Convexity Preserving Using GC Cubic Ball. Interpolation

Transcription

1 Appled Mathematcal Scences, Vol 8, 4, no 4, 87 - HIKARI Ltd, wwwm-hkarcom Convexty Preservng Usng GC Cubc Ball Interpolaton Samsul Arffn Abdul Karm, Mohammad Khatm Hasan and Jumat Sulaman 3 Department of Fundamental and Appled Scences Unverst Teknolog PETRONAS,Bandar Ser Iskandar 375 Tronoh, Perak Darul Rdzuan, Malaysa Jabatan Komputeran Industr, Unverst Kebangsaan Malaysa 436 UKM Bang, Selangor, Malaysa 3 Program Matematk dengan Ekonom, Unverst Malaysa Sabah Beg Berkunc 73, Kota Knabalu, Sabah, Malaysa Copyrght 4 Samsul Arffn Abdul Karm et al Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted Abstract Ths paper studes the use of cubc Ball nterpolaton for convexty preservng of scalar data The parameter r n the descrpton of the cubc nterpolant are subject to be constraned to guarantee the exstence of the convex cubc nterpolant for convex data The suffcent condton for the convexty-preservng of cubc Ball nterpolant wll be developed The degree smoothness attaned s GC Wth sutable choces of the shape parameter, r the method under consderaton s reducng to GC quadratc splne shape preservng nterpolaton for convex data Several numercal results ncludng wth the comparson wth exstng scheme wll be presented to test the applcablty of the proposed cubc nterpolant Keywords: Shape preservng, Interpolaton, Cubc Ball, Convexty, Contnuty

2 88 Samsul Arffn Abdul Karm et al INTRODUCTION Shape preservng data nterpolaton s mportant n scentfc vsualzaton and computer graphcs For example f the gven data s convex, t s necessary that the resultant nterpolant must be also convex everywhere Convexty occurs n nonlnear programmng such as parameter estmaton and to measure the senstvty of the duraton of a bond to changes accordng to the respectve nterest rates The natural choce for data nterpolaton n scences and engneerng applcatons s cubc splne polynomal Even though the cubc splnes are smoother wth C contnuty and well documented n many books n numercal analyss together wth Matlab mplementaton, t has weakness such as the cubc nterpolant not guarantee to produces the convex curves or surfaces for convex data There exst varous methods proposed by several researchers n order to control the convexty of the nterpolatng curves and/or surfaces For example, Hayman [] and Dougherty et al [] used cubc and quntc splne for the postvty-, monotoncty- and convexty-preservng Brodle and Butt [3] studed the convexty preservng usng cubc splne by nsertng extra knots (one or two) n the regon where shape volatons are found By nsertng extra knots, the computaton nvolve n generatng the convexty preservng nterpolatng curves wll also be ncreased Furthermore, t s not an easy task for the nexperence user to choose any sutable knots that need to be nserted Meanwhle, as an alternatve to the use of polynomal for convexty preservng, many ratonal splne schemes have been proposed by several authors Few examples ncludes, Gregory [4], Delbourgo[5], Delbourgo and Gregory [6], Sarfraz [7], Hussan and Hussan [8], Hussan et al [9], Zhang et al [] and Karm and Pah [] All of the ratonal splne schemes have one man dea: acheved the convexty preservng by developng constrant to produce the suffcent condton for the convexty of the ratonal nterpolant Ths can be acheved by constranng the shape parameters that appear n the descrpton of the ratonal nterpolant Sarfraz et al [] has proposed shape preservng GC cubc splne nterpolaton for postve, convex and monotone data Motvated by the work of Sarfraz et al [], the author n Karm [3] and Karm [4] have proposed cubc Ball nterpolaton for shape preservng nterpolaton of monotone and postve data The author derved the suffcent condton whch s slghtly stronger than the suffcent condton for postvty and monotoncty n Sarfraz et al [] The numercal results also show that the proposed cubc Ball nterpolant preserves very well the shape of the gven data sets In ths paper, the cubc Ball nterpolaton wll be used for convexty preservng for convex data The degree smoothness attaned s GC One of the man advantages of our proposed Cubc Ball nterpolant compared to the work of Sarfraz et al [] s that the cubc Ball nterpolant could be easly reduced to quadratc splne nterpolaton that have been dscussed n detals by Schumaker [6] and Lam [7] Smlar to the work by Sarfraz et al [] the proposed cubc Ball nterpolants not need any nsertng extra knots n the nterval n whch the shape volatons are found Fnally, the proposed GC cubc Ball nterpolant also ncluded GC quadratc splne polynomal as a specal

3 Convexty preservng 89 case We dentfy several features of our ratonal cubc splne for convexty preservng It s summarzed below: () In ths paper the cubc Ball nterpolant has been used for convexty preservng whle n Sarfraz et al [] the cubc splne s used nstead () Smlar to the work of Sarfraz et al [], the proposed cubc Ball nterpolant also has GC contnuty () Our defnton for the cubc nterpolant s dfferent from the defnton of cubc splne nterpolant by Sarfraz et al [] (v) Numercal comparson between the proposed cubc Ball schemes wth Sarfraz et al [] and Frstch and Carlson [3] (pchp) also has been done Furthermore nether dervatve modfcaton s requre nor nsertng extra knots to preserves the convexty, compare to the works of Dougherty et al [], Lam [7] and Schumaker [6] n whch they requres an extra knots n the nterval n whch the convexty of the data s not preserves For more detals on shape preservng nterpolaton and approxmaton, the readers can refer to [8] and [9] and the references cted theren The remander of the paper s organzed as follows Secton dscusses the cubc Ball nterpolant taken from Karm [4] Secton 3 dscusses the proposed nterpolant n the context of convexty preservaton together wth the mplementaton algorthm All numercal results wll be dscussed n Secton 4 A summary and conclusons are gven n Secton 5 GC CUBIC BALL INTERPOLANT In ths secton, the cubc Ball nterpolant proposed n Karm [3, 4] wll be dscussed Let { ( x, f),,, n} s a gven set of data ponts, where x < x < < x n Let ( f+ f) h x x, Δ ( x x ) and a local varable, θ where θ + h h x x, x+,,,, n, s x s x + hθ S θ () Now for ( ) ( ) ( ), where S ( θ ) A( θ) + Aθ( θ) + Aθ ( θ) + A3θ To make the cubc functon () be GC, one needs to mpose the followng nterpolatory propertes: ( ) ( + ) + () d ( ) () ( ) s x f, s x f, d () s x, s x + + r r+ where ( s ) ( x ) denotes the frst dervatve wth respect to x and d denotes the dervatve value whch s gven at the knot x,,,, n The parameters r ( r >, r ) wll be

4 9 Samsul Arffn Abdul Karm et al constrant n order to generate the postve cubc Ball nterpolant on entre gven nterval x, x+,,,, n Hence, the unknowns A,,,,3 have the followng values: hd hd + A f, A3 f +, A f +, A f+, r r+ Thus the cubc Ball nterpolant S GC x, xn n (), can be rewrtten as follows: S( x) S ( θ ), where hd hd S( θ ) f( θ ) f ( ) f ( ) f r θ θ r θ θ + + θ (3) + Obvously when r,,,, n, the cubc Ball nterpolant (3) reduced to cubc Ball polynomal n Hermte-lke form wth C contnuty whch s n general does not produced shape preservng nterpolaton curves In Sarfraz et al [], the shape parameter r r+,,,, n, thus ther cubc nterpolant has only one free parameter Throughout ths paper, we wll utlzed the dea from Sarfraz et al [], n whch the cubc Ball nterpolant has only one parameter r,,,, n Now Equaton (3) can be rewrtten nto Equaton (4) gven below: hd hd S( θ ) f( θ ) f ( ) f ( ) f r θ θ r θ θ + + θ (4) Snce the tested data may come from functon or any scalar data, usually the frst dervatve d,,, n, wll be not gven, hence ts must be estmated usng the methods dscussed n detals by [] and [] One possble choces of the method s arthmetc mean method (AMM) Ths method wll be utlzed for the estmaton of the frst dervatve because ths method s smple and sutable for convex data 3 CONVEXITY PRESERVING USING CUBIC BALL INTERPOLANT 3 Convexty preservng Cubc Ball bass functon has been proposed by Ball [] The cubc Ball nterpolant descrbed n the prevous secton wll not completely preserves the convexty of the data Ths fact can be seen clearly n Fgure and Fgure 7 respectvely In fact, the ordnary cubc splne also does not produce the convexty cubc nterpolant as can be seen clearly n Fgure and Fgure 8 Furthermore t can be seen clearly that the defaults cubc splne and cubc Ball are almost smlar In ths secton, the constrant for the convexty preservng wll be derved nto shape parameter r Our man dea s to construct cubc nterpolant S GC x, xn whch s convex on the whole nterval x, x n Ths can be acheved by choosng sutable values of shape parameter r The convexty-preservng property of cubc Ball nterpolant to a

5 Convexty preservng 9 gven convex (or correspondngly concave) data set wll be examned n detals Let us assume a strctly convex data set s gven such that Δ < Δ < < Δ n (5) (or Δ > Δ > > Δ n for concave data) For an nterpolant s ( x ) n Eq (4) to be convex, t s necessary for dervatve parameters d to satsfy d <Δ < Δ < d <Δ < d < <Δ < d (6) + n n (or d d d+ n dn >Δ > Δ > >Δ > > >Δ > for concave data) Now s ( x ) s convex f and only f s ( x), x [ x, x n ] (7) By dfferentatng S( x ) twce wrt θ () S ( x) b + cθ, (8) where d + d + d + d + 6Δ 6 Δ r b r, c h h () α θ βθ a bθ, S Now, by usng the transformaton ( ) + + ( x ) can be rewrtten as follows () S ( x) B, j( θ ) θ, (9) where B, j d 6 + d + d Δ,, + d + B j 6 Δ h r h r x, x + s: B and B, () Thus the suffcent condton for the convexty on [ ], j j If Δ d > and d + Δ > (for strctly convex data), the followng proposton hold Proposton (Convexty of Cubc Ball Interpolant) For a strctly convex data and Δ, the cubc Ball Interpolant (defned over the nterval [ x, x n ] s convex f n each subnterval [ x, x+ ],,,, n, the followng suffcent condtons are satsfed: (a) Case I: If Δ > and d >, d + >, d + d d + d < r < + + 3Δ 3Δ (b) Case II: If Δ < and d <, d + <, d + d d + d < r < 3Δ 3Δ + + () ()

6 9 Samsul Arffn Abdul Karm et al Proposton The followng choces of shape parameter r gves the suffcent condtons for the convex nterpolant on [ ] d + d r Δ when d >, d + >, Δ > and when d <, d + <, Δ < r d + + d + x, x +,,,, n Prove for Proposton The proof wll be shown only for (3a) By () and Δ >, by substtutng d + d r Δ + nequalty n () can be expressed as follows (3a) Δ (3b) d + d d + d < r < + + 3Δ 3Δ nto (), t can be shown that for the left hand sde of the d + d d + d 3d + 3d 4d d Δ 3Δ 6Δ d+ d > 6Δ due to (6) and Δ > Smlarly one can prove the rght hand sdes of the nequalty n () Furthermore, the nequaltes n (3b) can be proved smlarly usng the fact that d > d + whenever d <, d + < Ths completes the proof of Proposton Whenever Δ > and d <, d + >, then the suffcent condton for convexty n () or (3a) can be used for the convexty preservng purpose These results somehow are more comprehensve compared to the suffcent condton n Sarfraz et al [] Remark : For the case where the data are convex but not strctly convex (when Δ ) t would be necessary to dvde the data nto several strctly convex parts By settng d d + whenever Δ, then the resultng cubc Ball nterpolant wll be C at break ponts Remarks : The suffcent condtons for convexty n (), s the same as n Safraz et al [] Our proposed suffcent condtons for convexty s more comprehensve compared to [] and t s also vald for [] Proposton and Proposton already accommodate both convex data whenever Δ < or Δ >, and when n ( d ) ( d ) and sgn ( d ) sgn ( Δ ) The parameter value at ns r sgn sgn, 3 Shape Preservng Interpolaton Usng Quadratc Splne Polynomal The cubc Ball nterpolant n (4) can be reduced to quadratc splne polynomal wth convexty preservng by sutable choces of shape parameter The followng proposton summarzed the results n d Δ + d n n

7 Convexty preservng 93 Proposton 3 For a strctly convex data and Δ > (case for Δ < can be treated n the same manner), the cubc Ball nterpolant defned by Equaton (4) s reduces to quadratc splne polynomal (defned over the nterval x, xn whch s convex f n each subnterval x, x +,,,, n, d + d r Δ + and (4) can be wrtten as * S ( θ ) f( θ) Vθ( θ) f+ θ where V f + hdδ d + d + Proof for Proposton 3 (4) + +, (5) From the theory of cubc Ball bass functon [], let A A and assumng that then hd hd f + + f+, r r h d d f f r + + +, whch can be smplfed to d + d r Δ + Δ >, Hence, the result n (5) follows by substtutng (4) nto (4) Quadratc splne nterpolant n (5) can be used both for monotone and convex data nterpolaton Usng (5) the resultng quadratc nterpolant stll acheved GC contnuty Remarks 3: The quadratc splne polynomal proposed by Schumaker [6] and Lam [7] * can be obtaned by usng the fact that S ( θ ) s the quadratc splne wth C contnuty f and only f d + d + Δwhch mean that the shape parameter r n (4) s equal to Thus the proposed cubc Ball nterpolant n (4) reduces to quadratc splne nterpolaton n (5) for convexty preservng f r and d + d + Δ These results enable the user to preserves convex data wthout needed to supply an extra knots n whch shape volaton are found as dscussed n detals by Schumaker [6] Ths nterestng fact can only be acheved by usng cubc Ball nterpolant due to the propertes of cubc Ball bass functons [] An algorthm to generate GC convexty-preservng curves usng the results n Proposton and Proposton s gven as follows Algorthm for convexty-preservng usng GC cubc Ball nterpolant n Input the number of data ponts, n, and data ponts { x, f }

8 94 Samsul Arffn Abdul Karm et al For,,, n, estmate d usng arthmetc mean method (AMM) 3 For,,, n Calculate h and Δ 4 For,,, n Calculate the shape parameter r from () to (3) by pre-determned whether Δ < or n Δ > respectvely and d + dn rn Δ Generate the pecewse GC cubc Ball nterpolant nterpolatng convex curves usng Equaton (4) n 5 RESULTS AND DISCUSSIONS To test the capablty of the proposed GC cubc Ball nterpolant, two data sets taken from Sarfraz [7] and Brodle and Butt [3] were used for numercal computatons TABLE A convex data from [7] x 4 5 f TABLE A convex data from [3] x f Fgure : Default cubc Ball nterpolaton for data n Table Fgure : Default cubc splne nterpolaton for data n Table

9 Convexty preservng Fgure 3: Shape preservng nterpolaton cubc Ball nterpolant usng Proposton for data n Table Fgure 6: Shape preservng usng cubc Ball nterpolaton wth true dervatve r 5,5,5,5,5 ( ) Fgure 4: Shape preservng nterpolaton usng Sarfraz et al [] for data n Table Fgure 7: Default cubc Ball nterpolaton for data n Table Fgure 5: True functon for data n Table, f ( x), x [,] x Fgure 8: Default cubc splne nterpolaton for data n Table

10 96 Samsul Arffn Abdul Karm et al Fgure 9: Shape preservng nterpolaton cubc Ball nterpolant usng Proposton wth r 9,3,9,,8,96,96 for data n Table Fgure : Shape preservng cubc Ball nterpolaton usng Proposton wth r 9,3,9,,8, 96, (gray) and r 9,3,9,6,8,96, (blue) Fgure :Shape preservng nterpolaton cubc Ball nterpolant usng Proposton wth r 9,3,9,6,8,96, for data n Table Fgure : Shape preservng nterpolaton usng Sarfraz et al [] for data n Table Fgure 3: Shape preservng nterpolaton usng quadratc splne polynomal usng Proposton

11 Convexty preservng 97 Fgure 4 Shape preservng nterpolaton Fgure 4 Shape preservng nterpolaton usng pchp for data n Table usng pchp for data n Table Fgure and Fgure show the default cubc Ball nterpolaton and cubc splne nterpolaton for data set n Table Fgure 3 shows shape preservng nterpolaton wth cubc Ball nterpolant usng Proposton Meanwhle, Fgure 4 shows shape preservng nterpolaton wth Sarfraz et al [] schemes Fgure 5 shows the true functon for data n Table Fgure 6 shows shape preservng nterpolaton wth cubc Ball nterpolant usng true dervatve Fgure 7 and Fgure 8 show the default cubc Ball nterpolaton and cubc splne nterpolaton for data set n Table In Fgure 9 and Fgure, the shape preservng nterpolaton wth cubc Ball nterpolant usng Proposton are beng shown It can be seen clearly that, by changng the value from r 4 (Fgure 9) to r 4 6 (Fgure ) the postvty of the curve n gray s better resultng to enhanced postvty whle the blue curve s n tangent wth the horzontal axs Fgure shows shape preservng nterpolaton for data set n Table usng Sarfraz et al [] method Fgure shows shape preservng nterpolaton for data set n Table usng the mean result from our proposed Proposton The correspondng values of the shape parameters are r 9,3,9,,8,96 (gray) and r 9,3,9,6,8,96 (blue) respectvely The curve n gray color shows the convexty preservng s acheved as well as the postvty of the data has been preserved too Refer Karm [4] for postvty preservng by usng GC cubc Ball nterpolant Fgure 3, shows the shape preservng nterpolaton usng quadratc splne polynomal as ndcated n Proposton 3 Ths s one of our man contrbutons n ths paper By usng cubc Ball nterpolant to preserve the convex data, the proposed method wll be reduced to the GC quadratc splne polynomal subject to certan condton (as dscussed n Proposton 3) No extra knots are needed compare to the works by Schumaker [6] and Lam [7] Fnally, Fgure 4 and Fgure 5 show the convexty preservng by usng cubc splne polynomal dscussed n detals by Frstch and Carlson [3] and Dougherty et al [] It was mplemented through pchp functon n Matlab software It can be seen clearly that

12 98 Samsul Arffn Abdul Karm et al the proposed GC cubc Ball nterpolant gves better results and more vsual pleasng (graphcs) compare to the pchp Thus, the proposed cubc Ball nterpolant may gves us varety of shape preservng nterpolaton for the convex data by choosng sutable value of shape parameter r 5 CONCLUDING REMARKS The work n ths paper s concerned about the convexty-preservng for convex data sets usng GC cubc Ball nterpolant The proposed methods also have been beng accommodated for all types of convex data The scheme s smlar to the work of Sarfraz et al [] The proposed suffcent condtons for convexty preservng s more comprehensve compare to the suffcent condton by Sarfraz et al [] as shown n Proposton and Proposton respectvely One of the fndng n ths paper s that the proposed cubc Ball nterpolant smply reduced to quadratc splne polynomal for convexty preservng wth GC contnuty Indeed the scheme preserved the convex data wthout havng to nsert extra knots as compared wth the works by Schumaker [5] and Lam [6] The curve scheme can also be generalzed to the surface cases and scattered data nterpolaton The authors are keen to dscuss t n ther forthcomng papers ACKNOWLEDGMENTS The frst author would lke to acknowledge Unverst Teknolog PETRONAS (UTP) for the fnancal support receved n the form of a research grant: Short Term Internal Research Fundng (STIRF) No 76/ and 35/ REFERENCES Hyman, J Accurate convexty preservng cubc nterpolaton Informal report, Los Alamos Scencetfc Laboratory, Los Alamos, NM, 98 Dougherty,R LD, Edelman, A and Hyman, J M Nonegatvty-, Monotoncty-, or Convexty-Preservng Cubc and Quntc Hermte Interpolaton Mathematcs of Computaton, Vol 5, No 86, , Brodle, KW and Butt, S Preservng convexty usng pecewse cubc nterpolaton Computer Graphcs 5: 5-3, 99 4 Gregory, JA Shape preservng nterpolaton Computer Aded Desgn 8():53-57, 986

13 Convexty preservng 99 5 Delbourgo, R Shape preservng nterpolaton to convex data by ratonal functons wth quadratc numerator and lnear denomnator IMA Journal of Numercal Analyss 9: 3-36, Delbourgo, R and Gregory, JA Shape preservng pecewse ratonal nterpolaton SIAM J Sc and Statst Comput 6: , Sarfraz, M Vsualzaton of postve and convex data by a ratonal cubc splne nterpolaton Informaton Scences 46:-4 pp 39-54, 8 Hussan, MZ and Hussan, M Vsualzaton of 3D data preservng convexty Journal of Appled Mathematcs and Computng, 3() 397-4, 7 9 Hussan, MZ, Hussan, M and Tahra, SS Shape preservng convex surface data vsualzaton usng ratonal b-quartc functon European Journal of Scentfc Research, (), 39-37, 8 Zhang, Y, Duan, Q, Twzell, EH: Convexty control of a bvarate ratonal nterpolatng splne surfaces Computers & Graphcs, , 7 Karm, SAA, Pah, ARM Ratonal Generalzed Ball Functons for Convex Interpolatng Curves JQMA 5(): 65-74, 9 Sarfraz, M, Hussan, MZ and Chaudary, FS Shape Preservng Cubc Splne for Data Vsualzaton Computer Graphcs and CAD/CAM pp 85-93, 5 3 Karm, SAA GC Monotoncty Preservng usng Cubc Ball Interpolaton Australan Journal of Basc and Appled Scences, 7(): (3) 4 Karm, SAA Postvty Preservng usng GC Cubc Ball nterpolant, 3 Submtted 5 Karm, SAA, Kong, PV and Hashm, I, Postvty Preservng usng GC Ratonal Quartc Splne, AIP Conf Proc 5, (3); 6 Schumaker, LL On shape preservng quadratc splne nterpolant SIAM J Numercal Analyss : , Lam, MH Monotone and convex quadratc splne nterpolaton Vrgna Journal of Scence Volume 4(): -3, 99 8 Kvasov, B I Methods of Shape-Preservng Splne Approxmaton World Scentfc, Sngapore,

14 Samsul Arffn Abdul Karm et al 9 Sarfraz, M Interactve curve modelng: wth applcatons to Computer Graphcs, Vson and Image Processng Sprnger-Verlag, London, 8 Delbourgo, R and Gregory, JA The Determnaton of Dervatve Parameters for a Monotonc Ratonal Quadratc Interpolant IMA Journal of Numercal Analyss, 5:397-46, 985 Sarfraz, M, Mulhem, MA and Ashraf, F Preservng Monotonc Shape of the Data Usng Pecewse Ratonal Cubc Functons Computer & Graphcs ():5-4, 997 Ball, AA CONSURF Part I: Introducton to Conc Lftng Ttle Computer Aded Desgn, 6(4):43-49, FN Frtsch and RE Carlson Monotone pecewse cubc nterpolaton, SIAM J Numer Anal 7: 38-46, 98 Receved: August 5, 3