Type-2 Fuzzy Non-uniform Rational B-spline Model with Type-2 Fuzzy Data

Transcription

1 Malaysan Journal of Mathematcal Scences 11(S) Aprl : (2017) Specal Issue: The 2nd Internatonal Conference and Workshop on Mathematcal Analyss (ICWOMA 2016) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: Type-2 Fuzzy Non-unform Ratonal B-splne Model wth Type-2 Fuzzy Data Wahab, A. F., Yong, L. K., and Zulkfly, M. I. E. School of Informatcs and Appled Mathematcs, Unverst Malaysa Terengganu, Malaysa E-mal: fatah@umt.edu.my Correspondng author ABSTRACT Ths paper ntroduced type-2 fuzzy non-unform ratonal B-splne curve (F2NURBS) nterpolaton model for type-2 fuzzy data ponts. The curve model s defned usng type-2 fuzzy control ponts, crsp knot and weght. Later, the data ponts of type-2 fuzzy NURBS curve after type-reducton and defuzzfcaton s dscussed. Fnally, a numercal example and an algorthm to generate the model are shown at the end of ths paper. Keywords: Type-2 fuzzy non-unform ratonal B-splne, nterpolaton model, type-2 fuzzy control ponts, type-reducton, defuzzfcaton.

2 Wahab, A. F., Yong, L. K. and Zulkfly, M. I. E. 1. Introducton Real-lfe stuaton s full of vagueness and uncertantes. Neglectng those uncertantes would mean fal to descrbe the true stuaton of data ponts. Wth the method of fuzzy set theory, uncertantes can be transformed nto a form of set, whch s descrbed by some value of truth. The elements n the doman of uncertanty wll match wth the value of membershp n the close nterval 0 and 1. By usng ths approach, the data wll be n fuzzy data form. If uncertanty n uncertanty s consdered, then the data wll be n the form of type-2 fuzzy data set. For nterpolaton model, the data ponts are acqured from the object, and the NURBS nterpolaton could be used to reproduce the shape of the object such as curve or surface. In ths paper, type-2 fuzzy set theory and geometrc modelng wth uncertantes characterstc s nvestgated and vsualzed. Zadeh (1965) ntroduced fuzzy set theory to descrbe uncertantes by usng mathematcal representaton. Later, the concept of lngustc varable s ntroduced n Zadeh (1975) where lngustc varable can be used to measure words or sentences n language wthn values 0 and 1. Then Dubos et al. (2000) properly defned the word fuzzy. They also provded a systematc framework to handle uncertantes that occur n human thought. Moreover, the concept of fuzzy numbers was ntroduced by Dubos and Prade (1980). The operaton on sets and fuzzy numbers such as addton, multplcaton, ntersecton and unon s referred to Klr and Yuan (1995). Fnally, type-2 fuzzy set theory s defned n detals by Mendel (2001) where the uncertanty n ts membershp functon s defned as secondary membershp grade. There are some past research that combnes fuzzy set theory and geometrc modelng such as Jacas et al. (1997) that used fuzzy logc n CAGD for curve and surface desgn and Anle et al. (2000) dscussed about fuzzy B-splnes. In addton, researches on fuzzy Bézer curve have been done by Wahab et al. and Zakara et al. (2001) and have been appled on off-lne handwrtten sgnature verfcaton and fuzzy grd data. Fuzzy nterpolaton of ratonal cubc Bézer are proposed wth dfferent degree of sgnature blurrng n Zakara and Wahab (2012). Fuzzy nterpolaton of B-splnes curve s then ntroduced n Karm et al. (2013) and Wahab and Zakara (2012). Partcularly on NURBS, Wahab and Husan (2011a) and Wahab and Husan (2011b) shown fuzzy NURBS curve and surface wth fuzzy control ponts, fuzzy knot and fuzzy weght. Data ponts, knots and weghts are three mportant elements to generate NURBS curve or surface. However, when uncertantes occur n ether data ponts, knots or weghts, crsp NURBS model fals to express the problem. 36 Malaysan Journal of Mathematcal Scences

3 Type-2 Fuzzy Non-unform Ratonal B-splne Model wth Type-2 Fuzzy Data Recently, research has been done on type-1 fuzzy Bézer, type-1 fuzzy B-splne, type-1 fuzzy NURBS model as n Anle et al. (2000), Wahab et al., Karm et al. (2013) and Zakara and Wahab (2013). Nevertheless, nvestgaton on one of the common nterpolaton method, NURBS curve nterpolaton that can handle uncertantes n ether data ponts, knots or weghts have not been done yet. As a matter of fact, when researchers are dealng wth type-2 uncertanty data problem, the exstng type-1 fuzzy model s unable to defne the type-2 uncertanty (Zakara et al. (2013a)). Thus, research on type-2 fuzzy NURBS curve nterpolaton s necessary to be nvestgated. 2. Type-2 Fuzzy NURBS Curve Modelng The proposed type-2 fuzzy NURBS curve model provdes better curve nterpolaton that can handle complex uncertantes and enables t to reproduce a more desred and smoother curve. Partcularly, n curve nterpolaton, the developed model can handle type-2 uncertanty data through type-2 fuzzy data ponts n curve nterpolaton. The propose model allows the users to tuned the nterpolaton curve by adjustng knot and weght to obtan a perfect desred curve. As a result, the proposed method s an mportant contrbuton to CAGD especally n reverse engneerng. The development of type-2 fuzzy NURBS based on type-2 fuzzy control ponts was dscussed n Zakara et al. (2013a) and Zakara et al. (2013b). Undoubtedly, the advantage of type-2 fuzzy n geometrc modelng s t could defne uncertantes n uncertantes. Thus, type-2 fuzzy set s an approprate method to descrbe uncertantes n data ponts. 3. Type-2 Fuzzy Ponts Defnton 3.1. Let P = {x x type-2 fuzzy pont} and T 2 P = {P P pont} s a set of type-2 fuzzy pont wth P P X, where X s unversal set. The membershp functon µ P (P ) : P [0, 1] s defned as µ P (P ) = 1 and formulated by T 2 P = {(P, µ P (P ) P R}. Thus, 0 f P / X, µ P (P ) = c (0, 1) f P X, 1 f P X. (1) where µ P (P ) = µ P ( T 2L P ), µ P (P ), µ P ( T 2R P ) wth µ P ( T 2L P ) are left; µ P ( T 2R P ) Malaysan Journal of Mathematcal Scences 37

4 Wahab, A. F., Yong, L. K. and Zulkfly, M. I. E. are rght footprnt of membershp values wth µ P ( T 2L P ) = µ P ( L P and µ P ( L P of left footprnt. µ P ( R P ) are membershp grade of lower bound of rght footprnt; µ P ( R P ) are lower bound of the left footprnt; µ P ( L P ), µ P ( L P ) ) s the upper bound ) s the membershp grade of upper bound of rght footprnt. Type-2 fuzzy pont can also be wrtten as T 2 P = { T 2 P : = 0, 1, 2,..., n}. For all, T 2 P = T 2L P, P, T 2R P wth T 2L P = L P, L P where L P are lower bound of left footprnt; L P are upper bound of left footprnt of type-2 fuzzy pont and T 2R P = R P, R P where R P, and R P are lower and upper bound of type-2 fuzzy pont respectvely. Fgure 1 shows a process of obtanng type-2 fuzzy ponts/knot/weghts. Fgure 1: Process of obtanng type-2 fuzzy ponts/knots/weghts Defnton 3.2. Let T 2 P be the set of type-2 fuzzy ponts and T 2 P T 2 P where = 0, 1,..., n and n + 1 s the number of pont. α-cut operaton on type-2 fuzzy ponts T 2 P α T 2 P α = L = s defned as, P α ; L P α [(P L P α [ ( R P α, P, R ; L P α ; R P α P α ; R P α )α + L P α ; L P α P )α + R ], P, P α ; R P α ] (2) wth specfc α value chosen, where α (0, 1]. Type-2 fuzzy ponts after α cut can be calculated by usng Equaton (2). Moreover, Equaton (2) can also be appled when the ponts are type-1 fuzzy ponts, where L RP α = R P α. P α = L P α and 38 Malaysan Journal of Mathematcal Scences

5 Type-2 Fuzzy Non-unform Ratonal B-splne Model wth Type-2 Fuzzy Data Defnton 3.3. Let T 2 P be a set of (n + 1) type-2 fuzzy ponts, type-reducton method of T 2 P after α cut of T 2 P α, s defned as, P α = { P α = L P α, P, R P α ; = 0, 1,..., n} (3) where L P α s left type-reducton of α cut type-2 fuzzy ponts, L P α = 1 n 2 =0 L P α + L P α, P s the crsp pont and R P α s rght type-reducton of α cut of type-2 fuzzy ponts, R P α = 1 2 Defnton 3.4. P α n =0 R P α + R P α. s the type-reducton method after -cut process s appled for every type-2 fuzzy ponts, then defuzzfcaton process for P α s denoted by P α. If every P α P α, P α = { P α } for = 0, 1,..., n. So, the defuzzfcaton process can be formulated as, P α = 1 3 n L P α, P, R P α (4) =0 4. Type-2 Fuzzy NURBS Curve Interpolaton Smlar to type-2 fuzzy NURBS curve model, the contradcton of the crsp boundary n type-1 fuzzy model s nsuffcent to descrbe fuzzness that occur n the membershp grade. When researchers are uncertan of the membershp grade of type-1 fuzzy sets, n ths case, type-1 fuzzy NURBS curve nterpolaton, a type-2 fuzzy NURBS curve nterpolaton wth type-2 fuzzy data ponts s constructed nstead of type-1 fuzzy set to encounter type-2 fuzzy problem. The type-2 fuzzy data ponts are defned by type-2 fuzzy number and type-2 fuzzy relaton. Type-2 fuzzy data pont s sutable to handle complex uncertanty data as n Zakara et al. (2013a) and Zakara et al. (2013b). Defnton 4.1. Interpolaton of NURBS curve wth type-2 fuzzy data ponts are defned as, T 2F NrbI = { T 2 Nrb α T 2 Nrb α, α (0, 1]} (5) where T 2 Nrb α = ( L Nrb α, L Nrb α ), Nrb α, ( R Nrb α, R Nrb α ), ( L Nrb α, L Nrb α ) and ( R Nrb α, R Nrb α ) represent left and rght footprnt of type-2 fuzzy NURBS Malaysan Journal of Mathematcal Scences 39

6 Wahab, A. F., Yong, L. K. and Zulkfly, M. I. E. nterpolaton curve after α cut. Nrb α represents crsp NURBS nterpolaton curve. α cut level s defned by user and t s wthn the half open nterval (0, 1]. By assumng that only one α cut s appled, fve NURBS curve could be generated. Fnally, by rewrtng the symbol n Equaton (5) wll yeld L NrbI α (u) = n =0 R,p (ū k ) L P, (6) L NrbI α (u) = n =0 R,p (ū k ) L P, (7) n NrbI α (u) = R,p (ū k )P, (8) =0 R NrbI α (u) = n =0 R,p (ū k ) R P, (9) R NrbI α (u) = n =0 R,p (ū k ) R P, (10) Each set of data ponts, Q s the nput and P s the output. L P represents the lower bound of left footprnt and L P s the upper bound of left footprnt of control ponts whle R P s lower bound of rght footprnt and R P s the upper bound of rght footprnt. P s the control ponts generate by crsp data ponts and T 2 Q, T 2 Q = { L Q,L Q, Q, R Q,R Q } generated by type- T 2 2 fuzzy data ponts. Q preserve the same propertes as T 2 P and defned n Defnton 3.4. R,p (ū k ) s a ratonal B-splne bass functon and each of the NURBS curve s generated by usng Algorthm 1. For nstance, consder a set of fuzzfed type-2 fuzzy data ponts as shown n Table 1, wth crsp knot u = {0, 0, 0, 0, 0.35, 0.7, 1, 1, 1, 1} and crsp weght w = {1, 1, 1, 1, 1.5, 1}. Ths problem s vsualzed n Fgure Malaysan Journal of Mathematcal Scences

7 Type-2 Fuzzy Non-unform Ratonal B-splne Model wth Type-2 Fuzzy Data Table 1: Data ponts of type-2 fuzzy NURBS curve. LQ L Q R Q Q R Q q 0 (2.2,0) (2.5,0) (3,0) (3.5,0) (3.8,0) q 1 (0,5) (0,4.5) (0,4) (0,3.6) (0,3.2) q 2 (3,5.5) (3.5,5.7) (4,6) (4.2,6.2) (5,6.5) q 3 (6.9,4) (7.3,4) (8,4) (8.3,4) (8.6,4) q 4 (9.3,6) (9.5,6) (10,6) (10.2,6) (10.5,6) q 5 (7,9.1) (7,9.3) (7,10) (7,10.5) (7,10.9) Fgure 2: Type-2 fuzzy NURBS nterpolaton wth type-2 fuzzy data ponts. Table 2: Data ponts of type-2 fuzzy NURBS curve after α cut at 0.5. L Q 0.5 L Q 0.5 Q R Q 0.5 R Q 0.5 q 0 (2.6,0) (2.75,0) (3,0) (3.25,0) (3.4,0) q 1 (0,4.5) (0,4.25) (0,4) (0,3.8) (0,3.6) q 2 (3.5,5.75) (3.75,5.85) (4,6) (4.1,6.1) (4.5,6.25) q 3 (7.45,4) (7.65,4) (8,4) (8.15,4) (8.3,4) q 4 (9.65,6) (9.75,6) (10,6) (10.1,6) (10.25,6) q 5 (7,9.55) (7,9.65) (7,10) (7,10.25) (7,10.45) By applyng α cut at 0.5, type-2 fuzzy data ponts s obtan as n Table 2 and the NURBS curve can be vsualzed as n Fg. 3. Malaysan Journal of Mathematcal Scences 41

8 Wahab, A. F., Yong, L. K. and Zulkfly, M. I. E. Fgure 3: Type-2 fuzzy NURBS nterpolaton after α cut at 0.5. After applyng centrod type-reducton, type-1 fuzzy NURBS curve data ponts s obtaned as n Table 3 and the curve s vsualzed n Fg. 4. Table 3: Data ponts of type-2 fuzzy NURBS curve after type-reducton. LQ 0.5 R Q Q 0.5 q 0 (2.675,0) (3,0) (3.325,0) q 1 (0,4.375) (0,4) (0,3.7) q 2 (3.625,5.8) (4,6) (4.3,6.175) q 3 (7.55,4) (8,4) (8.225,4) q 4 (9.7,6) (10,6) (10.175,6) q 5 (7,9.6) (7,10) (7,10.4) Fgure 4: Type-1 fuzzy NURBS curve nterpolaton after type-reducton. 42 Malaysan Journal of Mathematcal Scences

9 Type-2 Fuzzy Non-unform Ratonal B-splne Model wth Type-2 Fuzzy Data By applyng centrod defuzzfcaton, the crsp data ponts can be referred n Table 4. The NURBS curve nterpolaton s vsualzed n Fgure 5. Table 4: Data ponts of type-2 fuzzy NURBS curve after defuzzfcaton. Q Q 0.5 q 0 (3,0) (3,0) q 1 (0,3.817) (0,4) q 2 (3.975,5.992) (4,6) q 3 (7.925,4) (8,4) q 4 (9.958,6) (10,6) q 5 (7,10) (7,10) Fgure 5: Crsp of type-2 fuzzy NURBS curve nterpolaton after defuzzfcaton. Below s the algorthm n order to generate the type-2 fuzzy non-unform ratonal B-splne (F2NURBS) curve nterpolaton model for type-2 fuzzy data ponts. Algorthm 1: 1. Step 1. Each upper and lower bound of the left and rght footprnt of type-2 fuzzy data ponts, L Q, L Q, Q, R Q, R Q of NURBS curve s nterpolated by usng Equaton (5). 2. Step 2. α cut s appled on type-2 fuzzy data ponts and calculated by usng Equaton (2). Each of the type-2 fuzzy control ponts of NURBS curve, L Q α, L Q α, Q, R Q α, R Q α s plotted by usng Equaton (2). Malaysan Journal of Mathematcal Scences 43

10 Wahab, A. F., Yong, L. K. and Zulkfly, M. I. E. 3. Step 3. Type-reducton s appled and calculated by usng Equaton (2). Each of the type-1 fuzzy data ponts, L Q α, Q, R Q α s plotted usng Equaton (5). 4. Step 4. Defuzzfcaton s appled and calculated by usng Equaton (4). Crsp of type-2 fuzzy NURBS curve, Qα s then obtaned and plotted by usng Equaton (5). 5. Concluson Type-2 fuzzy NURBS model that was dscussed n ths paper s focused on two dmenson curve only. Type-2 fuzzy NURBS curve preserve the same propertes as crsp NURBS curve snce both left and rght footprnt of the curve bult upon crsp NURBS curve. The type-2 fuzzy NURBS curve control ponts model s sutable for approxmate a curve model. Ths model could handle type- 2 uncertantes or complex uncertantes n control ponts, knots and weght to obtan desred curve model. In order to solve type-2 uncertantes of data ponts n NURBS curve nterpolaton, type-2 fuzzy NURBS curve nterpolaton method wth type-2 fuzzy data ponts method s used. After α cut and type reducton, type-1 fuzzy data ponts s obtaned and through defuzzfcaton, the result s crsp of type-2 fuzzy data ponts. Acknowledgement Ths work was supported by Research Management Center (RMC) Unverst Malaysa Terengganu (UMT) and Mnstry of Educaton (MOE) through ERGS Research Grant No References Anle, A. M., Falcdeno, B., Gallo, G., Spagnuolo, M., and Spnello, S. (2000). Modelng uncertan data wth fuzzy b-splnes. Fuzzy Sets and Systems, 113(3): Dubos, D., Ostasewcz, W., and Prade, H. (2000). Fundamentals of Fuzzy Sets. Kluwer Academc Publshers, London. Dubos, D. and Prade, H. (1980). Fuzzy Sets and Systems: Theory and Applcaton. Academc Press, London. 44 Malaysan Journal of Mathematcal Scences

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