The local support basis representation of the ‘weighted v-spline’
is derived in terms of double knot cubic B-splines, so providing a
convenient form for computing and analysing the representation

Gregory, JA

Zhou, J

English

Brunel University Research Archive

             TR/03/91     February 1991  The weighted -spline as a double vknot B-spline  John A. Gregory and Jianwei Zhou                                                                  BRUNEL UNIVERSITY SEP 1991  LIBRARY         w9198937  The weighted v-spline as a double knot B-spline  John A. Gregory Jianwei Zhou Department of Mathematics and Statistics, Brunel University, Uxbridge, Middlesex UB8 3PH, England.  Abstract  The local support basis representation of the ‘weighted v-spline’ i s  de r ived  in  t e rms  o f  doub le  kno t  cub ic  B-sp l ines ,  so  p rov id ing  a   conven ien t  fo rm fo r  comput ing  and  ana lys ing  the  represen ta t ion .   Keywords Spl ines ,  B-spl ines ,  tens ion .   1 Introduction In this note, the local support basis representation of the weighted v-spline, [Foley’87,88], is considered via a transformation to double knot cubic B-            spline form. The weighted v-spline is a C1 piecewise cubic curve which has        specific second derivative jump discontinuity constraints across the knots of                the spline. The constraints involve interval and point tension weights, which               can be used to influence the shape of the curve, and the transformation                 to double knot B-spline form provides a convenient computational tool for       handling and analysing the local support basis representation.     The weighted v-spline combines the v-spline, [Nielson’74], with the weight-                   ed spline, [Salkauskas’84], and thus includes these splines as special cases.            Since the v-spline is intimately connected with the -spline, [Barsky’81],  [Goodman’85], there is an immediate relationship between the local support             basis representation presented here with that of the -spline. The develop-            ment of the local support basis representation builds on the work of [Foley’86],                  for the weighted spline, and with the work of [Boehm’85] and [Lasser’88], for       ‘geometric splines’, where Bernstein-Bézier representations are discussed, see                 also [Dierckx and Tytgat'89]. Thus, for completeness, we briefly review the transformation to Bernstein-Bézier form in section 3. Here we will see that ββ      1 Weighted ν -Spline                  2  the control points of the Bernstein-Bezier representation include those of the double knot B-spline representation developed in section 2, a result that should urprise the thoughtful reader. In the final section 4, the deriva- not stion of the cal suppor asis representation which satisfies interpolation lo t b2    The double knot B-spline representation constraints at the knots is considered. Let { define a sequence of distinct knots with interval spacing }ΖΖ∈∈ iRIti :Ζ∈>−= + i,0tt:h i1ii . Then a weighted v-spline is a C1 parametric Function  p : such that mRIRI →,i,)tt(31iiΖ∈∈Ρ ∏+   (piecewise cubic curve)  (2.1) and Ζ∈=− −−+ itPvtPwtpw iiiiii ,)2(The case w)(()( )1()1)2( .   (2.2) i = wi-1 = 1 is that of the unweighted v-spline of [Nielson’74], when (2.2) defines a eometric GCg  across the kncurve is C2 constraint ot ti, that is the 2 under a reparameterization. The case Vi =0 is the weighted spline of [Salkauskas’84], se  als  [Foley’86]. he ene l case (2.2) is that e o T  g radescribed in [Foley’87,88]. For the analysis we take ΖΖ∈≠ iwi ,0 , and replace (2.2) by the equivalent form ,),(p)(p)(P (1)(2)(2) Ζ∈=− −+ itvtwt iiiii ))    (2.3) where .//1 iiiiii wvvwww == − ))     (2.4)         Assume that there exists a normalized local support basis { } Ζ∈ii )t(N  for the linear space of weighted v-splines, defined such that ),t,t(t,0)t(N 4iii +∉=  (local support)  (2.5) and 1)t(Nii=∑Ζ∈,  (partition of unity normalization).   (2.6) Then given the control points { }Ζ∈∈ j:Rc j , we wish to represent the Curve                              )t(Nc)t(p jjj∑Ζ∈=                  [ ,t,tt),t(Nc 1iijji3ij+−=∈= ∑ ]      (2.7)                                                   Weightedν -Spline                    3  in double knot cubic B-spline form. Indeed, the representation of a weighted v-spline as a double knot B-spline will demonstrate the existence of the local support basis representation. Thus, let   ,   (2.8)          Ζ∈+−⎥⎦⎤⎢⎣⎡−= +∧+∧+∧+∧∧∧+∧ j,)t(.,t,t,t,t,t)tt(:)t(B 314j3j2j1jjj4j4jdefine the normalized cubic B-splines, see [de Boor'78], on the double knot partition ⎭⎬⎫⎩⎨⎧ Ζ∈==∈ +∧∧∧ ,ttt:RRt j1j2j2j ,      (2.9) In particular,   [ ] +−−= ++++ 3211242 )(.,,,,,)(:)( tttttttttB jjjjjijj  and [ ] +−−= ++++++ 3221124 12 )(.,,,,,)(:)( tttttttttB jjjjjjjj  are the double knot B-splines with the local support of . )t,t( 2jj +The space of double knot cubic B-splines consists of C1 piecewise cubic curves which, in general, have second derivative discontinuities across the knots. We thus seek the representation of a weighted f-spline in the double knot cubic B-spline form    )t(Bb:)t(P 4jjzj∑∈=          ,                (2.10) [ ]141222,),(: ++−=∈= ∑ iijjiijttttBbwith control points { }Ζ∈∈ jRIb mj : . Following [de Boor’78, pl39] we have )/()(3),()( 31)1(3)1()1(jjjjjjjjttbbbtBbtp∧+∧−ΖΖ∈−−== ∑              (2.11) ),/()bb(2b),(b)(p 2)1( 1)1()2(2)2()2(jjjjjjjj tttBt∧+∧−ΖΖ∈−−== ∑              (2.12) where                                              (2.13) ⎪⎪⎭⎪⎪⎬⎫+−⎥⎦⎤⎢⎣⎡−=+−⎥⎦⎤⎢⎣⎡−=+∧+∧∧∧+∧+∧+∧+∧∧∧+∧,)(.,,,)(:)(,)(.,,,)(:)(12122232133tttttttBttttttttBjjjjjjjjjjjdefine the normalized quadratic and linear double knot B-splines. It is now a relatively simple exercise to show that the weighted v-spline constraint (2.3), applied to the double knot B-spline representation (2.10), gives ,bb)1(bb)1( 22122322222 iiiiiiii −−−−−−− +−=+− σσττ              (2.14) Weighted ν -Spline      4       Figure 1: Transformation to double knot B-spline control points where )/()(2:),/()(2: 2122112 pihhhpihhhw iiiiiiiii +−=+= −−−−∧− στ                          (2.15) with i1iii vh/w2h/2:pi∧−∧ ++= .                (2.16) Equation (2.14) can be interpreted as the requirement that two lines intersect at a point and it is the simple observation of Proposition 2.1 below that this intersection is the control point  of the local support basis representation 2ic −(2.7), see figure 1. Thus, given { }  the control points,c ii ΖΖ∈ { } ,b ii ΖΖ∈  are given by solving   ⎭⎬⎫=−+=+−−−−−−−−−,)1(,)1(121121222122iiiiiiiiiicbbcbbττσσ,ΖΖ∈i               (2.17) that is [ ] [ ][ ] [ ] ΖΒ∈⎭⎬⎫∆σ−+∆−=∆σ−∆−=−−−−−−−−− i,c/1c)/T(b,c/c/)T1(b1ii2i2ii1ii21ii2i2ii1i1i2                  (2.18) where 2i1ii T1: −− σ−−=∆ .                             (2.19) We now have: Proposition 2.1  Suppose  and 0pi ≠ Ζ∈≠∆ i,0i . Then (i) There exists a normalized local support basis { } Ζ∈itNi )(  for the space  of weighted v-splines.   Weighted ν -Spline                  5  (i  There xists the transformation (2.18) from the control points of the i)  e local support basis representation to those of a double knot B-spline represent  tation. Furthermore, if    (2.20) ,i,0vh/w2h/2:and0w i1iiiii Ζ∈>++=ρ>∧−∧∧then         (iii) The normalized local support basis is non-negative and the local sup-  port  basis representation [ ],,),( 1+∈ ii ttttP  lies in the convex hull of  { } . i 31jic −=       (iv) The local support basis representation is variation diminishing, in particular, any (hyper) plane of dimension m — 1 crosses [ ]1ii t,tt),t(P +∈  no more times than it crosses the control polygon joining the points { } . i3ijjc −=Proof. The proof is straightforward and for brevity we just outline its es-             sential features. For a given Ζ  let scalar values j, b,  be defined by (2.18), with right hand sides having scalar values  defined by  ∈j j,i21i2b − j,1ij,2i c,c −− Ζ∈δ= i,c j,ij,i      (2.21) Then bi,j,  has the support of the integer set {2j + 1,..., 2j + 4} and                (2.22) ZZi∈( ) ( ) ( )∑∑ ++=∈==4j21j2iij,iZZiij,i:j tBbtBbtNdefines a weighted v-spline function with the local support of (tj,tj+4). More-            over, our previous work then demonstrates the existence of the transforma-              tion (2.18). In particular, with scalar values { }Zici ∈= :1  we ob-                           tain { }from (2.18) and the partition of unity normalize-                  tion (2.6) follows from the partition of unity property of the double knot                   B-splines. The conditions (2.20) guarantee thatZibi ∈= :1,0,0 22 << −− ii στ  and                       then .,11 21 Ziiii ∈>−−=∆ −− στ  Hence the transformation (2.18)                     involves convex combinations and it follows that the in (2.22) are            non-negative. Hence NZZib ji ∈,, j(t) is non-negative and the convex hull property fol-       lows from the partition of unity normalization. Also, (2.18) defines a corner cutting process, see figure 1, and hence the variation diminishing property        follows from that of the double knot B-spline representation. □  Remark. In terms of the notation of the standard weighted v-spline con-            straint (2.2), condition (2.20) can be written as  ,ZZi),h/wh/w(2v,0w 1i1iiiii ∈+−>> −−    (2.23)  and it should be noted that negative values of the parameter vi are allowable        whilst still maintaining properties (iii) and (iv) of the Proposition.     6 Weighted ν -Spline                                                                                         Tension properties. The transformation (2.18) can be used to justify the         point and interval tension properties of the local support basis representation         of the weighted v-spline with respect to the parameters vi and wi. We first               observe that, in terms of the double knot B-spline representation,   ( ) .bhhhbhhhtP 1i2i1i1i2i2i1iii −−−−− +++=    (2.24) The analysis of the control point transformation (2.18) now leads to the        following results, where, for simplicity, it is assumed that the other shape parameters are held constant with respect to each limit process:  (i)Point tension . iν .cblimblim 2i1i2v2i2v ii −−∞→−∞→==   (2.25)  Thus, as  increases, P(tiν i) is pulled towards ci-2  in the limit a cusp                   tangent discontinuity is introduced at ci-2.  (ii)Interval tension wi.          (2.26) ,2i2i2w cblimi −−∞→= ,332lim 1111211112 −+−−−+−+−∞→ +++++++= iiiiiiiiiiiiiwchhhhhchhhhhbi   (2.27)   ,chh3hh2hchh3hhhblim 1i1ii1ii1i2i1ii1i1iii2wi−+−−−+−+∞→ +++++++=   (2.28)         (2.29) .cblim 1i1i2wi −+∞→=It follows that, as wi increases, P(t), [ ]1ii t,tt +∈  is pulled onto a straight line     segment between  and . Furthermore, the end points of this segmentare given by: 2ic − 1ic − ,chh3hhchh3hhh3)t(Plim 1i1ii1i1i2i1ii1i1iiiwi−+−−−+−+∞→ ++++++=   (2.30)  ,chh3hh3hchh3hh)t(plim 1i1ii1ii1i2i1ii1i1i1iwi−+−−−+−++∞→ ++++++=   (2.31) Here, the analysis confirms the results of [Foley’86].       Some simple examples of the point and interval tension behaviour for the       local support basis are given in figure 2.  3 The Bernstein-Bézier representation  For completeness, we briefly review the transformation of the weighted ν -        spline to Bernstein-Bézier (triple knot B-spline) form. This transformation   Weighted v-Spline 7 Figure 2: Examples of tension control behaviour  Weighted v-Spline                                                                                             8   could be easily derived from the double knot B-spline representation but       instead we generalize the discussion to the space of C° piecewise cubic gen-realized -spline curves  defined such that βnRIRI →:p     (3.1) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) .Zi,tPtPtP,tPtPi2i,3i1i,2i2i1i,1i1∈⎪⎭⎪⎬⎫β+β=β=−−+−+The derivation of the Bernstein-Bézier form of the local support basis repre-entation then follows from the work of [Boehm'85] and [Lasser'88].        It is well known that there is no loss of generality in taking integer knots        in (3.1), since a piecewise affine transformation onto integer knots gives con straint equations with parameters   ,hhˆ,hhˆ,hhˆ1i2ii,2:i,31i2ii,2:i,21iii,1:i,1−−−β=ββ=ββ=β    (3.2) On the general knot partition, the weighted ν -spline constraint (2.3) is re-   covered as the special case    .ˆ,ˆ,1 ,3,2,1 iiiii w=== βνββ     (3.3) The case  gives the standard i,12i,3 β=β β -spline of [Barsky '81], see [Goodman      ’85], when (3.1) defines a geometric GC2 constraint across the knot ti. In          fact, in the non-periodic case, the standard -spline and unweighted β ν -spline     are equivalent under a piecewise denned affine transformation, although we      will not pursue this equivalence here.      We now seek the representation of a generalized β -spline in the C° piece-  wise denned Bernstein-Bézier form  ( ) ( ) ( ) ,Zi,h/tt),(BbtP ii30k:kki3:tt 1i,i∈−=θθ=∑=++    (3.4) where               (3.5) ( ) ( ) .1k3B k3k:k−θ−θ⎟⎟⎠⎞⎜⎜⎝⎛=θImposing the constraints (3.1) on this form gives, after some elimination,           ( ) 1i3i1i3ii3 b1bb −− µ−+µ=      (3.6) and  ( ) ( ) ,bb1bb1 2i32i1i32i1i32i1i32i +−+−−−−− σ+σ−=τ+τ−    (3.7) where    ( ) ( ) ( ),kˆ12,kˆˆ12,ˆ1ˆii,1:2iii,3i,1:2ii,1i,1:iβ+−=σββ+−=τβ+β=µ −−    (3.8)   Weighted v-Spline                                                                                                     9     Figure 3: Transformation to Bernstein-Bézier form with         (3.9) .ˆˆˆ2 ,3,2,1 iiiik βββ ++=Equation (3.7) represents the intersection of two lines at the control point         of the local support representation, see figure 3. We then have 2ic −    (3.10) ( )[ ] [ ][ ] [ ] ,ZZic/)1(c/bc/c/1b,1ii2i2ii1i2i3,1ii2i2ii1i1i3 ∈⎭⎬⎫∆σ−+∆τ−=∆σ−∆τ−=−−−−+−−−−+where       .1 2i1i:i −− σ−τ−=∆      (3.11) Equations (3.10) and (3.6) thus define the transformation from the control        points Cj,  to those of the Bernstein-Bézier representation. ZZi∈    For the case of the weighted ν -spline, where the β -coefficients are defined      by (3.2), we obtain     ( )iiii hhh += −1/µ     (3.12) from (3.8) and coefficients ,  which are identical with those of the       double knot B-spline transformation defined by (2.15). Thus the interior       control points (3.10) of the Bernstein-Bézier representation are identical with      the control points (2.18) of the double knot B-spline representation (compare      also (2.24) with (3.6)). This is a property which could be observed directly      from a study of the behaviour of the double knot cubic B-splines. 2i−τ 2i−σ 4 Interpolation  using  the  local  support  basis We conclude by considering the derivation of the local support basis repre-sentation satisfying interpolation constraints at the knots.  For brevity, we  Weighted ν -Spline                                                                                            10  consider only clamped type end conditions for the interpolation problem. Thus, given the local support basis representation         (4.1) ( ) ( )∑−−==1n3jjj: tNctPof a weighted ν -spline on [t0, tn], we seek the cj, j = -3,..., n-1 such that                        ( ) ,n,....,0i,ytP ii ==  (4.2) with clamped end conditions     ( ) ( ) ( ) ( ) .dtP,dtP nn1001 ==    (4.3) The representation (4.1) could be derived from the work of [Foley'87], by onverting the piecewise Hermite representation used there to double knot            B-spline form. Here, however, we consider a direct derivation of the control     points Cj. Substituting the double knot B-spline representation (2.10) in            (4.2) and (4.3), with control points given by (2.18), gives ( ) ( )( ) ( 4.4,n,...,0i,yhhchc1h1hchi1ii1i1i2i1i2ii1i1i1i3ii3i1i2ii=+=∆σ−+⎥⎦⎤⎢⎣⎡∆τ−+∆σ−+∆τ−−−−−−−−−−−−−−) with end conditions ( ) ( )( ) .dhh31c11cc5.4,dhh31cc11cnn1n2nn1n1n3n3n2n2n1nn2n00110220113312++⎥⎦⎤⎢⎣⎡∆τ−−∆σ−+∆τ−=∆σ−+−∆σ−+⎥⎦⎤⎢⎣⎡∆τ−+∆σ−−=∆τ−−−−−−−−−−−−−−−−−−−−− (4.6)  Using (4.5) and (4.6) to eliminate c-3 and cn-1 then gives a tridiagonal system defined by (4.4) for i = 1,..., n — 1, where for i = 0 and i = n the equations become         ( )( )8.4.d31yc1c7.4,d31ycc11213312000102201nnnnnnnnn hh−−−−−−−−−−−−=∆−+∆−+=∆−+∆−στστ Let( ) .1:2iiiii pwhh += −−α  Then  > 0 for ρ2i−α i, satisfying (2.20) and simple substitution shows that ( ) ,01 3i2i3i2i2i2i3i >ατ−>ατ−α=ασ− −−−−−−−                  (4.10)      Weighted ν -Spline  11  ( ) ,01 1212221 >−>−=− −−−−−−− iiiiiii ασασαατ           (4.11)  It follows that under the transformation   (4.12) ∧α= ,c:c iiithe tridiagonal system in the unknowns ic), i = —2,..., n — 2 is diagonally                        dominant. We thus have:  Proposition 4.1 For parameters satisfying (2.20), or equivalently (2.23),  for i = 0,...., n—1, there exists a unique weighted v-spline on [t0,tn] satisfying the interpolation conditions (4.2) and (4.3).  Acknowldgements  This work was supported by the ACME Directorate of the Science and En-ineering Research Council, with the grants GR/E 25092, GR/F 80838, and          by a Ph. D. studentship from Siemens AG.  References  [1] Barsky, B. A. (1981), The beta-spline: A local representation based on                       shape parameters and geometric measures, Ph.D. dissertation, Dept.           Computer Science, The University of Utah. [2] Boehm, W. (1985), Curvature continuous curves and surfaces, Computer                       Aided Geometric Design 2, 313-323. [3] de Boor, C. (1978), A Practical Guide to Splines, Springer, Berlin. [4] Dierckx, P. and Tytgat, B. (1989), Generating the Bezier points of a         β -spline curve, Computer Aided Geometric Design 6, 279-291. [5] Foley, T. A. (1986), Local control of interval tension using weighted splines, Computer Aided Geometric Design 3, 281-294. [6] Foley, T. A. (1987), Interpolation with interval and point tension con-         trols using cubic weighted  v-splines, ACM Trans. Math. Softw. 13,68-96.  [7] Foley, T. A. (1988), A shape preserving interpolant with tension con-           trols, Computer Aided Geometric Design 5, 105-118. [8] Goodman, T. N. T. (1985), Properties of β -splines, J. Approx. Theory      44, 132-153.   Weighted ν -spline                   12  [9] Lasser, D. (1988), B-Spline-Bézier representation of Tau-Splines, NPS technical report 53-88-006, Naval Postgraduate School, Monterey, CA 93943. [10]  Nielson, G. M. (1974), Some piecewise polynomial alternatives to splines under tension, in: R.E. Barnhill and R. F. Riesenfeld, eds., Computer Aided Geometric Design, Academic Press, New York, 209-235. [11] Salkauskas, K. (1984), C1 splines for interpolation of rapidly varying   data, Rocky Mountain J. Math. 14, 239-250.                                                                              NOT TO BE REMOVED FROM THE LIBRARY                 

A Practical Guide to Splines,

A shape preserving interpolant with tension con- trols,

B-Spline-Bézier representation of Tau-Splines, NPS technical report 53-88-006,

C 1 splines for interpolation of rapidly varying data,

Curvature continuous curves and surfaces,

Generating the Bezier points of a β -spline curve,

Interpolation with interval and point tension con- trols using cubic weighted v-splines,

Local control of interval tension using weighted splines,

Some piecewise polynomial alternatives to splines under tension,

The beta-spline: A local representation based on shape parameters and geometric measures,

The weighted v-spline as a double knot B-spline

The weighted v-spline as a double knot B-spline

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