $\left(5,6\right),1.00$  $\left(5,7\right),0.84$  $\left(5,8\right),0.84$  $\left(5,9\right),1.00$ 
x_{6}  $\left(6,0\right),0.84$  $\left(6,1\right),0.67$  $\left(6,2\right),0.67$  $\left(6,3\right),0.84$  $\left(6,4\right),0.84$  $\left(6,5\right),0.84$  $\left(6,6\right),0.84$  $\left(6,7\right),0.67$  $\left(6,8\right),0.67$  $\left(6,9\right),0.84$ 
x_{7}  $\left(7,0\right),0.84$  $\left(7,1\right),0.67$  $\left(7,2\right),0.67$  $\left(7,3\right),0.84$  $\left(7,4\right),0.84$  $\left(7,5\right),0.84$  $\left(7,6\right),0.84$  $\left(7,7\right),0.67$  $\left(7,8\right),0.67$  $\left(7,9\right),0.84$ 
x_{8}  $\left(8,0\right),1.00$  $\left(8,1\right),0.84$  $\left(8,2\right),0.84$  $\left(8,3\right),1.00$  $\left(8,4\right),1.00$  $\left(8,5\right),1.00$  $\left(8,6\right),1.00$  $\left(8,7\right),0.84$  $\left(8,8\right),0.84$  $\left(8,9\right),1.00$ 
x_{9}  $\left(9,0\right),1.00$  $\left(9,1\right),0.84$  $\left(9,2\right),0.84$  $\left(9,3\right),1.00$  $\left(9,4\right),1.00$  $\left(9,5\right),1.00$  $\left(9,6\right),1.00$  $\left(9,7\right),0.84$  $\left(9,8\right),0.84$  $\left(9,9\right),1.00$ 
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Bidin, M.S.; Wahab, A.F.; Zulkifly, M.I.E.; Zakaria, R. Generalized Fuzzy Linguistic Bicubic BSpline Surface Model for Uncertain Fuzzy Linguistic Data. Symmetry 2022, 14, 2267. https://doi.org/10.3390/sym14112267
Bidin MS, Wahab AF, Zulkifly MIE, Zakaria R. Generalized Fuzzy Linguistic Bicubic BSpline Surface Model for Uncertain Fuzzy Linguistic Data. Symmetry. 2022; 14(11):2267. https://doi.org/10.3390/sym14112267
Chicago/Turabian StyleBidin, Mohd Syafiq, Abd. Fatah Wahab, Mohammad Izat Emir Zulkifly, and Rozaimi Zakaria. 2022. "Generalized Fuzzy Linguistic Bicubic BSpline Surface Model for Uncertain Fuzzy Linguistic Data" Symmetry 14, no. 11: 2267. https://doi.org/10.3390/sym14112267