Fuzzy B-spline Modeling of Uncertainty Data

Transcription

1 Appled Mahemacal Scences, Vol 6, 22, no 4, Fuzzy B-splne Modelng of Uncerany Daa Rozam Zakara Deparmen of Mahemacs, Faculy of Scence and Technology Unvers Malaysa Terengganu, Malaysa Abd Faah Wahab Deparmen of Mahemacs, Faculy of Scence and Technology Unvers Malaysa Terengganu, Malaysa Absrac In hs paper, we purposed a new mehod n modelng he uncerany daa based on he heory of fuzzy ses and he nerpolaon B-splne model curve The uncerany daa frsly defned by usng fuzzy number concep and fuzzy relaon by defned hem n real daa form The exsed coeffcen conrol of nerpolaon B-splne bass funcon s subsued wh he fuzzy daa pons(fdps) whch wll nroduce fuzzy nerpolaon B-splne curve model The fuzzfcaon mehod s appled n order o oban a fuzzy nerval of crsp fuzzy soluon curve based on he alpha-cu values Afer he fuzzfcaon mehod was appled, hen we used defuzzfcaon mehod o acheve he fnal soluon whch s he crsp fuzzy soluon curve Ths purposed model s appled n verfcaon of offlne handwrng sgnaure(ohs) as he hypohecal example Keywords: B-splne curve, fuzzy daa pons, fuzzy B-splne curve, fuzzfcaon, defuzzfcaon Inroducon One of he mporan role of mahemacal models s how hey were used n real world phenomena o descrbe he crera of her behavor Generally, n modelng he real world phenomena, he se collecon of daa are necessary needed as he represenave of he real world phenomena Bascally, n modelng he se collecon of daa hrough mahemacal model, wha s mporan s he needed

2 6972 Rozam Zakara and Abd Faah Wahab mahemacal funcon of hese colleced daa can be vsualzed n curve and surface forms There are many mahemacal funcons of curve and surface whch become he man fundamenal n modelng feld such geomerc modelng [9,,23] In geomercal modelng feld, curve and surface become he man facors n desgnng curve and surface [4] n order o represenave he real daa form These forms of desgnng curve and surface become he sources for users o undersand he desgnng curve and surface of represenave real daa form before he users makng he analyss, decson and concluson [6] Ths shape form can be modeled f he represenave daa have he complee daa se whch can be used o reconsruc he requred shape However, here s major problem n shape reconsrucon due o uncerany, mprecse and vague of he real daa forms whch are represenave he real phenomena Ths suaon make us unable o model he uncerany daa hrough exsed funcons of curves and surface In addon, he uncerany daa makng us fal o undersand he naure of hs knd of daa Thus, one of mehods whch s used o defne he uncerany ssues s fuzzy se heory whch was nroduced by [24] Ths heory become basc heory n order o handle he uncerany maers Then, hs heory was expanded along he arsng he uncerany ssues Therefore, for defnng he uncerany daa n geomercal modelng, he ssues of uncerany daa can be solved by usng he defnons of fuzzy number conceps whch was dscuss by [4] When he uncerany daa ssues had been defned whch become fuzzy daa, hen here are several sudes were exended due o he ssues modelng he uncerany daa n geomercal modelng feld such fuzzy Bezer curve and surface [2,4,8], fuzzy nerpolaon Bezer curve [8,2], fuzzy nerpolaon raonal Bezer curve [,3,9,2] and fuzzy B-splne curve and surface [5,,2,3] Our approach o solve he problem s based on he negraon beween nerpolaon B-splne curve and fuzzy se heory especally fuzzy number n handlng he uncerany daa The resul s fuzzy nerpolaon B-splne curve(fibsc) whch gve user o modelng he fuzzy daa afer been defned hrough fuzzy number n curve form where s funcon s B-splne funcon Then, fuzzfcaon and defuzzfcaon mehods of FIBsC are nroduced where he fuzzfcaon mehod s he α -cu operaons whch used o reduce he fuzzy daa nerval and defuzzfcaon mehod s used o oban crsp fuzzy soluon curve of FIBsC The remander of he paper s organzed as follows: frs, he dscussng abou how o defne uncerany daa hrough fuzzy se heory whch ncludes fuzzy number concep and fuzzy relaon Then, a bref descrpon on B-splne curve and nerpolaon B-splne curve are gven ogeher wh he paramerzaon mehods of nerpolaon B-splne curve In he fourh secon, we develop he

3 Fuzzy B-splne modelng of uncerany daa 6973 mehod of FIBsC based on he heores and defnons whch had been gven before Example s gven hrough verfcaon of OHS and fnally, dscusson and some concludng remarks are gven and also fuure developmens wll be oulned 2 Prelmnares Snce he heory of fuzzy ses was nroduced by Zadeh [24], here were exsed ohers heores were nroduced due o requremen n handlng he uncerany problem such as fuzzy number whch dealng wh he uncerany n numbers form [5,6,25] Therefore, n hs secon, we dscussed he fuzzy number concep n defnng he uncerany real daa Defnon Le R be a unversal se whch R s a real number and A s subse o R Fuzzy se, A n R(number around A n R) called fuzzy number whch explaned hrough he α -level se(srong and normal α -cu) ha s f for every α (,], here exs se A α n R unl Aα = { x R: μ A ( x) > α} and α Aα = x R: μ A ( x) α { } α Defnon 2 If rangular fuzzy number represen as A = ( adc,, ) and A α be a α -cu operaon of rangular fuzzy number, hen crsp nerval by α -cu operaon s obaned as A [ a α α α =, c ] = [( d a) α + a, ( c d) α + c] wh α (,] where he membershp funcon, μ ( x ) gven by A for x< a x a for a x d d a μ ( x) = A () c x for d x c c d for x> c

4 6974 Rozam Zakara and Abd Faah Wahab μ b α Fgure Trangular fuzzy number, A = ( adc,, ) Fg shows he rangular fuzzy number where pon b s a crsp pon and a and c lef fuzzy pon and rgh fuzzy pon The α symbol means he α values of rangular fuzzy α -cu In order o defne he uncerany real daa, we used he defnon of fuzzy relaon o conver he defnon of fuzzy number o become FDPs The defnon of fuzzy relaon and hen he defnon of FDPs are gven as follow Defnon 3 Le XY, R be unversal ses hen R = (( xy, ), μ ( xy, )) ( xy, ) X Y R s called a fuzzy relaon on X Y [25] { } Defnon 4 Le XY, R and A = {( x, μ ( x) x X A } and B= {( y, μ ( y) y Y B } are wo fuzzy ses Then R = { ( x, y), μ ( x, y),( x, y) X Y R } s a fuzzy relaon on A and B f μ ( x, y) μ ( x), ( x, y) X Y and μ ( x, y) μ ( y), ( x, y) X Y [25] R A R Defnon 5 Le XY, R and M = {( x, μ ( x) x X M } and N = {( y, μ ( y) y Y N } are wo fuzzy daa Then, he fuzzy relaon beween boh fuzzy daa s gven by P = { ( x, y), μ ( x, y),( x, y) X Y } P = and D= { P P s daa pon} are he se of FDPs whch s D D X Y R wh R s unversal se and μ ( D) : D [,] s membershp funcon defned as Defnon 6 Le D {( x, y), x X, y Y xand y are fuzzy daa} P a a α d c α c B x

5 Fuzzy B-splne modelng of uncerany daa 6975 μ P( D) = where can be formulaed as D= ( D, μd( D) ) D R f D R μp( D) = c (,) f D R f D R { } Therefore, wh μd( D) = μp( D ), μp( D), μp( D ) where μd( D ) and μd( D ) are lef-grade and rgh-grade membershp pons values respecvely Ths can be overwren as D= { D = ( x, y) =,,, n} (3) for all, D = D, D, D wh D, D and D are lef FDP, crsp daa pon and rgh FDP respecvely The defnng process and he form of FDP afer defned usng ype- fuzzy relaon can be shown as Fg 2 and Fg 3 respecvely (2) μ ( x ) A μ ( x ) A x Ordnary pon daa 2 x Fuzzy pon daa μ ( x ) A x Crsp pon daa Fgure 2 Process of defnng fuzzy daa Fg 2 shows ha he process of defnng fuzzy daa from ordnary pon o fuzzy pon The crsp pon s he ordnary pon whch s membershp funcon s equal o μ( xy, ) y x 8 Fgure 3 FDP form afer defned by fuzzy relaon

6 6976 Rozam Zakara and Abd Faah Wahab Fg 3 shows ha he FDP formed by usng he defnons fuzzy relaon and fuzzy number Ths FDP was consruced wh he combnaon of xy-axs hrough fuzzy relaon Fg 2 shows ha he process of how FDP was defne whch s defned a x-axs In Fg 3, he consrucon FDP n xy-coordnae whch represen he real FDP form Ths real form FDP was obaned hrough he defnon fuzzy number and fuzzy relaon whch beng used n defne he real uncerany daa pon 3 α -Cu Operaon and Defuzzfcaon Mehod of Fuzzy Daa Pon The nex proses n geng crsp fuzzy soluon of FDP s he α -cu operaon(fuzzfcaon process) whch s appled o oban a new FDP(α -FDP) wh a new fuzzy nerval of α -FDP Therefore, hs followng defnon s he fuzzfcaon process for FDP Defnon 7 Le D be he se of FDPs wh D D Then Dα s he α -cu operaon of FDPs whch s gven by D = D, D, D α α (4) α where D α, D and D α are α -cu of lef FDPs, crsp daa pons and α -cu of rgh FDPs wh =,, 2,, n Uncerany daa Uncerany daa Uncerany daa D = D D D {,,, } n α D D D 2 D α D α level se Fgure 4 FDPs and he nerval D, P, D α α α a D α D α D α α -level se n (,] Fg 4 shows ha he α -cu operaon has been appled agans FDPs whch gves he new fuzzy nerval, Dα, P, D α α a α -level se values n (,] When he new fuzzy nerval has been obaned, hen he crsp fuzzy soluon s whn n

7 Fuzzy B-splne modelng of uncerany daa 6977 The Def 7 s exended o he ceran cases ncluded he α -cu operaon of fuzzy dagonal pons besde he α -cu operaon of FDPs a x- and y- axs Therefore, he defnon of hs cases can be gven as follow whch s defned based on he α -cu of rangular fuzzy number Defnon 8 From Fg, le say here exss wo FDPs n rangular forms whch s consruced by Dx = ax, bx, cx, and Dy = ay, by, cy where b x and b y are crsps FDP a x- and y- axs respecvely whch s vewed by Fg 5 Then, α -FDP s obaned based on he α -cu operaons whch can be defned hrough he Def 2 such as: ) α -FDP a x-axs α α D = ( b a ) α + a, D = ( c b ) α + c (5) ( x x x x x x x x) 2) α -FDP a y-axs α α D = ( b a ) α + a, D = ( c b ) α + c ( y y y y y y y y) c y (6) n R α y b y n L a y b x α x a x Fgure 5 Two FDP wh her nersecon pons afer fuzzfcaon process Fg 5 shows ha he nersecon of wo FDP whch nvolves wo dfferen suaon The wo dfferen suaon are he FDP a x-axs nersecon wh he FDP a y-axs From he llusraon based on he Fg 5, here exss a suaon where he FDPs nersec a dagonal whch produced fuzzy dagonal daa pons among wo dfferen axs Therefore, n order o oban α -FDP n dagonal form, hen he fuzzfcaon process need o be exended from he prevous mehods whch are gven as follow 3) α -cu of FDP dagonal a ( m, n ) L L m L m R c x

8 6978 Rozam Zakara and Abd Faah Wahab ( Dx ( ml) = ( bx ax) α + ax, Dy ( nl) = ( by ay) α + ay) 4) α -cu of FDP dagonal a ( mr, n L) D ( m ) = ( c b ) α + c, D ( n ) = ( b a ) α + a ( x R x x x y L y y y) 5) α -cu of FDP dagonal a ( ml, n R) D ( m ) = ( b a ) α + a, D ( n ) = ( c b ) α + c ( x L x x x y R y y y) 6) α -cu of FDP dagonal a ( mr, n R) D ( m ) = ( c b ) α + c, D ( n ) = ( c b ) α + c ( x R x x x y R y y y) From Def 8, he formula n geng he α -FDPs a dagonal had been consruced by usng he defnon of α -cu operaon Also, hs defnon can be appled for FDP n 3D form(surface daa) Afer we obaned he α -FDPs, he fnal seps n geng he crsp fuzzy daa soluon s he defuzzfcaon mehod Defuzzfcaon mehod s used as he fnal soluon o oban crsp fuzzy soluon whch wll gve he sngle value oupu Ths mehod s beng used afer he fuzzfcaon mehod has been appled Therefore, he defnon of defuzzfcaon for FDP s gven as follow Defnon 9 Le α be he α -cu for every FDPs, D wh =,,, n Then D named as defuzzfcaon FDPs for D f for every D D, D = { D } for =,,, n () D + D + D = where for every D = whch =,,, n The 3 defuzzfcaon process can be llusraed by Fg 6 μ ( x ) A μ ( x ) A (7) (8) (9) () x x Ordnary daa pon Defuzzfy daa pon μ ( x ) A μ ( x ) A x x Crsp daa pon Fuzzy daa pon Fgure 6 Defuzzfcaon process of FDP Fg 6 shows how o oban crsp fuzzy soluon of FDP hrough defuzzfcaon mehod Ths mehod used afer α -FDP was obaned

9 Fuzzy B-splne modelng of uncerany daa Inerpolaon B-splne Curve Model As we know, he nerpolaon curve means ha all daa pons are nerpolaed by a sngle curve whch he curve mus goes hrough all he daa pons Therefore, nerpolaon B-splne curve can be defned va Def m Defnon Gven a ls of daa pons d, n, he B-splne nerpolaon problem of order k s o fnd: () he kno vecor T = ( T, T,, Tn+ k, Tn+ k) ; (2) he parameer value for each d, n, and ; (3) he conrol pons such ha he resulng B-splne curve n Bs() = PN () (2) = has he propery Bs( ) = d, n [7,8,22] The compuaon of B-splne nerpolaon s sraghforward Frs, we choose kno vecor T = ( T, T,, Tn+ k, Tn+ k) so ha he B-splnes Nk, () can be defned In open curve cases, he frs k T s are equal and he las k T s equal, such ha, T = T = = Tk and Tn+ = Tn+ 2 = = Tn+ k because we wan he nerpolang curve Bs() o have he propery Bs( sar ) = P and Bs( end ) = Pn [4] For he oher kno values, equally-spaced values can be assgns, such ha, T+ T = consan, for example Then, he paramerzaon mehod of choce assgns an approprae parameer value o each daa pon P Once we have done he paramerzaon, he followng equaon holds PN, k( ) + PN, k( ) + + PN n n, k( ) = d, n (3) In he marx form, he equaon becomes AP = d, where N, k( ) N, k( ) L Nn, k( ) N, k( ) N, k( ) Nn, k( ) A L =, M M M M N, k( n) N, k( n) Nn, k( n) L, k P = ( P, P,, P ) T n and d = ( d, d,, d ) T n [8] Then, he nerpolaon B-splne curve can be llusraed n Fg 7

10 698 Rozam Zakara and Abd Faah Wahab D 2 D 7 D 5 D D 8 D 4 D 3 D 6 D Fgure 7 Inerpolaon B-splne curve of nne pons daa Fg 7 shows ha he nerpolaon B-splne curve whch hs curve perform n cubc form where nerpolaed nne daa pons 4 Paramerzaon Mehod of Inerpolaon B-splne Curve The form and qualy of nerpolang B-splne curve, Bs() s depend o he paramerc values, for every gven daa pons, d There are several paramerzaon echnques whch commonly used are he unformly spaced mehod, he chord lengh mehod and he cenrpeal mehod [8,22] We defne he chord lengh mehod: =, = + ds, for k =,, n, (4) k k k 2 2 k ds = ( x) + ( y), for k =,, n where xk = xk xk, yk = yk yk D 2 D 7 D 5 D D 8 D 4 D 3 D 6 D Fgure 8 Inerpolaon B-splne curve usng chord lengh paramerzaon Fg 8 shows ha he nerpolaon B-splne curve where s paramerzaon s based on he Eq 4 whch also known as chord lengh paramerzaon

11 Fuzzy B-splne modelng of uncerany daa 698 The unformly spaced mehod s defned by dvdng he doman [ x, x n] no equal subnervals, = x, xn x k = k, for k n, (5) n = x n n D 2 D 7 D 5 D D 8 D 4 D 3 D 6 D Fgure 9 Inerpolaon B-splne curve usng unformly spaced paramerzaon Fg 9 shows ha he nerpolaon of cubc B-splne curve usng unformly spaced mehod paramerzaon whch menoned by Eq 5 Then, he cenrpeal mehod s gven by =, 2 = + ( ds ), for k =,2,, n, (6) k k k 2 2 dsk = ( x) + ( y), for k =,,, n where x = x x, y = y y k k k k k k D 2 D 7 D 5 D D 8 D 4 D 3 D 6 D Fgure Inerpolaon B-splne curve usng cenrpeal paramerzaon Fg shows ha he nerpolaon of nne pons daa formed by cubc B-splne curve usng cenrpeal paramerzaon

12 6982 Rozam Zakara and Abd Faah Wahab 5 Fuzzy Inerpolaon B-splne Curve In hs secon, we dscuss abou he consrucon of fuzzy Inerpolaon B-splne curve n order modelng he uncerany pons daa Therefore, he uncerany pons daa s defned hrough fuzzy number conceps and fuzzy relaon as n Def 6 Before we defne he fuzzy Inerpolaon B-splne curve, we would lke o defne fuzzy B-splne curve frs 5 Fuzzy B-splne Curve Fuzzy B-splne curve whch had been dscuss n [2,4,7] Therefore, fuzzy B-splne can be defned as n Def n Defnon Le { P } be a se of fuzzy conrol pons relave o crsps kno = sur sequences, 2,, m= k+ 2( n ) A fuzzy B-splne curve s a funcon Bs() from a real lne o he se of real fuzzy number and defned by sur k+ h Bs() = PB, k() (7) where [2] = P are fuzzy conrol pons and B, () s are crsp B-splne basc funcon k P P 3 P Fgure Fuzzy B-splne curve wh four fuzzy conrol pons Fg shows ha he fuzzy B-splne curve whch s nerpolae frs and las fuzzy conrol pons(fuzzy daa) s obaned The fuzzy conrol pons s defned a x-axs whch means he uncerany daa pons a x-axs are defned usng fuzzy number and fuzzy relaon conceps 52 Fuzzy Inerpolaon B-splne Curve In hs par, we dscussed he consrucon of fuzzy nerpolaon B-splne curve based on he defnon of nerpolaon B-splne curve, fuzzy conrol pons and P 2

13 Fuzzy B-splne modelng of uncerany daa 6983 fuzzy B-splne curve Then, we appled fuzzfcaon process of rangular α -cu agans he fuzzy nerpolaon B-splne curves whch gves us he fuzzy nerval of fuzzy nerpolaon B-splne curve We used he cenrpeal paramerzaon o model he fuzzy nerpolaon B-splne curve because has less of bendng curve from he oher paramerzaon mehods as n shown before The defnon of fuzzy nerpolaon B-splne curve can be gven n Def 2 by usng Def m Defnon 2: Le d be a ls of fuzzy daa pons wh n, he fuzzy nerpolaon B-splne curve can be defned as sur k+ h sur Bs() = PB, k() whch Bs( ) = d, (8) = where s crsp kno sequences, 2,, m= k+ 2( n ), P are fuzzy conrol pons and Bk, () s basc funcon of B-splne Therefore, Fg shows ha he llusraon of fuzzy nerpolaon B-splne curve D 2 D 7 D 5 D D 8 D 4 D D 3 D 6 Fgure 2 Fuzzy Inrpolaon B-splne curve of nne fuzzy daa pons Fg 2 shows ha he modelng of fuzzy nerpolaon B-splne curve whch nerpolaed nne fuzzy daa pons Also, hs modelng fuzzy nerpolaon B-splne curve of non-symmerc fuzzy daa pons means ha he fuzzy nerval values beween crsp pons and lef-rgh fuzzy pons are no equally same Then, fuzzfcaon mehod of rangular α -cu s appled agans he fuzzy daa pons o gve a new fuzzy daa pons Thus, he defnon of rangular α -cu s gven as follow Defnon 3 Le us assume ha membershp funcon of rangular fuzzy number A denoed by A= (, c δ, β ) s defned as

14 6984 Rozam Zakara and Abd Faah Wahab c x f c-δ x c δ x c μa( x) = f c x c+ β β oherwse where c s he cenre and δ > s he lef spread, β > s he rgh spread of A If δ = β, hen he rangular fuzzy number s called symmerc fuzzy number and denoed by (, c δ ) The rangular fuzzy number can be shown n Fg 3 as he llusraon c δ Fgure 3 Trangular fuzzy number Fg 3 shows ha he rangular fuzzy number whch was menoned n Def 3 Ths rangular fuzzy number s used o defne he fuzzy conrol pon as menoned before Then, for he α -cu conceps of rangular fuzzy number can be defned n Def 4 c + β Defnon 4 Le α [,], hen from ( α) ( α) ( c δ) ( c δ) ( c+ β) ( c+ β) = α, = α c ( c δ) ( c+ β) c we obaned ( α ) ( c δ) = ( c ( c δ)) α + ( c δ) ( α ) ( c+ β ) = (( c+ β) c) α + ( c+ β) herefore ( α) ( α) Aα = [( c δ), c,( c+ β) ] = [( c ( c δ )) α + ( c δ), c, (( c+ β) c) α + ( c+ β)] c

15 Fuzzy B-splne modelng of uncerany daa 6985 Thus, he α -cu of rangular fuzzy number can be appled n he process fuzzfcaon of fuzzy conrol pons whch s resul can be shown n Fg 4 D 2 D 7 D 5 D D 8 D 4 D D 3 D 6 (a) D 2 D 7 D 5 D D 8 D 4 D D 3 D 6 (b) D 2 D 7 D 5 D D 8 D 4 D D 3 D 6 (c) Fgure 4 New fuzzy nerpolaon B-splne curve afer α -cu of rangular fuzzy number operaon wh he value of α are (a) 3, (b) 6 and (c) 9 respecvely From Fg 4 (a), (b), and (c) shows ha he operaon of α -cu rangular fuzzy number agans he fuzzy conrol pons and shows hem n curve form When he fuzzfcaon process s done, hen n order o fnd he fuzzy crsp soluon of fuzzy nerpolaon B-splne curve, we used defuzzfcaon mehod as he fuzzy crsp soluon Based on Def 9 whch are appled o Fg 4 (a), (b) and (c), we obaned hese resul and hen modelng hem n Fg 5

16 6986 Rozam Zakara and Abd Faah Wahab ( D, D) 2 2 ( D8, D8) 7 7 ( D5, D5) 4 4 ( D3, D3) 6 6 (a) (a) ( D2, D2) 7 7 ( D5, D5) ( D, D) ( D, D) ( D8, D8) ( D3, D3) 4 4 ( D6, D6) (b) ( D2, D2) 7 7 ( D5, D5) ( D, D) ( D, D) ( D8, D8) ( D3, D3) 4 4 ( D6, D6) (c) (Pnk=Defuzzfcaon, Blue=Crsp) Fgure 5 Defuzzfcaon process agans fuzzy nerpolaon B-splne curve Fg 5 shows ha he defuzzfcaon process of fuzzy nerpolaon B-splne curve has been appled based on Fg 4 (a), (b) and (c) The defuzzfy curve s marked by green color and he crsps curve s marked by red color If he α values end o one of fuzzfcaon curve, hen defuzzfcaon curve s end o crsp curve Ths s he relaon beween fuzzfcaon and defuzzfcaon 6 Example of Fuzzy Inerpolaon B-splne Curve Applcaon Afer he consrucon of fuzzy nerpolaon B-splne curve was obaned, hen we need some example of applcaon o apply n fuzzy nerpolaon B-splne curve model Therefore, we choose verfcaon offlne handwrng sgnaure (OHS) model as he basc and smple example

17 Fuzzy B-splne modelng of uncerany daa Verfcaon of Offlne Handwrng Sgnaure The verfcaon of offlne handwrng sgnaure usng fuzzy geomerc curve had been dscussed n [,9] usng fuzzy nerpolaon raonal Bezer curve model Ths model had been developed n pecewse raonal curve Therefore, we demonsraed he fuzzy nerpolaon B-splne curve n modelng OHS The crsp OHS can be gven n Fg 6 D D 2 D D 6 D 8 D D 3 D 5 D D 7 D 9 D 4 D 2 Fgure 6 Crsp OHS model Fg 6 shows ha he crsp OHS model whch s model by usng nerpolaon B-splne curve wh cenrpeal paramerzaon Ths OHS model s he sgnaure model whch doesn use global nerpolaon B-splne curve The sgnaure model s modelng by pece of nerpolaon B-splne curve such ha he C -connuy are a D 6, D 8 and D If here exss uncerany of OHS, hen fuzzy se heory especally fuzzy number concep s used o defne he uncerany daa of OHS whch called fuzzy daa By FDPs defnon, hen he modelng of fuzzy OHS model hrough fuzzy nerpolaon B-splne curve can be pcured as Fg 7

18 6988 Rozam Zakara and Abd Faah Wahab D D 2 D D 6 D 8 D D 3 D 5 D 7 D 9 D D 4 D 2 Fgure 7 Fuzzy OHS whch modeled hrough fuzzy nerpolaon B-splne curve Fg 7 shows ha he fuzzy OHS whch s modeled usng fuzzy nerpolaon B-splne curve wh welve fuzzy daa pons The fuzzy daa pons of fuzzy OHS had dfferen suaon of FDPs whch has FDPs a x-axs, y-axs and dagonal Therefore, for fuzzfcaon of all hese knd of fuzzy daa, we appled he α -cu rangular fuzzy number whch has menoned before n order o oban a new fuzzy daa pons of fuzzy OHS afer α -cu operaon Thus, we obaned he new fuzzy daa whch s modeled by fuzzy nerpolaon B-splne curve of fuzzy OHS by Fg 8 wh he values of α s Fgure 8 The new fuzzy OHS(α -fuzzy OHS) of fuzzy nerpolaon B-splne curve

19 Fuzzy B-splne modelng of uncerany daa 6989 The α -fuzzy OHS s obaned by Fg 8 based on he operaon of α -cu agans FDPs n Fg 5 When he α -fuzzy OHS s acheved, hen he nex sep s o fnd crsp fuzzy soluon of he fuzzy OHS Based on Def 9, he crsp fuzzy soluon of fuzzy OHS can be gven hrough Fg 9 ( D, D) ( D, D) ( D5, D5) 2 2 ( D3, D3) ( D6, D6) ( D9, D9) ( D, D ) ( D, D ) Green=Defuzzfcaon, Red=Crsp Fgure 9 Crsp fuzzy soluon(defuzzfcaon) of α -fuzzy OHS Fg 9 shows ha he defuzzfcaon of α -fuzzy OHS(green curve) s ploed ogeher wh crsp OHS(red curve) Based on ha fgure, he crsp fuzzy soluon of fuzzy OHS whch s modeled by fuzzy nerpolaon B-splne curve have a lle error wh he crsp OHS Ths s happen because ha he ceran FDPs of fuzzy OHS had non-symmerc FDPs and he ohers are symmerc FDPs 7 Dscusson and Concluson The consrucon of fuzzy B-splne nerpolaon curve had been dscovered based on s requred n modelng of he uncerany daa va geomerc modelng The fuzzy se heory once more s used o develop a new model based on he expended of fuzzy geomerc feld The fuzzy B-splne nerpolaon curve whch s used n modelng of fuzzy OHS n verfcaon of he sgnaure s one of he applcaon of hs fuzzy model Ths fuzzy B-splne nerpolaon curve also can be exended o varey of FDPs forms and also can be appled n many feld n order o model he FDPs whch defned by fuzzy number and fuzzy relaon conceps

20 699 Rozam Zakara and Abd Faah Wahab Acknowledgemen The frs auhors would lke o acknowledge Research Managemen and Innovaon Cenre (RMIC) Unvers Malaysa Terengganu and Mnsry of Hgher Educaon (MOHE) Malaysa for her fundng(frgs, vo59244) and provdng he facles for dong hs research References [] W Abd Faah, Z Rozam, Fuzzy Inerpolaon Raonal Cubc Bezer Curves Modelng of Blurrng Offlne Handwrng Sgnaure wh Dfferen Degree of Blurrng, Appled Mahemacal Scences, 6 (22), [2] W Abd Faah, MA Jamaluddn, AM Ahmad, Fuzzy Geomerc Modelng, Proceedngs Inernaonal Conference on Compuer Graphcs, Imagng and Vsualzaon, 29 CGIV 29, 29, pp [3] W Abd Faah, Z Rozam, MA Jamaluddn, Fuzzy Inerpolaon Raonal Bezer Curve, Sevenh Inernaonal Conference on Compuer Graphcs, Imagng and Vsualzaon (CGIV 2) (2), [4] W Abd Faah, MA Jamaluddn, AM Ahmad, MT Abu Osman, Fuzzy Se In Geomerc Modelng, Proceedngs Inernaonal Conference on Compuer Graphcs, Imagng and Vsualzaon, 24 CGIV 24 (24), [5] AM Anle, B Falcdeno, G Gallo, M Spagnuolo, S Spnello, Modelng unceran daa wh fuzzy B-splnes, Fuzzy Ses and Sysems, 3 (2), [6] S Azzan, AM Ahmad, MP Abd Rahn, Vsualzaon of Ranfall Daa Dsrbuon Usng Qunc Trangular Bezer Paches, Bullen of he Malaysan Mahemacal Scences Socey, 32 (29), 37-5 [7] H Caglar, N Caglar, K Elfaur, B-splne nerpolaon compared wh fne dfference, fne elemen and fne volume mehods whch appled o wo-pon boundary value problems, Appled Mahemacs and Compuaon, 75 (26), [8] L Choong-Gyoo, A unversal paramerzaon n B-splne curve and surface nerpolaon, Compuer Aded Geomerc Desgn, 6 (999), [9] G Farn, Curves and Surfaces for CAGD: A Praccal Gude, 5h ed, Academc Press, USA, 22 [] G Farn, J Hoschek, M-S Km, Handbook of Compuer Aded Geomerc Desgn, Elsever Scence BV, The Neherlands, 22 [] G Gallo, M Spagnuolo, Uncerany codng and conrolled daa reducon usng fuzzy-b-splnes, Proceedngs Compuer Graphcs Inernaonal, 998 (998),

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