Shape-Preserving Rational Bi-Cubic Spline for Monotone Surface Data

Transcription

1 WSEAS TRASACTIOS on ATHEATICS uammad Abbas Amad Abd ad od an H Aang Jamaludn d Al Sape-Preservng Ratonal B-Cubc Splne for onotone Surface Data UHAAD ABBAS a AHAD ABD AJID a OHD AI HJ AWAG b JAALUDI D ALI a a Scool of atematcal Scences b Scool of Dstance Educaton Unverst Sans alasa 8 US Penang ALAYSIA Emals: m.abbas@uos.edu.pk mad@cs.usm.m mnan@usm.m amaluma@cs.usm.m Abstract: - In ts paper e etended te ratonal cubc functon to ratonal b-cubc functon tat presents a smoot vsuall pleasant and nteractve ve of monotonct preservng surface. oreover t nvolves s free parameters n ts descrpton. Tese free parameters are arranged n suc a a ere to of tese are constraned to preserve te monotonct le te remanng oter four free parameters are left free to desgner for te refnement of monotone surface as desred. Te sceme under dscusson s C fleble smple local and economcal as compared to estng scemes. umercal eamples are provded to demonstrate tat te antcpated sceme s nteractve and smoot. Ke-Words: - Sape preservng nterpolaton Ratonal cubc functon Ratonal B-cubc functon onotone surface onotone surface data ree parameters. Introducton Splne nterpolaton plas a sgnfcant role n Computer Grapcs Computer Aded Geometrc Desgn Engneerng ont Desgnng Data Vsualzaton Sape preservaton Sape control and so man oters. Te data ave some specal sape caracterstcs e.g. monotonct postvt convet and of course t s often needed to generate a monotonct preservng nterpolatng curve and surface accordng to te gven monotone data. Te aspraton of ts paper s to preserve te eredtar attrbute of data tat s te monotonct. onotonct s a substantal sape caracterstc of data. an pscal stuatons est ere enttes are taken onl monotone for nstance monotonct s appled n te specfcaton of Dgtal to Analog Converters (DACs Analog to Dgtal Converters (ADCs and sensors [7]. Tese devces are used to control sstem applcatons ere onl monotonct s acceptable. Ertrocte sedmentaton rates (ESR n cancer patents urc acd level n gout patents appromaton of couples and quas couples n statstcs rate of dssemnaton of drug n blood emprcal opton of prcng models n fnance and te curves and surfaces of Doseresponse n bocemstr and parmacolog [4] are good eamples of monotone data. As ell as n Engneerng te stud of tensle strengt of te materal c gve te monotone data because te tensle strengt of a materal can be defned as te mamum force tat a materal can tstand before breakng Te forced appled usuall s called stress and s studed alongsde te stretc of te materal referred as stran []. Te problem of monotonct preservng nterpolaton as been consdered b a number of autors [-7] and references teren. Beatson and Zegler [] presented a vsualzaton of monotone data arranged over a rectangular grd b C monotone quadratc splne. ecessar and suffcent constrants on functonal and dervatve values ere derved. Carlson and rtsc [] upgraded ter results of unvarate monotone nterpolaton to bvarate monotone nterpolaton for regular data. Te nterpolatng functon as determned b frst partal dervatves and frst med partal dervatves (tst at mes ponts. Te derved necessar and suffcent condtons on tese dervatves. As a result on a sngle rectangular element b-cubc polnomal s monotone. E-ISS: Issue 7 Volume Jul

2 WSEAS TRASACTIOS on ATHEATICS uammad Abbas Amad Abd ad od an H Aang Jamaludn d Al Cascola and Roman [6] etended URBS (on-unform Ratonal B-Splne verson of te ratonal nterpolatng splne t tenson parameters for rectangular topolog case. Te sceme alloed te user to nteractvel modf te resultng surface b a set of tenson parameters. Costantn and ontanella [7] developed a metod for constructng sape preservng surfaces nterpolatng arbtrar sets of data on rectangular grd b usng monotone and or conve splnes avng degree n and order of contnut k.te nterpolatng splnes ere obtaned b usng Bernsten polnomals of sutable contnuous pecese lnear functons. Te presented ork s useful n developng algortms for te constructon of sape preservng splnes nterpolaton for arbtrar set of data ponts. loater and Peña [8] defned and stud te tree knds of monotonct preservaton of sstems of bvarate functons on a trangle b usng Bernsten polnomals t some geometrc applcatons. Hussan and ara [] developed te scemes t to sape parameters to vsualze te monotone data n te ve of monotone curves and surfaces. Te etended to partall blended ratonal b-cubc functon (Coons patces. Te scemes are local but unfortunatel dd not gve te fleblt to te user for te refnement of curves and surfaces as desred. Hussan et al [] etended te GPRC (General Pecese Ratonal Cubc functon to partall blended ratonal b-cubc functon (Coon patces t steen sape parameters (n eac patc. Te autors derved te constrants for te egt sape parameters to preserve te sape of monotone data le te remanng egt free parameters ere left to user for te refnement of surfaces. Te constraned parameters n te sceme depend on eac oter. Hussan et al [] etended te ratonal cubc functon to partall blended ratonal b-cubc functon t telve free parameters. Data dependent suffcent constrants ere derved for te four of tese parameters to preserve te monotonct le te remanng of tese free parameters ere left free for user s coce. Te sceme [ ] are epensve and tme consumng due to bunc of free parameters. Sarfraz et al [6] presented a sceme to vsualze te sape of monotone data b b-cubc functon. Te autors derved smple constrants on te free parameters to preserve te sape of te data. Te monotonct sceme does enable dervatve specfcaton but fals to mantan smootness of surface. In ts paper e ave presented an effcent monotone data vsualzaton sceme. Te C ratonal cubc functon t tree free parameters s etended nto ratonal b-cubc functon. Te ratonal b-cubc functon nvolves s free parameters n eac patc for ts representaton. Tese free parameters are arranged n suc a manner ere to of tese are constraned to preserve te sape of monotone data le te remanng free parameters are left free to desgner s coce for refne of surfaces as desred. Our sceme as a number of advantages over te estng scemes. Te developed sceme as been demonstrated troug dfferent numercal eamples and observed tat te sceme s not onl C local computatonall economcal eas to compute tme savng but also vsuall pleasant as compared to estng scemes [-]. In [-4] te scemes do not allo te user to refne te monotone surface as desred le for more pleasng surface (and stll avng te monotone sape preserved an addtonal modfcatons s requred ts task s more easl done n ts paper b a smple adustment of free parameters n te ratonal b-cubc nterpolaton on desgner s coce. Data dependent suffcent constrants for free parameters are derved to preserve te monotonct of monotone data. Ts sceme orks ell for bot equall and unequall space data. In [] te autor ere derved te smple data dependent constrants for free parameters but tese constrants parameters are depend on eac oter le n ts paper te sceme s computatonall tme savng due to ndependence of te constrants parameters. In [679] te autors developed te scemes to aceve te desred sape of data b nsertng etra knots beteen an to knots n te nterval le e preserve te sape of monotone data b onl mposng constrants on free parameters tout an etra knots. E-ISS: Issue 7 Volume Jul

3 WSEAS TRASACTIOS on ATHEATICS uammad Abbas Amad Abd ad od an H Aang Jamaludn d Al In ts paper tere s no need of necessar and suffcent condtons on functonal and dervatve values lke [] and []. Te remanng part of te paper s arranged as follos: Reve of ratonal cubc functon t tree sape parameters s dscussed n secton. Te ratonal b-cubc functon s gven n secton. Te coce of dervatve for te constructon of smoot monotone surface s dscussed n secton 4. onotonct preservng ratonal b cubc surface nterpolaton s dscussed n secton. nall numercal eamples and concluson are gven n sectons 6 and 7 respectvel. u f ( θ ( f ud θ( θ p ( θ ( f vd θ ( θ v f θ q ( θ u ( θ θ( θ vθ Remark.: or te values of free parameters set as: u v and ten te C pecese ratonal cubc functon (4 reduces to standard cubc Hermte splne.. Abbas et al [] developed te follong result for te monotonct-preservng of D monotone data. Reve of Ratonal Cubc Splne uncton Let{ ( f :... n} be te gven set of data ponts suc as < < <... <. Te ratonal cubc n functon t tree free parameters [] n eac I... n s defned as: subnterval [ ] ( θ θβ S ( ( q ( θ ere θ ( and u v are te postve free parameters. Te follong nterpolator condtons are mposed for te C contnut of te pecese ratonal cubc functon ( S ( ( f S f ( S ( d S ( d ere S ( denotes te dervatve t respect to and d denotes te dervatves estmated at knots. Te C contnut condtons defned n equaton ( clam te follong values of unknon β β uf β f ud ( β f vd β vf Usng equaton ( te pecese ratonal cubc functon ( reformulated as: p ( ( θ S (4 q ( θ Teorem. [. Abbas et al []] Te C pecese ratonal cubc functon (4 preserves te monotonct of monotone data f n eac subnterval I [ ]... n te free parameters satsf te follong suffcent condtons ud vd ud vd > ma u > v >. Te above result can be rertten as: ud vd ud vd l ma l > u > and v >. ere ( f f Ratonal B-Cubc Splne uncton In ts secton e etended a C pecese ratonal cubc functon (4 to ratonal b-cubc S over te rectangular functon ( doman [ ab ] [ cd ] ntervals [ ab ] and [ ] Ω. Te partton of arbtrar cd can be defned as π : a < < <... < n b π : c < < <... < m d respectvel. Te ratonal b-cubc functon over eac rectangular patc [ ]... n... mdefned as: t S ( S ( B( θ B ( ( φ ( ere E-ISS: Issue 7 Volume Jul

4 WSEAS TRASACTIOS on ATHEATICS uammad Abbas Amad Abd ad od an H Aang Jamaludn d Al ( [ ] (6 B ( θ b ( θ b ( θ b ( θ b ( θ (7 B ( φ b垐 ( φ b ( φ b垐 ( φ b ( φ (8 t ( θ u θ( θ b ( θ q ( θ θ v θ ( θ b ( θ q ( θ θ( θ u b ( θ q ( θ θ ( θ v b ( θ q ( θ ( φ u垐 φ( φ b ( φ q ( φ φ v垐 φ ( φ b ( φ q ( φ φ( φ u b ( φ q ( φ b ( φ q ( φ φ ( φ v q( θ u ( θ θ( θ v θ q ( 垐? φ u ( φ φ( φ v φ ere (. θ ( and ϕ Substtutng te equatons (6-(8 n ( ten te ratonal b-cubc functon ( defned as: ( θ P θ( θ Q θ ( θ R θ T S ( ere P t k ( φ ( θ u θ( θ θ v k q ( φ k φ A k (9 ( A u u A u ( 垐 u A u 垐 ( v A u v k k ( φ φ Bk k Q q ( φ suc tat B u ( u ( B ( u u ( u ( B u v ( u B v ( u k k ( φ φ Ck k R ( q ( φ t C u ( v C ( v u ( v C ( v ( v v C v ( v T k ( φ k q ( φ k φ D ere D u v D 垐 v ( u D v 垐 ( v D v v k 4 Determnaton of dervatves ( Usuall te dervatves and and are not knon so must be calculated eter from te gven data or b some oter sources. Let us denote and as te frst order dervatves t E-ISS: Issue 7 Volume Jul

5 WSEAS TRASACTIOS on ATHEATICS uammad Abbas Amad Abd ad od an H Aang Jamaludn d Al respect to and respectvel at te data pont. Smlarl let te med dervatves be denoted b. Artmetc mean metod as proposed n [] s te tree-pont dfference appromaton based on artmetc calculaton for te monotone curve manpulaton. Ts metod can be orented and etended for te D data vsualzaton as follos: 4. Artmetc ean etod for D data ( ( ( n n n n n ( n n (... n... m 垐 ( ( 垐 垐 ( m m m m m ( 垐 m m ( 垐... n... m 垐... n ;... m ere. Tese artmetc mean metods are computatonall economcal and sutable for vsualzaton of data. onotonct-preservng Ratonal B-Cubc Splne Interpolaton. Te ratonal b-cubc functon ( does not preserve te sape of monotone surface data. So t s requred to assgn sutable constrants for te free parameters b some matematcal treatment to preserve te monotonct of monotone surface data. Teorem.: Te ratonal b-cubc functon ( preserves te monotonct of monotone surface te free parameters satsf te follong suffcent condtons as: u u > and v v > data f n eac rectangular patc [ ] > ma 垐垐 > ma 垐垐 Te above results are rertten as: l ma m ma l > Proof: ( :... n... m Let { } be a monotone surface data arranged over rectangular doman Ω [ n] [ m]. Te necessar condtons for monotonct of monotone surface data are: < > > (4 enever ( < ( m > E-ISS: Issue 7 Volume Jul

6 WSEAS TRASACTIOS on ATHEATICS uammad Abbas Amad Abd ad od an H Aang Jamaludn d Al < > > ( enever ( < ( and te free parameters are: u > v > > (6 u垐? > v > > Te ratonal b-cubc functon Sdefned ( n ( preserves te monotonct of monotone surface data f te follong condtons are satsfed as S ( > ( Ω (7 S ( > Te functons S ( and S( represent te frst order partal dervatons of ratonal b-cubc functon (. r. t. and respectvel ( θ θ S ( ( ( ( ϕ ere q θ q ( ϕ ϕ t u u u ( u ( u v ( ϕ ϕ t u u. {( } (8 垐 ( ( u ( 垐 ( ( u ( u v. {( } ( ϕ ϕ t ( u ( u u v 垐 ( (. u ( u v u v ( ( ( (. v? ( u v u v ( ( ( ( u v u v ( ϕ ϕ t ( v ( u u v 垐 ( (. u ( u v u v ( ( E-ISS: Issue 7 Volume Jul

7 WSEAS TRASACTIOS on ATHEATICS uammad Abbas Amad Abd ad od an H Aang Jamaludn d Al 垐 ( (. v ( u v u v ( ( ( ( v v u v 4 ( ϕ ϕ 4 t {( } v u. 4 ( ( ( ( ( 4 v ( 4 v {( } v v. 4 ( ϕ ϕ ere v u v ( v ( v u S ( q ere ( θ θ ( q ϕ ( θ ( ( ϕ ϕ t u u ( { } u u垐 (9 ( ( u u u v ( v ( u u v u v ( 4 u v ( ϕ ϕ t ( ( ( u ( u ( (. u ( u v ( ( ( (. v ( u v ( ( ( ( 4 v ( v ( ϕ ϕ t ( u E-ISS: Issue 7 Volume Jul

8 WSEAS TRASACTIOS on ATHEATICS uammad Abbas Amad Abd ad od an H Aang Jamaludn d Al ( ( ( u ( (. v ( u v ( ( ( (. v ( u v ( ( ( v ( 4 ( v ( ϕ ϕ t v u {( } ( ( v u u v u v ( v ( v u v v v {( } 4 v u rstl from equaton ( te ratonal functon S ( > f ( θ θ > We ave satsfed and and ( q ( θ q ( ϕ >. ( q ( θ q ( ϕ > f equaton (6 s ( θ θ > f >. > f We ave > and > f equatons (4 and ( are satsfed. > f > f Hence > f > > > ma > f > ma > f > ma > ma > f > ma > ma 4 > f > ma > ma > ma ( ( ( ( (4 ( (6 (7 (8 E-ISS: Issue 7 Volume Jul

9 WSEAS TRASACTIOS on ATHEATICS uammad Abbas Amad Abd ad od an H Aang Jamaludn d Al Lastl > f > ma (9 After reducng te smlar terms n above epressons of and from equatons (-(9 e ave > ma ( > ma ( Secondl from equaton (7 te second ratonal functon S ( > f Bot ( θ θ > and q ( q We ave ( q ( θ ( ϕ >. ( θ q ( ϕ > f equaton (6 s satsfed. ( θ θ > f >. > f >... We ave > and > f equatons (4 and ( are satsfed. > f > > and > f > ma 4 > f > Hence > f > ma ( > f > ma ( > f > ma (4 Lastl > f > ma ( Combnng te above epressons of and from equatons (-( e get > ma (6 > ma (7 Jonng te equatons ( and (6 e ave > ma (8 Te above result can be rertten as: l ma l > Combnng te equatons ( and (7 e ave Rertng te above result as: > ma (9 E-ISS: Issue 7 Volume Jul

10 WSEAS TRASACTIOS on ATHEATICS uammad Abbas Amad Abd ad od an H Aang Jamaludn d Al m ma m > Table.: D monotone data Table. : onotone surface data E-ISS: Issue 7 Volume Jul

11 WSEAS TRASACTIOS on ATHEATICS uammad Abbas Amad Abd ad od an H Aang Jamaludn d Al g.: B-cubc Hermte Splne g.4: Sape-preservng ratonal b-cubc surface nterpolaton g.: z ve of gure. g.: z ve of gure.4 g.: Dfferent ve of gure. g.6: Dfferent ve of gure.4 E-ISS: Issue 7 Volume Jul

12 WSEAS TRASACTIOS on ATHEATICS uammad Abbas Amad Abd ad od an H Aang Jamaludn d Al g.7: B-cubc Hermte Splne g.: Sape-preservng ratonal b-cubc surface nterpolaton t u 垐. u. v. v. g.8: z ve of gure.7 g.: z ve of gure. g.9: Dfferent ve of gure.7 g.: Dfferent ve of gure. E-ISS: Issue 7 Volume Jul

13 WSEAS TRASACTIOS on ATHEATICS uammad Abbas Amad Abd ad od an H Aang Jamaludn d Al 6. umercal Eamples In ts secton a numercal demonstraton of monotonct preservng sceme gven n secton s presented. Eample. A monotone surface data set taken n Table. gure. s dran b b-cubc Hermte splne tat does not preserve te monotonct of monotone surface data. gure. and gure. are representng dfferent ve of gure.; e remark tat tese fgures do not preserve te sape of data. To overcome ts fla gure.4 s generated b te monotonct preservng ratonal b-cubc surface sceme developed n secton t te values of free parameters set as: u. u. v. and v. to preserve te monotonct of monotone data. gure. and gure.6 represent dfferent ve of gure.4; t s clearl son tat tese fgures preserve monotonct everere. A promnent dfference n te smootness t a vsuall pleasant ve can be seen n tese fgures due to te lbert bestoed to te desgner on te values of sape parameters. Eample. Te b-cubc Hermte splne sceme as been used to dra te gure.7 usng te monotone surface data set taken n Table. A more clear ve of nonmonotonct can be seen n gure.8 and gure.9 c are z and dfferent ve of gure.7 respectvel. On te oter and te effcenc of te monotonct preservng ratonal b-cubc sceme developed n secton can be seen n gure.. A more compreensble ve of monotonct can be seen n gure. and gure. c are z and dfferent ve of gure. respectvel. A remarkable dfference n te smootness t a sape preservng pleasant grapcal ve s vsble n te fgures dran b te nel developed sceme due to te freedom granted to te desgner on te values of sape parameters. 7 Concludng Remark In ts paper e ave etended a C pecese ratonal cubc functon to ratonal b-cubc functon t s free parameters n eac rectangular patc to preserve te monotonct of D monotone data. Te free parameters are arranged n suc a a tat to of tem are constraned parameters (not depend on te oter parameter lke [] to preserve te sape of D monotone data le te remanng are left free for desgner 's coce for te refnement of monotone surface as desred. Te developed surface sceme as been tested troug dfferent numercal eamples and t s son tat te sceme s not onl local and computatonall economcal but also vsuall pleasant. Acknoledgments Te autors are gl ndebted to anonmous referees for te nsprng comments and te constructve suggestons c ave led to an mproved manuscrpt. References: []. Abbas A. A. ad.. H Aang J. d. Al onotonct Preservng Interpolaton usng Ratonal Splne Internatonal multconference of Engneers and computer scentsts vol ( pp [] R. K. Beatson Z. Zegler onotonct preservng surface nterpolaton SIA Journal of umercal Analss ( (98 pp [] S. Butt Sape preservng curves and surfaces for Computer Grapcs P.D. Tess Scool of Computer Studes Te Unverst of Leeds UK (99. [4] G. Belakov onotonct preservng appromaton of multvarate scattered data BIT umercal atematcs 4(4 ( pp [] R. E. Carlson.. rtsc onotone pecese bcubc nterpolaton SIA Journal of umercal Analss ( (98 pp [6] G. Cascola L. Roman Ratonal nterpolants t tenson parameters Proceedngs of Curve and Surface Desgn T. Lce. azure and L. L. Scumaker(eds. Sant-alo asboro Press Brentood T. ( pp. 4-. [7] P. Costantn. ontanella Sape preservng bvarate nterpolaton SIA Journal of umercal Analss 7( (99 pp E-ISS: Issue 7 Volume Jul

14 WSEAS TRASACTIOS on ATHEATICS uammad Abbas Amad Abd ad od an H Aang Jamaludn d Al [8]. S. loater J.. Peña onotonct preservaton on trangles atematcs of Computaton 69( ( pp [9] LU. Han L. L. Scumaker ttng monotone surfaces to scattered data usng C pecese cubcs SIA Journal of umercal Analss 4( (997 pp []. Z. Hussan. Hussan Vsualzaton of data preservng monotonct Appled atematcs and Computaton 9 (7 pp []. Z. Hussan. Hussan. Sarfraz Vsualzaton of onotone data b Ratonal B-cubc Interpolaton Trans. on Comput. Sc. VIII LCS 66 ( pp []. Hussan Data Vsualzaton usng splne functons P.D. Tess Department of atematcs Unverst of te Punab Laore-Pakstan ( []. Z. Hussan. Sarfraz onotone pecese ratonal cubc nterpolaton Internatonal Journal of Computer atematcs 86( (9 pp [4]. Z. Hussan. Hussan onotonc surfaces for Computer Grapcs Journal of prme researc n atematcs Vol. (6 pp []. Z. Hussan D. Zad. Hussan A. A. ad onotone Surface Data Vsualzaton European Journal of Scentfc Researc 8( (9 pp [6]. Sarfraz S.Butt.Z. Hussan Surfaces for te vsualzaton of scentfc data preservng monotonct IA atematcs for Surfaces VII Conference September - (997 pp [7] B. S. Kolts Understandng lneart and monotonct Analog Zone (7 ttp://.analogzone.com/nett8.pdf E-ISS: Issue 7 Volume Jul