Appl. Math. If. Sc. 7, No. 2, 741-747 (2013) 741 Appled Mathematcs & Iformato Sceces A Iteratoal Joural A Exteso of a New Kd of Graphcs Fttg Method Shaohu Deg 1, Suje Gua 2, Guozhao Wag 1 ad Shaobo Deg 3,4 1 Departmet of Mathematcs, Zhejag Uversty, Zhejag 310027, Cha 2 School of Iformato Egeerg, NaChag Isttute of Techology, Nachag, 330099 P.R. Cha 3 Key Laboratory of tellget formato, Isttute of Computg Techology, Chese Academy of Sceces, Bejg, 100190.P.R. Cha 4 Graduate Uversty of Chese Academy of Sceces, Bejg, 100180, P.R. Cha Receved: 25 Aug. 2012; Revsed 21 Nov. 2012; Accepted 3 Dec. 2012 Publshed ole: 1 Mar. 2013 Abstract: The progressve teratve approxmato method (PIA) for fttg s developed recet years. The prevous research s maly restrcted to the ormalzed totally postve (NTP) bass. I ths paper, we propose a ew exteded method whch ca ufy the classc PIA method, weghted PIA method (WPIA) ad local PIA (LPIA) method to oe formula by troducg the trasform matrx. I addto, the method preseted by us has the followg advatages: frstly, t ca be appled to ay bass; secodly, compared wth other exteded methods, the covergece rate of our method ca be depedet of the tal parameterzatos; thrdly, for the NTP bass, the exteded PIA method ca be accelerated greatly wth the fastest covergece rate. Fally, we preset two umercal examples to show that our algorthm s effcet. Keywords: Progressve-teratve, Data fttg, Trasform matrx 1. Itroducto Data fttg s used wdely the feld of CAGD ad CG. From the hstory pot of vew, ths topc has bee extesvely studed. May kds of methods are well establshed, for example, the least-square approxmato, the Newto terpolato ad other umercal methods. However, the potetal dsadvatage of the tradtoal fttg method s that t lacks of geometrc tuto, whch s mportat to the CAGD. Recetly, a ew kd of data fttg method, called the progressve teratve approxmato, has receved much atteto whch ca compesate for that problem, furthermore t has the vrtue of smplcty, o eed to solve a system of lear equatos ad umercal stablty ad so o, hece, t s more attractve whe comparg wth the tradtoal methods. The PIA method was proposed by Q et al. [1] ad de- Boor [2] respectvely. Afterwards, L et al. [3, 4] showed that the PIA method ca be exteded to the o-uform B- sple bass fuctos ad surface, furthermore, to ay ormalzed totally postve bass fuctos. Lkely, Maekawa et al. [5] made a mprovemet to the PIA method, whch ca be vewed as the geometrc form of the PIA method. Some ew progress has bee made, for stace, Che et al. [6] exteded ths method to tragular Bézer surfaces. Very recetly, L [7] developed a adaptve algorthm, whch s more flexble data fttg. So far, three kds of teratve formats of the PIA method have bee preseted: Q [1] preseted the classcal PIA teratve format; L [8] devsed a local PIA teratve format whch ca ft the data pots locally; Lu [9] preseted weghted teratve format to speed up the covergece of the method. Although there s sgfcat dfferece betwee the three kds of PIA methods vew of the teratve form, we fd that they have the ufed teratve format by troducg the trasform matrx. I addto, we preset aother teratve format whch ca be appled to ay bass fuctos cludg the NTP bass. Especally, as for the NTP bass, the covergece rate of the exteded method gve by us ca be accelerated greatly. Correspodg author: e-mal: paperdeg@126.com
742 Shaohu Deg et al. : A Exteso of a New Kd of Graphcs Fttg Method 2. Relatoshp betwee the PIA method ad power seres expaso of the collocato matrx 2.1. Prelmary Gve a sequece of data pots {P } =0 ad a set of ormalzed totally postve bass fuctos {B (t) = 0,1,...,}, each P s assged a parameter value t, = 0,1,, where {t } =0 s a real creasg sequece, that s, t 0 < t 1 < < t. For k = 0,1,, the terato of the PIA method cossts of the followg three steps: Step1. Geerate a startg curve as: R 0 (t) = B(t)P 0 where P 0 = [P0 0,P0 1,,P0 ] T ad P 0 = P, = 0,1,,. Step2. Compute the adjustg vectors: k = P R k (t ), = 0,1,,. Step3. Adjust the cotrol pots: P k+1 = P k + k, = 0,1,,. Repeatg ths process, we get a curve sequece R k (t) = B(t)P k,k = 0,1,. where P k = [P k 0,Pk 1,,Pk ] T. We have the followg teratve formula: k = (I B) k 1,k = 1,2, (1) where k = [ k 0, k 1,, k ] T. 2.2. Power seres expaso of the verse of the collocato matrx I ths secto, we reveal the relatoshp betwee the PIA method ad the power seres expaso of the matrx verse. We frstly troduce the followg theorem. Theorem 2.1. Let B ad U be the square osgular matrces ad suppose ther spectral radus satsfes: ρ(i UB) < 1, the we have the followg power seres expaso: B 1 = U + k=1 (I UB) k U, (2) where I s the detty matrx. Proof. Sce matrces B ad U satsfy: ρ(i UB) < 1, t s ot dffcult to see that the matrx I UB s osgular. Let s cosder the followg matrx equato: s =0 (I UB) (UB) = I (I UB) s+1. I terms of ρ(i UB) < 1, takg the lmt, we have Furthermore, t gves lm (I s UB)s+1 = 0, =0 (I UB) (UB) = I. Hece, we ca obta the cocluso: B 1 = U + k=1 (I UB) k U. It should be oted that there always exsts such matrx U for ay gve osgular matrx B that the seres of the matrx coverges. So, we preset the followg lemma whch guaratees the exstece of the matrx U. lemma 2.1. For ay gve osgular matrx, the matrx U always exsts. I geeral, we ca take U = BT ρ(bb T ), where ρ(bb T ) s the spectral radus of the matrx BB T. Theorem 2.1 gves aother way to fd the matrx verse, the ma challege of ths method s that the covergece rate s ucerta. I geeral, dfferet choce of the matrx U leads to dfferet covergece rate of the power seres expaso of the matrx B 1 whch depeds o the spectral radus ρ(i UB) < 1. I addto, the choce of the matrx U s ot uque. For some specal form of matrx B, we ca choose some specal matrx U. Corollary 2.1. Let B be a osgular square matrx ad suppose the egevalues of the matrx B satsfy: 0 < λ (B) < 1, = 0,1,,. If we take U = I, the we have the followg matrx expaso: B 1 = k=0 (I B) k. Corollary 2.2. Let B be a osgular square matrx ad suppose the egevalues of the matrx B satsfy: 0 < λ (B) < 1, = 0,1,,. If we take U = ai wth 0 < a < 1 λ max (B), the we have the followg matrx expaso: B 1 = ai + a =1 (I ab), where λ max (B) s the maxmal egevalue of the matrx B. It s obvous that dfferet value a leads to dfferet covergece rate of the matrx expaso. Therefore, we ca optmze the value a to speed up covergece of the matrx seres. To ths ed, we troduce the followg corollary. Corollary 2.3. Let B a osgular square matrx ad suppose the egevalues of the matrx B satsfy: 0 < λ (B) <
Appl. Math. If. Sc. 7, No. 2, 741-747 (2013) / www.aturalspublshg.com/jourals.asp 743 2 1, = 0,1,,. If we take a = 1+λ m (B), the t wll speed up the covergece of the matrx seres. Furthermore, we have the followg matrx expaso: B 1 = 2 1 + λ m (B) I + 2 1 + λ m (B) =1 ( ) 2 I 1 + λ m (B) B. where λ m (B) s the mmal egevalue of the matrx B. Proof. The proof s smlar to the [9], here we leave detals to the terested readers. From the three corollares above, we fd that f dfferet form of the matrx U s take, the dfferet power seres expaso of the matrx verse ca be obtaed. If we take some specal form of the matrx U, some specal matrx expaso ca be obtaed. Corollary 2.4. Let B be a osgular square matrx. Suppose the matrx U satsfes: [ ] I 0 UB =, (3) C 21 C 22 ad ts spectral radus satsfes:ρ(i UB) < 1, the we have [ ] B 1 I = + C 21 C 22 (I UB) U. =1 2.3. Iterpolato expressed by the matrx expaso The terpolato problem s usually stated as: gve a sequece of data pots {P } =0 ad a set of bass fuctos {B (t)} =0, each P s assocated wth a parameter value t, wth 0, where o two t are the same. Geerally, the terpolato problem ca be expressed the matrx form B 0 (t 0 ) B 1 (t 0 ) B (t 0 ) X 0 P 0 B 0 (t 1 ) B 1 (t 1 ) B (t 1 ) X 1...... = P 1.. B 0 (t ) B 1 (t ) B (t ) X We ca shorte the above equato to BX = P, where B s the collocato matrx ad P = [P 0,P 1,,P ] T. It s well kow that whe the matrx B s osgular the terpolatg curve ca be expressed as the followg form: P(t) = B(t)B 1 P. The terpolato problem ca always be reduced to solve a lear system equatos. May umercal methods have bee establshed to solve t. However, we ca obta a specal form of the terpolatg curve by troducg the trasform matrx whch s vtal to PIA method. P Defto 2.1. Let U be a osgular square matrx, {B (t)} =0 be a set of bass fuctos ad {P } =0 be a set of data pots. The matrx U s called the trasform matrx f we make a trasform to the gve bass fuctos ad the gve data pots, that s, B j (t) = P = j=0 j=0 u j B j (t) u j P j, (4) where u j s the j th etry of the matrx U. Accordg to Theorem 2.1, we ca choose the trasform matrx U to satsfy ρ(i UB) < 1, the the partcular form of the terpolatg curve ca be obtaed. Theorem 2.2. Gve a set of data pots {P } =0 ad a set of bass fuctos {B (t)} =0, each P s assocated wth a parameter value wth 0 t 0 < t 1 < < t 1, suppose U s the trasform matrx whch satsfes ρ(i UB) < 1, the the terpolatg curve ca be expressed the form of matrx expaso. Furthermore, the terpolatg curve s depedet of the choce of the trasform matrx U. R(t) = B(t) =0 where B s the collocato matrx. (I UB) UP, (5) Proof. Sce the matrx U satsfes: ρ(i UB) < 1, t follows that power seres of the matrx coverges (I UB). =0 I terms of Theorem 2.1, t s ot dffcult to see that: R(t) = B(t) =0 = B(t)B 1 P. (I UB) UP Furthermore, the above cocluso shows that the terpolatg curve s depedet of the choce of the trasform matrx U. Remark: It should be oted that the terpolatg curve R(t) = B(t) =0 (I UB) UP s the same oe as that obtaed by solvg the system of the lear equatos, here, we express t aother form. Furthermore, we fd that usg the Eq.(5), we ca derve the three kds of PIA methods. Let us rewrte the Eq.(5) as R(t) = B(t)UP + B(t) =1 (I UB) UP. We call the UP the tal cotrol pots, ad the formula =1 (I UB) UP the teratve cotrol pots. The followg theorem shows the er relatoshp betwee the PIA method ad the matrx expaso.
744 Shaohu Deg et al. : A Exteso of a New Kd of Graphcs Fttg Method Theorem 2.3. Gve a set of data pots ad a set of NTP bass of fuctos {B (t)} =0, the trasform matrx U satsfes ρ(i UB) < 1, f we set the tal cotrol pots UP = P, the the PIA method s equvalet to the tradtoal terpolato method, furthermore we have: (a) f we take U = I, the we ca get the classcal PIA teratve format. 2 (b) f we take U = 1+λ m (B) I, the we ca get the WPIA teratve format. (c) f the matrx U s take as Eq.(3), the we ca get the LPIA teratve format. Proof. Sce the tal pots are set to P, the terpolatg curve ca be expressed as the followg form: R(t) = B(t)P + B(t) =1 (I UB) UP. (6) We deote B = UB, P = UP, the Eq.(6) ca be rewrtte as R(t) = B(t)P + B(t) =1 (I B) P. Takg m + 1 ad m + 2 frst terms of the above equato, we get ad R m (t) = B(t)P + B(t) m =1 m+1 R m+1 (t) = B(t)P + B(t) =1 (I B) P, (I B) P. Suppose the cotrol pots of the approxmatg curve R m (t) s Q m, R m (t) ca be rewrtte as R m (t) = =0 B (t)q m. (7) Suppose the cotrol pots of the R m+1 (t) s Q m+1, by the same way, we ca get R m+1 (t) = =0 B (t)q m+1. (8) The, we subtract Eq. (8) from Eq. (7), we ca obta R m+1 (t) R m (t) = =0 B (t)(q m+1 Q m ) = B(t)(I B) m+1 P. Furthermore, we deote k, = 0,1,, by the adjustg errors, lke the teratve formula Eq. (1), we ca have B(t)(I B) m+1 P = =0 B (t) m. Thus, we get =0 B (t)(q m+1 Q m ) = =0 B (t) m. Accordg to {B (t)} =0 are learly depedet, we get: Q m+1 Q m = m, = 0,1,,,m = 0,1, Sce k = (I UB) k 1, t follows that f we take U = I, we obta the classcal PIA method, Smlarly, f U s take to the correspodg forms, we ca get cases (b) ad (c), whch completes the proof. 3. Exteded PIA method Theorem 2.3 shows that the PIA method essece s equvalet to the followg teratve process: =0 (I UB) U B 1, where U s the trasform matrx. If U s properly chose, some partcular effect ca be obtaed, for stace, the LPIA ad the WPIA method. I ths secto, we exted the PIA method to ay bass fuctos. The prmary challege extedg the PIA method s the selecto of the trasform matrx U. To get the most beeft from the choce of ths matrx, we would lke to fd optmal U, mmzg the spectral radus ρ(i B). The followg lemma ca be foud the book [10]. Lemma 3.1. Let A be a matrx, the the matrx A ca be dagoalzed as: A = QR, where Q T Q = I ad R s the upper tragular matrx. Let B be the collocato matrx, accordg to Lemma 3.1, the matrx B ca be dagoalzed as: λ 0 r 12 r 1 λ 1 r 2 B = Q.... λ = QR. We ow defe the trasform matrx U to be: 1 / λ 0 1 / λ 1 U = QT = HQ T.... 1 / λ (9)
Appl. Math. If. Sc. 7, No. 2, 741-747 (2013) / www.aturalspublshg.com/jourals.asp 745 The exteded PIA method cossts of the followg steps: Step1. Iput the data pots P, = 0,1,,. Step2. Make a trasform to the gve bass fuctos ad data pots. B(t) = UB(t) T P = UP. Step3. Geerate a startg curve as: R(t) = B(t)P. Step4. Compute the error vectors: k = P R(t ), = 0,1,,. Step5. Adjust the cotrol pots: P k+1 = P k + k, = 0,1,,. The, we get the teratve curve sequece: { R (t) }. Theorem 3.1. Gve a set of bass fuctos {B (t)} =0 ad a set of data pots {P} =0, suppose the trasform matrx s chose as Eq. (9), the the exteded PIA method coverges ad ths algorthm termates at most + 1 steps. Proof. Accordg to the algorthm gve by us, the teratve curve sequece { R (t) } ca be wrtte as the followg form: R (t) = B(t) =0 (I B) P, where B = UB ad P = UP. Sce the matrx B ca be dagoalzed as B = QR wth QQ T = I, t follows that I B = I HQ T QR = I HR. It s ot dffcult to see that the ma dagoal of the matrx I HR s equal to zero, furthermore we have: ρ(i B) = 0. Sce B s a ( + 1 + 1) square matrx, t follows that: ( I B ) +1 = 0. The, the error vectors +1 = 0, = 0,1,,. So, the exteded PIA method ca termate at most + 1 steps. 4. Numercal examples I ths secto, we apply our method to two dfferet kds of bass fuctos. I order to compare the covergece rate wth the prevous research, we use the same examples proposed Delgado [11] ad Che [12]. For the example 1, we use the Berste bass {B 10 (t)} 10 =0 whch s the typcal NTP bass. For example 2, we cosder the Wag-Ball bass {W 4(t)}4 =0 whch s o-ntp bass whe = 4. Example 1. Cosder the plaar Bézer curve of degree 10, ts cotrol pots are sampled from the Lemscate of Geroo gve by the parametrc form (x(t),y(t)) = (cos(t),s(t)cos(t)), whch are sampled by the followg way t [0,2π] P = (x(u ),y(u ),u = π 2 +, = 0,1, 10. 10 Example 2. Cosder four degree Wag-ball bass fuctos, whch s defed as follow: (W 4 0,W 4 1,,W 4 4 ) = ((1 t) 2,2(1 t) 3,4(1 t) 2 t 2,t 2 ) ts cotrol pots are sampled from the followg parametrc curve by the the way P = (cos(t ),s(t )), t = π, = 0,1,,4. 4 For smplcty, we use the uform parameter. Each P s assged a parameter value t, t = /, = 0,1,,, that s, the uform dstrbuto of the parameters. Let s cosder example 1. Through computato, the trasform matrx U s: 0.88 0.31 0.09 0.02 0.01 0.0 0.00 0.00 0.00 0.00 0 0.69 1.57 1.18 0.55 0.18 0.05 0.01 0.00 0.00 0.00 0 0.53 2.32 1.79 2.81 1.83 0.75 0.19 0.03 0.00 0.00 0 0.39 2.38 4.55 0.47 4.85 4.43 2.09 0.52 0.05 0.01 0 0.28 2.06 5.76 5.80 3.00 6.40 8.55 4.35 0.85 0.03 0 0.18 1.58 5.46 9.00 4.20 7.73 6.78 13.85 6.55 0.54 0 0.11 1.07 4.25 8.96 9.24 0.16 11.47 6.82 19.26 5.26 0 0.06 0.62 2.71 6.60 9.26 5.66 4.28 12.48 9.58 20.11 0 0.04 0.36 1.66 4.38 7.08 6.47 0.90 6.72 12.11 16.49 0 0.10 1.11 5.63 17.14 35.00 50.40 52.50 40.00 22.50 10.00 0 0 0 0 0 0 0 0 0 0 0 1.0 It should be oted that the classcal PIA method depeds o the ρ(i B), ad furthermore, the tal parameterzatos determe the covergece rate of the PIA method. Hece, although the classcal PIA method usually have the desrable covergece rate most cases, the covergece rate s ucerta whch chages alog wth the tal parameterzatos. The example 1 shows that aother advatage of our method s that t s depedet of the choce of the tal parameters, whch has the fastest covergece rate, because the spectral radus ρ(i B) = 0. I order to make a comparso, we set the precso ε = 0.01, ad the error s measured by the formula ε = max 0 { P R (t ) }. We frstly apply the classcal PIA to example 1, ad we fd that the umber of teratos requred s 190. Secodly, we use the WPIA method preseted by Lu, the covergece rate s accelerated early
746 Shaohu Deg et al. : A Exteso of a New Kd of Graphcs Fttg Method (a) (b) (c) Fgure 1 Bézer bass.(a) Ital curve,(b) After 8 teratos,(c) After 11 teratos. (d) (e) (f) Fgure 2 Wag-ball bass. (a)ital curve, (b) After 3 teratos,(c) After 5 teratos. two tmes. Fally, we use the exteded PIA method: the Fg.1(a),(b),(c) shows the fttg effect at dfferet level of teratos. We fd that the error s 2.42e 007 oly after 11 teratos. Whe compared wth the classcal PIA method ad WPIA method, the covergece rate of our method has bee accelerated almost te tmes. At the same tme, the example 2 shows that our method ca also be appled to o-ntp bass, Fg.2(a),(b),(c) show that the fttg effect at dfferet level of teratos. I fact, The Fg.2(b) shows that the desrable fttg effect have already bee gaed oly after 3 teratos. Furthermore, we fd that the error s : 1.5101e 015. I Fg.2(c) after 5 teratos. So, for the o-ntp bass, the exteded method s also qute effectve. By computato, the trasform matrx s 0.7232 0.4068 0.1808 0.0452 0 2.4822 3.3870 2.2139 0.3763 0 U = 0.2870 2.4197 3.4442 3.4077 0 0.5000 2.6667 6.0000 8.0000 0. 0 0 0 0 1.000 5. Coclusos I ths paper, we lk the PIA method wth the power seres expaso of the matrx. Through troducg the trasform matrx, we preset a more geeral teratve format of PIA method whch cludes the classcal PIA, LPIA ad WPIA. I addto, we gve a specal form of trasform matrx whch ca be appled to both NTP bass ad ay other bass fuctos. Compared wth other form of the PIA methods, umercal examples show that our method has the fastest covergece rate wth oly a lttle computatoal complexty. Ackowledgemet The research of ths paper s supported by the Natoal Nature Scece Foudato of Cha (Nos. 60970079 ad 60933008) Refereces [1] D. X. Q, Z. Ta, Y. Zhag, J. Feg, The method of umerc polsh curve fttg, Acta Mathematca Sca, 18, 173-184(1975). [2] C. D. Boor. How does Agee s smoothg method work?, ftp://ftp.cs.wsc.edu./approx/agee.pdf [3] H. W. L, H. J. Bao, C. S. Dog, Costructg teratve o-uform B-sple curve ad surface to ft data pots, Scece Cha Seres F: Iformato Sceces, 47, 315-331(2004).
Appl. Math. If. Sc. 7, No. 2, 741-747 (2013) / www.aturalspublshg.com/jourals.asp [4] H. W. L, H. J. Bao, G. J. Wag, Totally postve bases ad progressve terato approxma-to, Computer ad Mathematcs wth Applcatos, 50, 575-586(2005). [5] T. Maekawa, Y. Matsumoto, K. Namk, Iter-polato by geometrc algorthm. Computer Aded Desg, 39, 313323(2007). [6] J. Che, G. J. Wag, Progressve teratve approxmato for tragular Bzer surfaces, Computer-Aded Desg, 43, 889895(2011). [7] H. W. L, Adaptve data fttg by the progressve-teratve approxmato, Computer Aded Geometrc Desg, 29, 463-473(2012). [8] H. W. L, Local progressve-teratve approxmato format for bledg curve ad patches, Computer Aded Geometrc Desg, 27, 322-339(2010). [9] L. Z. Lu, Weghted progressve terato approxmato ad covergece aalyss, Computer Aded Geometrc Desg, 27, 129-137(2010). [10] B. Stephe, Matrx methods for egeers ad scetsts. McGraw-Hll, Berkshre, UK, 1979. [11] J. Delgado, J. M. Peoa, Progressve teratve approxmato ad bases wth the fastest covergece rates, Computer Aded Geometrc Desg, 24,10-18(2007). [12] J. Che, G. J. Wag, Two kds of geeralzed progressve teratve approxmato, Atca Automatca Sca 38, 135139(2012). 747 Shaohu Deg receved the MS degree appled mathematcs from Zhejag Uversty 2009, He s curretly PhD Caddate Zhejag Uversty. Hs research terests are the areas of CAGD ad CG. Suje Gua Suje Gua, the research terests are computer graphcs, mathematcal logc ad modal logc. Guozhao Wag receved the MS degree appled mathematcs from Zhejag Uversty 1982, He s curretly a professor ad doctoral tutor Zhej-ag Uversty. Hs research terests are the areas of CAGD ad CG. Shaobo Deg the research terests are computer graphcs, Mathematcal logc ad modal logc. He s curretly PhD Caddate key Laboratory of tellget formato, Isttute of Computg Techology,Chese Academy of Sceces. c 2013 NSP