Shape Preserving Positive and Convex Data Visualization using Rational Bi-cubic Functions

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1 Shape Preservng Postve and Conve Data Vsualzaton usng Ratonal B-cubc unctons Muhammad Sarfraz Malk Zawwar Hussan 3 Tahra Sumbal Shakh Department of Informaton Scence Adala Campus Kuwat Unverst Kuwat E-mal: prof.m.sarfraz@gmal.com 3 Department of Mathematcs Unverst of the Punab Lahore Pakstan E-mal: malkzawwar.math@pu.edu.pk Abstract Ths paper s concerned wth the problem of postve and conve data vsualzaton n the form of postve and conve surfaces. A ratonal b-cubc partall blended functon wth eght free parameters n ts descrpton s ntroduced and appled to vsualze the shape of postve data and conve data. The developed schemes n ths paper have unque representatons. Vsual models of surfaces attan C smoothness. Kewords: Data vsualzaton; ratonal cubc functon; ratonal b-cubc partall blended functon; postve surface; conve surface.. Introducton Splnes are basc tools for the vsualzaton of the shaped data. Although ordnar splnes smoothl nterpolate the gven data ponts but do not fulfll the hdden shape propertes of the data. Dealng wth dfferent shapes lke monoton postvt. or convet data vsualzaton has sgnfcant role for varous real lfe applcatons. Ths paper specfcall concentrates on the vsualzaton of postve and conve data n the form of surfaces. There are man felds where the enttes onl have meanng when ther values are postve. or nstance n probablt dstrbutons the representaton of data s alwas postve. Smlarl when we deal wth the samples of populatons the data are alwas n postve fgures. Convet s also an mportant shape feature and plas a sgnfcant role n varous dscplnes and applcatons. or eample nonlnear programmng n engneerng scentfc applcatons such as desgn optmal control parameter estmaton and appromaton of functons are few of them to menton. One can observe that n gure b-cubc Hermte functon nterpolates the postve data but nherent shape feature.e. postvt s mssng n the resultant surface representaton. Smlarl n gure 7 b-cubc Hermte functon nterpolates the conve data but nherent shape feature.e. convet s lost n the ultmate surface representaton. Ths motvates to come up wth the schemes whch can preserve the nherent features of the data.

2 In recent ears researchers [-6] have publshed a sgnfcant number of artcles n the feld of shape perseverng data vsualzaton. Asm and Brodle [] dscussed the problem of drawng postve curve through postve data. A pecewse cubc Hermte nterpolaton s used to ft a postve curve. In an nterval where the postvt s lost authors added etra knots to cubc Hermte nterpolant to obtan desred postve curve. Brodle et al [] adopted modfed quadratc Shepard MQS method for the nterpolaton of scattered data of an dmenson. The restrcted the range of nterpolatng functon between and. Postvt of postve scattered data s acheved b MQS method but the graphcal results are not vsuall pleasant. Brodle et al [3] presented an algorthm for the vsualzaton of postve data. The derved suffcent condtons n terms of frst partal dervatves and med partal dervatves at the grd ponts. The also generalzed the case of lnearl constraned nterpolaton. Cascola and Roman [4] constructed NURBS wth tenson parameters to control the shape of nterpolator surfaces. The presented some technques to reconstruct the shape preservng bvarate NURBS n whch shapes of resultng surfaces can be modfed b changng the values of tenson parameters. Duan et al [5] constructed a bvarate ratonal nterpolatng functon based on the functon values and partal dervatves. The attached s parameters n the descrpton of bvarate nterpolatng functon to keep the nterpolatng surface n the orgnal shape. angum et al [6] presented an eplct epresson of a weghted blendng nterpolator based on the functon values. In [6] postve parameters and weght coeffcents are freel selected accordng to the needs of practcal desgn and the nterpolator s C n the whole nterpolatng regon. usawa et al [7] brefl descrbed the mplementaton and vsualzaton through some eamples of applcatons to scentfc arts such as archeolog sculpture fne arts and nformaton aesthetcs. Hussan et al [8 9] dscussed vsualzaton of 3D conve data. In [8] the used ratonal b-cubc functon whle n [9] the used ratonal b-quartc functon to vsualze conve data. The mposed condtons on free parameters n the descrpton of ratonal functons to vsualze conve data. Kouba and Pasadas [] presented an appromaton problem of parametrc curves and surfaces from the Lagrange or Hermte data set. The also dscussed the nterpolaton problem of mnmzng some functonal on a Sobolev space that produced the new noton of nterpolatng varatonal splne. Pah et al [] presented two algorthms for the postvt preservaton of scattered data. The nterpolatng surface comprses cubc Bezer trangular patches wth suffcent condtons mposed on the ordnates of the Bazer control ponts n each trangle to guarantee preservaton of postvt. Renka [3] dscussed the constructon of C conve surface nterpolaton. or ths purpose Renka ntroduced a ortran-77 software package. Sarfraz et al [4] constructed a ratonal cubc functon wth two parameters n ts representaton and etended the ratonal cubc functon to ratonal b-cubc partall blended functon. The vsualzed the shape of postve data n the form of postve curves and surfaces b ntroducng constrants on parameters. Wang and Tan [5] generated a ratonal b-quartc surface b usng tensor product method. The developed an algorthm to preserve the shape of monotone data b pecewse ratonal b-quartc functon. Zhang et al [6] dealt out the problem of convet control of nterpolatng bvarate surfaces. Suffcent and necessar condtons are derved to obtan conve surface through conve data based on the functon values.

3 In ths paper a ratonal b-cubc functon s used to vsualze the shaped data. Two schemes are developed; one for the surface vsualzaton of postve data and the other for the surface vsualzaton of conve data. Important features of ths paper are as follows: Developed schemes are desgned n such a wa that no addtonal knots are necessar to control the shape as n []. There s no lmtaton on the nterpolatng functon for lng n a specfed nterval as n []. In ths paper constrants are derved on free parameters whch provde more control to the user to control the shape of nterpolatng surface as compared to the scheme n [3]. Developed schemes are equall applcable for the data wth dervatve or wthout dervatves whle schemes n [4 5] developed the work f partal dervatves at the knots are known. In [9] ratonal b-quartc functon s used to preserve the shape of conve surface data whle n ths paper we used ratonal b-cubc functon to vsualze the conve surface data. In [4] authors clamed that postve surfaces generated b ther schemes are C contnuous and smooth. Smlarl the surfaces generated b the proposed schemes developed n ths paper are also C contnuous. Developed schemes are useful for both equall spaced as well as unequall spaced data whle n [5] authors derved results for equall spaced data. Generated surfaces are unque n ther representaton. Ths paper s organzed as follows. Secton s a revew of curve nterpolant of []. Secton 3 deals wth the pecewse ratonal b-cubc partall blended functon. Sectons 4 and 5 deal wth the tasks of vsualzaton for postve and conve data respectvel. Secton 6 concludes the paper.. Ratonal Cubc uncton { } Let f 3 n nterval [ ] the ratonal cubc functon K be gven set of data ponts where < < K < n. In each S s defned as: where U f S S 3 3 U W T V θ θ θ θ θ θ θ θ θ θ W hd f T hd f V f

4 and θ h 3 K n. h The ratonal cubc functon has followng propertes: S f S f S d S d where S denote the frst order dervatve wth respect to and d denotes dervatve value at the knot. Remark: It s observed that when the ratonal cubc functon reduces to the S C. standard cubc Hermte polnomal [ ] ollowng Theorems and follow from []: n Theorem : The ratonal cubc functon s postve n each nterval [ ] f the shape parameters satsf the followng constrants: hd f κ f κ ma f hd f. Theorem : The ratonal cubc functon s conve n each nterval [ ] f the shape parameters satsf the followng constrants: 3. Ratonal B-cubc uncton d d d d d δ δ ma A pecewse ratonal cubc functon s etended to ratonal b-cubc partall blended functon S over the rectangular regon [ ab ] [ cd ] partton of [ ab ] and % π : c < <... < m d be the partton of [ ] partall blended functon s defned over each rectangular patch [ ]... n ;... m as: Ω. Let π : a < <... < n b be cd the ratonal b-cubc.

5 where S T PQ 3 S S S S S S S S θ θ Q q φ q φ P p p wth p θ θ p 3 θ 3 θ q φ φ q φ 3 φ h h h h θ φ. S θ S S and S boundar of rectangular patch [ ] φ. are ratonal cubc functon defned over the as: wth A S A h A h A A θ q q θ θ θ θ θ wth B S B h 4 B h B 3 4 B q θ θ 4 θ θ 4 q θ θ θ θ θ θ 4 5

6 wth C S C h C h C C φ q φ q φ φ φ φ φ wth D S D h 4 D h D 3 4 D q φ 4 φ φ 4 q φ φ φ φ φ. φ Choce of Dervatves or most of the applcatons the dervatve parameters are unknown. There are man methods for the appromaton of these dervatve parameters. In ths artcle the are calculated b Arthmetc Mean Method. The descrpton of ths method s as follows. Let and denote the frst dervatve wth respect to and respectvel and be the med dervatve at the data pont. m m h h h m m m h h m m... ;... n n h m n n n n h h n n h

7 ... m;... n h... m ;... n h h h. h h 4. Postve Ratonal B-cubc uncton { :... n ;... m} Let rectangular grd [ ] be the set of postve data ponts defned over I... n ;... m. Due to Cascola and Roman [4] b-cubc partall blended surface patch nherts all the propertes of network of boundar curves. Therefore b-cubc partall blended surface patch defned n 3 wll be I f each of the boundar curves postve n each rectangular patch [ ] S S and S S S f S f S f S f s postve. Now the curves h h All the above dscusson can be summarzed as: and h and h and and h h h h. Theorem 3: The ratonal b-cubc partall blended functon defned n 3 vsualzes postve I f surface data n the vew of postve surface n each rectangular patch [ ] the shape parameters and satsf the followng condtons:

8 h δ ma h h η ma h h δ ma h h η ma h where δ η δ η. The above dscusson can be summarzed n the followng algorthm for computaton purposes: Algorthm Step. Enter the n m postve data ponts... n;... m. Step. Estmate the dervatves and at knots. Step 3. Calculate the values of shape parameters and usng Theorem 3. Step 4. Substtute the values of... n;... m and and... n ;... m n ratonal bcubc functon 3 to obtan postve ratonal b-cubc functon..5 cos5.4 Table. A data set generated b

9 z-as. z-as as -as gure. B-cubc Hermte uncton 3 3 -as 3 -as gure 4. Postve ratonal b-cubc Surface z-as.5. z-as as as gure. z-vew of gure. gure 5. z-vew of gure z-as.5. z-as as as gure 3. z-vew of gure gure 6. z-vew of gure 4.

10 4.. Demonstraton: A postve data set s taken n Table whch s generated b the.5 cos5.4 functon. 6 3 correct to four decmal places defned over the rectangular grd [ 3] [ 3]. gure s generated usng b-cubc Hermte functon whch nterpolates the data ponts but does not vsualze the shape of the data. gures and 3 are the z-vew and z-vew of gure respectvel. gure 4 s generated b the postve ratonal bcubc functon developed n Secton 4 wth.5. gures 5 and 6 are the z-vew and z-vew of gure 4 respectvel. One can clearl observe that that gure 4 preserves the shape of data taken n Table. The numercal results correspondng to the surface n gure 4 for dfferent parameters are shown n Table. All the values n Table are truncated up to four decmal places. Table. Numercal results of gure Conve Ratonal B-cubc uncton { :... n ;... m} Let over rectangular grd [ ] be the collecton of conve data ponts defned I... n ;... m. The data wll be conve f t satsfes the followng necessar condtons:

11 < < < < < < < < < < where h. h Due to Cascola and Roman [4] b-cubc partall blended surface patch nherts all the propertes of network of boundar curves. Therefore the b-cubc partall blended surface patch I f each of the defned n 3 wll be conve n each rectangular patch [ ] S S and S boundar curves S s conve.e. S S S and S. The second ordered dervatves of the boundares curves S S are as follows: where S 6 T 6 θ θ 3 q θ { } { } { } { } { } T h T 5 h T 7 3 h 3 T 3 7 h 4 T 5 h 5 { } T h. 6 Smlarl S S and 8 S f T s and q θ '...6 q θ f and T s f '

12 where S 6 and 6 θ θ q θ 3 S { } { } { } { } { } { } S h S 5 h S 7 3 h 3 S 3 7 h 4 S 5 h 5 S h 6 Smlarl 9 S f S s and q θ '...6 q θ f and S s f ' and where S 6 U φ φ q φ { } { } { } { } U h U 5 h U 7 3 h 3 U 3 7 h 4 3 6

13 { } U 5 h 5 { } U h 6 Smlarl ' S f U s...6 and q φ ' q φ f and U s f and where 6 W φ 3 S φ q φ { } { } { } { } W h W 5 h { } W 7 3 h 3 W 3 7 h 4 W 5 h 5 6 { } W h 6 ' S f W s...6 and q φ ' q φ f and W s f and. All the above dscusson can be summarzed n the form of followng theorem:

14 Theorem 4: The ratonal b-cubc partall blended functon defned n 3 preserves the shape of I f conve data n the vew of conve surface n each rectangular patch [ ] the shape parameters and satsf the followng condtons: α ma β ma α ma β ma where α β α β. The above dscusson can be summarzed n the followng algorthm for computaton purposes: Algorthm : Step. Enter the n m conve data ponts... n;... m. Step. Estmate the dervatves and at knots. Step 3. Calculate the values of shape parameters and usng Theorem 4. Step 4. Substtute the values of... n;... m and... n ;... m n ratonal b-cubc functon 3 to obtan the conve ratonal b-cubc functon. Table 3. A data set generated b

15 5 5 z-as 5 z-as as - - -as gure 7. B-cubc Hermte uncton -as as gure. Conve ratonal b-cubc Surface. 5 5 z-as z-as as as gure 8. z-vew of gure 7. gure. z-vew of gure. 5 5 z-as z-as as as gure 9. z-vew of gure 7. gure. z-vew of gure.

16 5.. Demonstraton: 4 A conve data set s taken n Table 3 whch s generated b the functon defned over the rectangular grd [ ] [ ]. gure 7 s generated b usng b-cubc Hermte functon whch nterpolates the data ponts but does not preserve the shape of conve data. gures 8 and 9 are the z-vew and z-vew of gure 7 respectvel. gure s generated b the conve ratonal b-cubc functon developed n Secton 5 wth and. gures and are the z-vew and z-vew of gure respectvel. It can be observed that gure preserved the shape of data taken n Table 3. Table 4. Numercal results of gure The numercal results correspondng to the surface n gure for dfferent parameters are shown n Table 4. All the values n Table 4 are truncated up to four decmal places.

17 6. Concluson A ratonal b-cubc partall blended functon b-cubc/b-quadratc s ntroduced to preserve the shape of postve and conve data. Eght free parameters are attached n the descrpton of ratonal b-cubc partall blended functon. Smple data dependent constrants are derved on free parameters to preserve the shape of postve and conve data n the vew of postve and conve surfaces respectvel. Developed schemes work for both equall and unequall spaced data. In the developed schemes there s no constrant on dervatves the are equall useful for the data wth or wthout dervatves. Developed schemes are local andc. References [] M. R. Asm & K. W. Brodle Curve drawng subect to postvt and more general constrants Computers and Graphcs [] K. W. Brodle M. R. Asm & K. Unsworth Constraned vsualzaton usng the Shepard nterpolaton faml Computer Graphcs orum [3] K. Brodle P. Mashwama & S. Butt Vsualzaton of surface data to preserve postvt and other smple constrants Computers and Graphcs [4] G. Cascola & L. Roman Ratonal nterpolants wth tenson parameters Curves and Surface Desgn Tom Lche Mare-Laurence Mazure and Larr L. Schumakereds [5] Q. Duan Y. Zhang & E. H. Twzell A bvarate ratonal nterpolaton and the propertes Appled Mathematcs and Computaton [6] B. angum S. Qnghua & Q. Duan A bvarate blendng nterpolator based on functon values and ts applcatons Journal of Computatonal Informaton Scence [7] N. usawa K. Brown Y. Nakaama J. Hatt & T. Corb Vsualzaton of scentfc arts and some eamples of applcatons Journal of Vsualzaton [8] M. Z. Hussan & M. Hussan Vsualzaton of 3D data preservng convet Journal of Appled Mathematcs and Computng [9] M. Z. Hussan M. Hussan & S. S. Tahra Shape Preservng Conve Surface Data Vsualzaton usng ratonal b-quartc functon European Journal of Scentfc Research [] M. Z. Hussan M. Sarfraz & S. S. Tahra Shape Preservng Ratonal Cubc Splne for Postve and Conve Data accepted n Egptan Informatcs Journal. [] A. Kouba & M. Pasadas Appromaton b nterpolatng varatonal splnes Journal of Computatonal and Appled Mathematcs [] A. R. M. Pah T. N. T. Goodman & K. Unsworth Postvt preservng scattered data nterpolaton n: Proceedngs of the th IMA Mathematcs of Surfaces Conference Loughborough UK September [3] R. J. Renka Interpolaton of scattered data wth a C convet preservng surface ACM Transactons on Mathematcal Software [4] M. Sarfraz M. Z. Hussan & A. Nsar Postve data modelng usng splne functon Appled Mathematcs and Computaton [5] Wang & J. Tan Shape preservng pecewse ratonal bquartc surfaces Journal of Informaton and Computatonal Scences [6] Y. Zhang Q. Duan & E. H. Twzell Convet control of a bvarate ratonal nterpolatng splne surfaces Computers and Graphcs

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