Visualization of Data Subject to Positive Constraints

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1 ISSN England UK Journal of Informaton and Computng Scence Vol. No. 6 pp Vsualzaton of Data Subect to Postve Constrants Malk Zawwar Hussan and Mara Hussan Department of Mathematcs Unverst of the Punab Lahore- Pakstan. (Receved March 6 Accepted June 4 6) Abstract. In ths paper frst free parameters are constraned n the descrpton of ratonal cubc functon [6] to preserve the shape of data that les above the straght lne. Then ratonal cubc functon s etended to ratonal bcubc partall blended functon (Coons patches). A local postvt preservng scheme s developed for postve data b makng constrants on free parameters n the descrpton of ratonal bcubc partall blended patches. We also develop at the end the constrants for vsualzng a data that le above the plane. Kewords: Vsualzaton Ratonal Functon Interpolaton Postve Surfaces Free Parameters.. Introducton Scentfc vsualzaton s the representaton of data graphcall for ganng understandng and nsght nto the data. Sometmes t s also referred to vsual data analss. Vsualzaton nvolves research n computer graphcs mage processng hgh performance computng meteorologcal montorng maps data plots drawng and man other areas. It enrches the process of scentfc dscover and fosters profound and unepected nsghts but ts recent emphaszed s on computer graphcs and vsualzaton n scentfc computng. In Computer Graphcs envronment a user s usuall n need of an nterpolatng scheme whch possesses certan characterstcs lke shape preservaton shape control to vsualze the data n a pleasant wa. The propertes those are used to quantf shape are postvt monotonct and convet. Problem of postvt s descrbed as: f entres n the sample data are postve then nterpolatng curve and surface s postve. Preservng postvt s partcularl mportant n vsualzng enttes that cannot be negatve e.g. amount of ranfall populaton volume area denst and concentraton of sugar n blood. Some nterest has been shown n ths area n [-4] and references there n. Brodle Mashwama and Butt n [] preserved the postvt of D postve data b the rearrangement of data. The nserted one or more knots where requred to preserve the shape of data. Pah Goodman and Unsworth n [] have dscussed the problem of postvt preservng scattered data nterpolaton. Nadler n [9] also dscussed the problem of non-negatve nterpolaton. The have consdered non-negatve data arranged over a trangular mesh and have nterpolated each trangular patch usng a bvarate quadratc functon. Zawwar n [5] preserved the shape of D data b ratonal bcubc tensor product surface. In secton Ratonal cubc functon [6] used n ths paper s descrbed. In secton a scheme s developed to preserve the shape of data that s lng above the straght lne. In secton 4 ratonal cubc functon [6] s etended to ratonal bcubc partall blended patches (Coons Patches). The advantageous feature of these partall blended patches s that the nhert all the propertes of the network of boundar curves [4]. In secton 5 dervatve appromaton scheme s ntroduced. In secton 6 a scheme s developed to preserve the shape of postve data b makng constrants on free parameters n the descrpton of ratonal bcubc partall blended patches. In secton 7 constrants for vsualzng a data that le above the plane are developed. The secton 8 dscusses and demonstrates the schemes developed n secton 6 and 7. Secton 9 concludes the paper.. Ratonal Cubc Functon In ths secton we ntroduce the pecewse ratonal cubc functon [6] used n ths paper. Let ( f ) E-mal address: malkzawwar@math.pu.edu.pk Publshed b World Academc Press World Academc Unon

2 5 M. Z. Hussan M. Hussan: Vsualzaton of Data Subect to Postve Constrants =... n be gven set of data ponts where < < < n. Pecewse ratonal cubc functon s defned over each nterval I = [ ] as: p ( θ ) S ( ) = q ( θ ) () where p ( θ ) = v f ( θ) [( u v v ) f v hd ]( θ) θ [( u v u ) f u hd ]( θ) θ uf θ q ( θ) = v ( θ) u v ( θ) θ uθ ( ) h = θ =. h The ratonal cubc functon () has the followng propertes: S ( ) = f S ( ) = f S ( ) = d S ( ) = d. () () S () ( ) denotes the dervatve wth respect to and denotes dervatve values (gven or estmated b d v () some method) at knot. S( ) C [ n ] has u and as free parameters n the nterval [ ]. We note that n each nterval I when we take u = and v = the pecewse ratonal cubc functon reduces to standard Cubc Hermte. Zawwar and Jamaludn n [6] have developed the followng result: Theorem.. The pecewse ratonal cubc functon () preserves postvt f free parameters satsf the followng condton n each nterval [ ] hd u > Ma. f. Vsualzaton of D Constraned Data hd v > Ma. f In ths secton we consder data lng above a straght lne and constrants are developed on free parameters for vsualzng ths data. Let ( f ) =... n be gven data ponts lng above the straght lne = m c.e. f m c =... n. For each subnterval [ ] above relaton can be epressed as: The curve wll le above the straght lne f ratonal cubc functon () satsfes the condton: u and S ( ) > m c [ n ]. () p ( θ ) S( ) = > m c. () q ( θ ) The equaton of straght lne n parameter θ s defned as: a ( θ ) bθ where a = m c and b = m c. The parametrc form of equaton () n terms of parameter θ s: p ( θ ) a( θ) bθ θ []... n. q ( θ ) > = (4) q ( θ ) > f u > and v >. Multpl both sdes of (4) b q ( θ ) and rearrangng we obtan: v JIC emal for contrbuton: edtor@c.org.uk

3 Journal of Informaton and Computng Scence Vol. (6) No. pp wth A = vf va U ( θ ) > ( ) ( ) = θ A = U θ θ (5) A = ( uv v) f vhd uva vb A = ( uv u) f uhd uvb ua A = uf ub U ( θ ) > f A > =. A > f A > f A > f v >. f hd b u >. ( f a) f hd a v >. ( f b) A > f u >.. All ths dscusson s summarzed n the followng theorem: Theorem.. The ratonal cubc functon () preserves the shape of data that les above the straght lne f n each subnterval [ ] free parameters u and v satsf the followng condtons: f hd b u = l Ma l >. ( f a) 4. Bcubc Partall Blended Ratonal Functon f hd a v = m Ma m >. ( f b) The pecewse ratonal cubc functon () s etended to bcubc partall blended ratonal functon S( ) over rectangular doman D= [ m] [ n]. Let π : a= < < < m = b be partton of [a b] and π : c= < < < n = d be partton of [ c d]. Ratonal bcubc functon s defned over each rectangular patch [ ] [ ] where =... m ; =... n as: where T S ( ) = AFB (6) S( ) S( ) F = S( ) S( ) S( ) S( ) S( ) S( ) A = [ φ ( ) φ ( )] B φ φ = [ ( ) ( )]. ( ) ( ) φ φ φ ( ) and φ ( ) are the Cubc Hermte blendng functons. S( ) S( ) S ( ) and S ( ) are ratonal cubc functons defned on the sdes of [ ] [ ] usng () wth S( ) F F F F u v S( ) F F F F u v S( ) F F F F u v S ( ) F F F F u v JIC emal for subscrpton: nfo@c.org.uk

4 5 wth wth wth A = v F M. Z. Hussan M. Hussan: Vsualzaton of Data Subect to Postve Constrants ( θ) θ A = = (7) q ( θ ) S( ) A = ( u v v ) F v hf = ( u v u ) F u hf A A = u F q ( θ ) = ( θ) v u v ( θ) θ θ u. B = v F ( θ) θ B = = (8) q( θ ) S( ) B = ( u v v ) F v hf B = ( u v u ) F u hf B = u F q ( θ ) = ( θ) v u v ( θ) θ θ u. ( φ) φ C = = (9) q( φ) S( ) C = v F C = ( u v v ) F v h F C = ( u v u ) F u h F C = u F q ( φ ) = ( φ) v u v ( φ) φ φ u. wth D = F 5. Choce of Dervatves ( φ) φ D = = () q4( φ) S( ) v D = ( u v v ) F v h F D = ( u v u ) F u h F D = u F q ( φ ) = 4 ( φ) v u v ( φ) φ φ u. In most applcatons the dervatve parameters d F F and F are not gven and hence must be determned ether from gven data or b some other means. These methods are the appromaton based on varous mathematcal theores. An obvous choce s mentoned here: 5. Arthmetc Mean Method Arthmetc mean method s the three-pont dfference appromaton based on arthmetc manpulaton. Ths method s defned as: 5.. Arthmetc Mean Method for D Data JIC emal for contrbuton: edtor@c.org.uk d d =Δ Δ Δ ( ) ( h ) h n n =Δ n ( Δn Δn ) ( hn hn ) d Δ Δ = h h

5 Journal of Informaton and Computng Scence Vol. (6) No. pp where =... n Δ =. f f h 5... Arthmetc Mean Method for D Data where F h h =Δ ( Δ Δ ) F =Δ ( Δ Δ ). ) m m m m m ( h h) ( hm hm Δ Δ =. = ; =. ( ) h ( ) hn F = Δ Δ Δ F n = ( Δ n Δ n Δ n h ) ( h h n h ). n Δ Δ F =. =... m; =... n. F F F F F =. =... m ; =... n. h h h h F m n F F h Δ = and Δ F F = h and sutable for vsualzaton of shaped data. 6. Vsualzaton of D Postve Data. These arthmetc mean methods are computatonall economcal Let ( F ) be postve data defned over rectangular grd = m ; = n s.t. F >. I = [ ] [ ] The bcubc partall blended surfaces patches (6) nhert all the propertes of network of boundar curves [4]. The bcubc partall blended surface (6) s postve f boundar curves S( ) S( ) S ( ) and S ( ) defned n (7) (8) (9) and () are postve. A ( θ) θ A > = S( ) > f and q ( θ ) >. > = f S q ( ) θ > f u > and v >. ( θ) θ A > = f A > =. hf hf u > Ma v > Ma. F F ( ) > f ( θ) θ B > and q ( θ ) >. = q ( ) θ > f u > and v >. ( θ) θ B > = f B > =. JIC emal for subscrpton: nfo@c.org.uk

6 54 M. Z. Hussan M. Hussan: Vsualzaton of Data Subect to Postve Constrants D B hf hf > = f u > Ma v > Ma. F F C ( φ) φ C > = S ( ) > f and q ( φ ) >. q ( ) φ > f u > and v >. ( φ) φ C > = f C > =. hf hf > = f u > Ma v > Ma. F F S ( ) > f ( φ) φ D > and q ( φ ) >. 4 = q ( ) 4 φ > f u > and v >. ( φ) φ D > = f D > =. hf hf > = f u > Ma v > Ma. F F Ths leads to the followng theorem: Theorem 6.. The bcubc partall blended ratonal functon defned n (6) vsualze postve data n the vew of postve surface f n each rectangular patch I = [ ] [ ] free parameters u v u v u v u and satsf the followng condtons: v hf hf u = l Ma l > v = m Ma m >. F F hf hf u = n Ma n > v = o Ma o >. F F hf hf u = k Ma k > v = t Ma t >. F F hf hf u = g Ma g > v = w Ma w >. F F 7. Vsualzaton of D Constraned Data Let ( F ) be data defned over the rectangular grd lng above the plane I = [ ] [ ] =... m ; =... n JIC emal for contrbuton: edtor@c.org.uk

7 Journal of Informaton and Computng Scence Vol. (6) No. pp e. Z = C A B F > Z. The correspondng surface generated b bcubc partall blended ratonal functon (6) wll also le above the plane f each of the boundar curve S( ) S( ) S ( ) and S( ) le above the plane. The boundar curve Substtutng value of ( ) S ( ) S wll le above the plane f S( ) > ( ) Z Z. () n above equaton = ( θ) q ( θ ) θ θ θ A > ( θ) Z θz. () q ( ) θ > f u > and v >. Multplng both sdes of above equaton b q ( ) θ and after some rearrangement () can be rewrtten as U ( θ ) > where wth M ( ) = ( ) M = U θ θ θ M = v ( F Z ) M = ( u v v ) F v hf u v Z v Z M = ( u v u ) F u hf u v Z u Z M = u ( F Z ). U ( θ ) > f ( ) M >. = ( θ) θ M > M > = = f. hf F Z hf F Z > = f u > Ma v > Ma. ( F Z ) ( F Z ) The boundar curve S( ) wll le above the plane f free parameters u and v satsf the followng condtons hf F Z hf F Z u > Ma v > Ma. ( F Z ) ( F Z ) The boundar curve Substtutng value of S ( ) S ( ) θ wll le above the plane f θ S φ Z φ Z () ( ) > ( ). (4) n above equaton = ( φ) q ( φ) φ C > ( φ) Z φz. q ( ) φ > f u > and v >. Multplng both sdes of above equaton b q ( ) φ and after some rearrangement (5) can be rewrtten as U ( φ ) > where (5) JIC emal for subscrpton: nfo@c.org.uk

8 56 M. Z. Hussan M. Hussan: Vsualzaton of Data Subect to Postve Constrants wth L U( φ) = ( φ) φ L (6) = L = v ( F Z ) L = ( u v v ) F v h F u v Z v Z L = ( u v u ) F u h F u v Z u Z L = u ( F Z ). U ( φ ) > f ( ) L > = ( φ) φ L > L > = = f. hf F Z hf F Z > = f u > Ma v > Ma. ( F Z ) ( F Z ) The boundar curve S( ) wll le above the plane f free parameters and satsf the followng condtons: φ φ u v hf F Z hf F Z u > Ma v > Ma. ( F Z ) ( F Z ) Therefore we can conclude above dscusson n the followng theorem: Theorem 7.. The bcubc partall blended ratonal functon (6) generates surface that les on same sde of plane as that of data f n each rectangular patch I = [ ] [ ] free parameters u v u v u v u and satsf the followng condtons: v hf F Z hf F Z u = k Ma k > v = t Ma t >. ( F Z ) ( F Z ) hf F Z hf F Z u = r Ma r > v = w Ma w >. ( F Z ) ( F Z ) hf F Z u = l Ma l > ( F Z ) hf F Z v = m Ma m >. ( F Z ) hf F Z u = n Ma n > ( F Z ) hf F Z v = o Ma o >. ( F Z ) JIC emal for contrbuton: edtor@c.org.uk

9 Journal of Informaton and Computng Scence Vol. (6) No. pp Demonstraton We shall llustrate the shape preservng schemes developed n Secton 6 and 7 wth some eamples. A postve data set s consdered n Table. Table Fg. Fg. Fgure s produced usng Cubc Hermte Splne whch faled to preserve postve shape of data. Fgure s produced b postvt preservng scheme developed b Zawwar and Jamaludn n [6]. The data set for second eample shown n Table s lng above the straght lne: =. Table Fg. Fg. 4 Fgure s produced usng Cubc Hermte Splne. Ths curve does not le above the lne flaw s recovered ncel n Fgure 4 usng scheme of Secton wth l = m =.. The data set n Table s lng above the straght lne: = /. =. Ths JIC emal for subscrpton: nfo@c.org.uk

10 58 M. Z. Hussan M. Hussan: Vsualzaton of Data Subect to Postve Constrants Table Fg. 5 Fg. 6 Fgure 5 s produced usng Cubc Hermte Splne. Ths curve does not le above the lne Ths flaw s recovered ncel n Fgure 6 usng scheme of Secton wth l = m =.. The postve data set n Table 4 s generated from the followng functon: F.6.6 ( ) = = /. Table 4. / The postve surface generated b scheme n Secton 6 s shown n Fgure 7 wth l = m = n = o = k = t = g = w =.4. The postve data set n Table 5 s generated from the followng functon: ( ) F( ) =.5.. Table 5. / The postve surface generated b scheme n Secton 6 wth l m n o k t g w = = = = = = = =.5 JIC emal for contrbuton: edtor@c.org.uk

11 Journal of Informaton and Computng Scence Vol. (6) No. pp s shown n Fgure 8. Fg. 7 Fg. 8 The data set n Table 6 s of the followng plane: Z = Table 6. / The data n Table 7 s of the functon: F Sn ( ) = /4. 6. Fg. 9 JIC emal for subscrpton: nfo@c.org.uk

12 6 M. Z. Hussan M. Hussan: Vsualzaton of Data Subect to Postve Constrants Table 7. / Ths data s lng above the plane defned n Table 6. Fgure 9 s produced b surface scheme developed n Secton 7 wth k = t = r = w = l = m = n = o = From fgure t s clear that surface s lng above the plane as requred. 9. Concluson.6. In ths paper the problem of postve nterpolaton of curves and surfaces s dscussed. Smple constrants are developed on free parameters n the descrpton of ratonal cubc and ratonal bcubc functon to vsualze postve data. Choce of the dervatve parameters s left at the wsh of the user as well. The method s ver eas to mplement as compared to methods alread developed ([][5][7]) local computatonall economcal and vsuall pleasng.. References [] M. R. Asm and K. W. Brodle. Curve drawng subect to postvt and more general constrants Computers and Graphcs 7: [] K. W. Brodle P. Mashwama and S. Butt. Vsualzaton of surface data to preserve postvt and other smple constrants. Computers and Graphcs 995 9(4): [] S. Butt and K. W. Brodle. Preservng postvt usng pecewse cubc nterpolaton. Computers and Graphcs 99 7(): [4] G. Cascola and L. Roman. Ratonal nterpolants wth tenson parameters. Curve and Surface Desgn Tom Lche Mare-Laurence Mazure and Larr L. Schumaker (eds) 4-5. [5] M. Z. Hussan. Shape preservng curves and surfaces for computer graphcs Ph.D. Thess Unverst of the Punab Lahore Pakstan. [6] M. Z. Hussan and J. M. Al. Postvt preservng pecewse ratonal cubc nterpolaton. Matematka 6 (). [7] A. Kouba and M. Pasadas. Varatonal bvarate nterpolatng splnes wth postve constrants Appl. Numer. Math 44: [8] C. Mann. On shape preservng C Hermte nterpolaton. BIT 4: [9] E. Nadler. Non-negatvt of bvarate quadratc functons on a trangle. Computer Aded Geometrc Desgn 99 9: [] A. R. M. Pah T. N. T. Goodman and K. Unsworth. Postvt preservng scattered data nterpolaton. Proc. th IMA Mathematcs of Surfaces Conference Loughborough U. K. September [] M. Sarfraz M. Z. Hussan and S. Butt. A ratonal Splne for vsualzng postve data. Proc. IEEE Internatonal Conference on Informaton Vsualzaton London U. K. Jul [] M. Sarfraz S. Butt and M. Z. Hussan. Vsualzaton of shaped data b a ratonal cubc Splne nterpolaton Computers and Graphcs 5: [] J. W. Schmdt and W. Hess. Postvt nterpolaton wth ratonal quadratc Splnes. Computng 987 8: [4] J. W. Schmdt and W. Hess. Postvt of cubc polnomal on ntervals and postve Splne nterpolaton BIT 988 8: 4-5. JIC emal for contrbuton: edtor@c.org.uk