Pseudo-polyharmonic div-curl splines and elastic splines

Pseudo-polyharmonic div-curl splines and elastic splines

M. N Benbourhim∗ and A. Bouhamidi†

Abstract

Vector field reconstruction is a problem that arises in many scientific applications. Inthis paper we study a div-curl approximation of vector fields by pseudo-polyharmonic splinesand elastic splines. This leads to the variational smoothing and interpolating spline problemswith minimization of an energy involving the rotational and the divergence of the vector field.

1 Introduction and notations

The theory of pseudo-polyharmonic splines, also called the (m, s)-splines, was introduced byDuchon [17, 18, 19]. The pseudo-polyharmonic splines lead to a native space of Sobolev typedenoted by Xm,s(Rn), see also [3]. The space Xm,s(Rn) is endowed with a semi-scalar producttogether with its associated semi-norm, and it is complete for this semi-norm. In this paper,we introduce the pseudo-polyharmonic div-curl splines and elastic splines and we investigatethe problem of interpolating and smoothing of a vector fields by these splines. The appropriatevectorial space is naturally the vector space Xm,s(Rn; Rn) which is the space of vector-functionswhose components belong to the scalar space Xm,s(Rn).

Vector field approximation is a problem arising in many scientific applications. These include,for example, fluid mechanic, electromagnetic, meteorology, optic flow analysis. When consideringthe approximation of vector fields, a key problem is how to correlate its components. It hasbeen observed, particularly for meteorological problems, that if no inter-component correlationis assumed the approximating field may give unrealistic results, see [15].

In [2], Amodei and Benbourhim introduced a family of vector splines based on the bi-harmonic thin plate splines for two variables and minimizing some energy. The energy usedin [2] has a divergence and a rotational terms, each multiplied by a fixed real positive parameterthat controls its relative weight. The value of the parameters depends on the particular field tobe approximated and some a priori knowledge about the underlying field is needed. The problemof div-curl splines approximation by minimization of certain energy related to the divergenceand the curl of vector fields was also studied for some situations by Benbourhim and Bouhamidi[7] and by Dodu and Rabut [21].

Div-curl vector splines provide an effective tool for the problem of reconstruction of a vectorfield by using the interpolation or the approximation of the vector field from a set of scatteredobserved data. The div-curl vector splines have been used successively in many applications like

∗Laboratoire MIP-UMR 5640, Universite Paul Sabatier, UFR MIG, 118, route de Narbonne, F-31062 Toulouse

Cedex 04, FRANCE. E-mail:[email protected].†L.M.P.A, Universite du Littoral Cote d’Opale, 50 rue F. Buisson BP699, F-62228 Calais Cedex, France.

E-mail:[email protected].

1

Benbourhim and Bouhamidi/Pseudo-polyharmonic div-curl splines and elastic splines 2

reconstruction of wind velocity in meteorology [1], optical flow motion estimates [26, 27], humanheart motion analysis [12] as well as image processing [13, 28].

Throughout this paper, we use the following notations. The letter n denotes an integer ≥ 1.For any x, y ∈ Rn, we denote by xT y the Euclidian scalar product of x and y and |x| denotethe associated Euclidian norm in Rn, where xT denotes the transpose of the vector x. We usethe standard notation, for any multi-index α = (α1, . . . , αn) ∈ Nn, we write |α| = α1 + · · ·+ αn,α! = α1! · · ·αn!. For any x = (x1, . . . , xn) ∈ Rn, xα = xα1

1 · · ·xαnn . We use ∂i to denote the

partial derivative ∂∂xi

, for i = 1, . . . , n and Dα or ∂α denote the partial derivative ∂α = ∂|α|

∂α1

1···∂αn

n

of order α. Let Ck(Rn) be the space of real valued functions of class Ck on Rn. Let D(Rn)denotes the space of compactly supported and infinitely differentiable real valued functions onRn. Its topological dual space is the space D′(Rn) of distributions on Rn. Let S(Rn) be theSchwartz space of C∞ real valued functions which, with all derivatives of all order, are rapidlydecreasing at infinity. The topological dual space of S(Rn) is the space S ′(Rn) of tempereddistributions on Rn. Let m > 0 be a positive integer, we denote by Πm−1(R

n) the space of allpolynomials defined over Rn of total degree at most equal to m − 1. The corresponding vectorreal-valued spaces are denoted by Ck(Rn; Rn), D(Rn; Rn), D′(Rn; Rn), S(Rn; Rn), S ′(Rn; Rn)and Πm−1(R

n; Rn), respectively.The outline of our paper is as follows. In Section 2, we recall some basic properties of the

scalar functional space Xm,s(Rn). Section 3 is devoted to the study of the properties of theassociated vectorial functional space Xm,s(Rn; Rn). In Section 4, we study the interpolating andapproximating div-curl problem and we give the expression of the pseudo-polyharmonic vectorialsplines. The elastic splines are studied in Section 5. Section 6 is a short section where we givebriefly some extension of our work. In Section 7, we give a numerical example to illustrate theeffectiveness of our approach.

2 The scalar functional space Xm,s(Rn)

Let s be a given real number and consider the following space introduced by Peetre [23]

Hs(Rn) = v ∈ S ′(Rn) | v ∈ L1loc(R

n),

∫

Rn

| ξ |2s | v(ξ) |2 dξ < +∞.

The space Hs(Rn) is equipped with the scalar product together with its associated norm denotedby

(u | v)0,s =

∫

Rn

|ξ|2su(ξ)u(ξ)dξ and | u |0,s=√

(u | u)0,s. (2.1)

We have the following result

Proposition 2.1 Suppose that s < n/2. Then the space Hs(Rn), endowed with the scalarproduct ( . | . )0,s is a Hilbert space, contained in S ′(Rn) with continuous injection.

Proof.see [20, 3]. 2

For any integer m ≥ 1 and for any real s, we consider the Beppo-Levi space [20, 16] associatedto the space Hs(Rn) denoted by

Xm,s(Rn) =v ∈ D′(Rn) : ∀α ∈ Nn, | α |= m, ∂αv ∈ Hs(Rn)

.


The space Xm,s(Rn) is equipped with the following semi-scalar product together with its semi-norm given by

(u | v)m,s =∑

|α|=m

m!

α!

∫

Rn

| ξ |2s F(∂αu)(ξ)F(∂αv)(ξ)dξ,

| u |m,s =√

(u | u)m,s,

(2.2)

for all u, v ∈ Xm,s(Rn). We have the following result

Proposition 2.2 Suppose that s < n/2. The space Xm,s(Rn) has the following properties

1- | u |m,s= 0 ⇐⇒ u ∈ Πm−1(Rn)

2- The space Xm,s(Rn), endowed with the semi-scalar product ( . | . )m,s is a semi-Hilbertspace and it is contained in S ′(Rn).

Proof. See [20, 3]. 2

Hereafter and in all the remainder of this paper, we assume that m ∈ N? and s ∈ R satisfythe hypothesis

−m +n

2< s <

n

2. (2.3)

Let N be a positive integer and let L : Xm,s(Rn) → RN denote a linear operator such that[L(p) = 0 and p ∈ Πm−1(R

n)]

=⇒ p ≡ 0, (2.4)

and consider the scalar product

(u | v)L,m,s = (u | v)m,s + 〈Lu|Lv〉RN (2.5)

defined on Xm,s(Rn). Its associated norm is denoted by || . ||L,m,s. Here 〈 .|. 〉RN denotes theusual Euclidian scalar product in RN .

Proposition 2.3 The space Xm,s(Rn) endowed with the scalar product given by (2.5) has thefollowing properties

1- The space Xm,s(Rn) is a Hilbert space and its topology is independent of the choice of L.

2- The following continuous inclusions hold,

Xm,s(Rn) → S ′(Rn), Xm,s(Rn) → Hm+sloc (Rn),

and the last one implies that Xm,s(Rn) → Ck(Rn) for all integer k such that n2 +k < m+s.

3- The space D(Rn) + Πm−1(Rn) is dense in Xm,s(Rn).

Proof. See [20, 3]. 2

Let s ∈ R such that (2.3) holds and set ν = (m + 1) + s − n/2. We consider the functionKm,s defined by

Km,s(x) =

cν |x|

2ν log(|x|) if 2ν ∈ 2N∗

cν |x|2ν if 2ν 6∈ 2N∗ (2.6)

It is well known that the function Km,s defines a tempered distribution on Rn also denoted byKm,s.


Remark 2.1 The exact value of the constant cν appearing in the expression (2.6) is of purelytheoretical importance. In fact, it is such that the Fourier transform Km,s satisfies the followingrelation

|ξ|2(m+1)+2sKm,s(ξ) = 1. (2.7)

The constant cν is not needed in practice for the computation of the spline.

The Fourier transform of Km,s is given by (see [25])

Km,s(ξ) =

Fp

[| ξ |−2(m+1)−2s

]+ c∆νδ if 2ν ∈ 2N?,

Fp

[| ξ |−2(m+1)−2s

]if 2ν 6∈ 2N?,

(2.8)

where c is a real constant. We recall that the notation Fp

[| ξ |−2(m+1)−2s

]stands for the finite

part of the distribution | ξ |−2(m+1)−2s and

〈Fp

[| ξ |−2(m+1)−2s

], ϕ〉 = finite part of

∫

Rn

| ξ |−2(m+1)−2s ϕ(ξ)dξ,

for all function ϕ ∈ D(Rn). We also recall that a compact support measure µ is said to beorthogonal to the space Πm−1(R

n) if it satisfies the following condition

〈µ, q〉 :=

∫

Rn

q(x)dµ(x) = 0, ∀q ∈ Πm−1(Rn). (2.9)

We have the following proposition

Proposition 2.4 For any measure µ with compact support orthogonal to the space Πm−1(Rn),

the function ∂α(µ ∗ Km,s) belongs to Xm,s(Rn) for all multi-index α such that |α| = 2.

Proof. We will prove that the tempered distribution F [∂α(µ ∗ Km,s)](ξ) belongs to L1loc(R

n)and | ξ |s F [∂α(µ ∗ Km,s)](ξ) belongs to L2(Rn) for all multi-index α such that |α| = m + 2.The Fourier transform of Km,s is given by the expression (2.8). Since the measure µ is withcompact support, then its Fourier transform µ is a C∞ function on Rn. Let h denote thedistribution obtained by the product of the C∞ function ξ 7→ (iξ)αµ(ξ) by the distribution ∆νδ,with 2ν = 2(m + 1) + 2s − n ∈ 2N?. We have

h = (iξ)αµ(ξ)∆νδ =∑

|β|=ν

ν!

β!µ(ξ)(iξ)α∂2βδ.

According to Schwartz [25, p122], we have ξα∂2βδ = (−1)2|β|−|α|(2β)!(2β−α)! ∂2β−αδ if α ≤ 2β and

ξα∂2βδ = 0 otherwise. Since

F [∂α(µ ∗ Km,s)](ξ) = (iξ)αµ(ξ)Km+1,s(ξ),

then there exist constants cα,β such that

〈h, ϕ〉 =∑

|β|=να≤2β

cα,β∂2β−α(µϕ)(0), ∀ϕ ∈ D(Rn).


Using Leibniz’s formula, we obtain that

〈h, ϕ〉 =∑

|β|=να≤2β

∑

γ≤2β−α

cα,β,γ∂γµ(0)∂2β−α−γϕ(0), ∀ϕ ∈ D(Rn) (2.10)

where cα,β,γ are constants. In the relation (2.10) the multi-index γ satisfies |γ| ≤ 2|β| − |α| =2ν − |α| = m + 2s−n. But s < n

2 , it follows that |γ| ≤ m− 1. As µ is orthogonal to Πm−1(Rn),

then all the derivatives of order | γ |≤ m − 1 vanish at the origin (∂γµ)(0) = 0. Finally, thedistribution h is null. It follows that the Fourier transform of ∂α(Km,s ∗ µ) is given by

F [∂α(µ ∗ Km,s)](ξ) = C (iξ)α µ(ξ)Fp

[| ξ |−2(m+1)−2s

](2.11)

Using again the property that µ has all the derivatives of order ≤ m− 1 null at the origin, thenthere exists a constant C > 0 such that in a neighborhood N of the origin we have

| µ(ξ) |≤ C| ξ |m.

It follows that in a neighborhood N of the origin we get

∃C > 0, | ξ |−2(m+1)−2s | ξαµ(ξ) |≤ C| ξ |−2s.

As s < n2 , the distribution F [∂α(µ ∗ Km,s)] is, in fact, a locally integrable function, the finite

part symbol Fp in (2.11) is useless. Thus

∫

N| ξ |2s| F [∂α(µ ∗ Km,s)] |

2 dξ ≤ C

∫

N| ξ |−2sdξ (2.12)

The last integral in the right hand side of (2.12) is finite, because 2s < n.Now, the Fourier transform of a compactly supported measure is a bounded function, thus

in the outside of a neighborhood N of the origin, we have∫

Rn\N| ξ |2s| F [∂α(µ ∗ Km,s)] |

2 dξ ≤ C

∫

Rn\N| ξ |−2m−2sdξ. (2.13)

The last integral in the right hand side of (2.13) is finite, because 2m + 2s > n. Finally, thefunction | ξ |sF [∂α(µ ∗ Km,s)] is square integrable on Rn. 2

3 The vector functional space Xm,s(Rn; Rn)

Now, we define the following space

Xm,s(Rn; Rn) = (Xm,s(Rn))n,

with n = 2, 3, endowed with the semi-scalar product

(u | v)m,s =n∑

i=1

(ui | vi)m,s (3.1)

and its associated semi-norm

| u |m,s=√

(u | u)m,s (3.2)


Let N be a positive integer and let L : Xm,s(Rn; Rn) → RN×n denote a linear operator suchthat [

L(p) = 0 and p ∈ Πm−1(Rn; Rn)

]=⇒ p ≡ 0. (3.3)

The set RN×n is the space of real matrices of size N × n. Let 〈 .|. 〉RN×n denote the usual scalarproduct defined in RN×n by

〈X|Y 〉RN×n =n∑

j=1

XTj .Yj ,

where Xj = (X1j , · · · , XNj)T denotes the jth column of the matrix X.

Now, we consider the following scalar product defined in Xm,s(Rn; Rn) by

(u | v)L,m,s = (u | v)m,s + 〈Lu|Lv〉RN×n . (3.4)

Its associated norm is denoted by || . ||L,m,s.The following proposition is an immediate consequence of Proposition 2.3

Proposition 3.1 The space Xm,s(Rn; Rn) endowed with the scalar product given by (3.4) hasthe following properties

1- The space Xm,s(Rn; Rn) is a Hilbert space and its topology is independent of the choice ofL.

2- The following continuous inclusions hold,

Xm,s(Rn; Rn) → S ′(Rn; Rn), Xm,s(Rn; Rn) → Hm+sloc (Rn; Rn),

and the last one implies that Xm,s(Rn; Rn) → Ck(Rn; Rn) for all integer k such thatn2 + k < m + s.

3- The space D(Rn; Rn) + Πm−1(Rn; Rn) is dense in Xm,s(Rn; Rn).

Let us now introduce some operators which appear naturally when we study the div-curlapproximation problem. Let

div u = ∇ · u =

n∑

i=1

∂iui (3.5)

and

rot u = ∇× u =

(0, ∂1u2 − ∂2u1) for n = 2(∂2u3 − ∂3u2, ∂3u1 − ∂1u3, ∂1u2 − ∂2u1) for n = 3

(3.6)

denote the divergence and the rotational operators, respectively. The notation ∇ = (∂1, · · · , ∂n)T

stands for the nabla operator.Let a and b denote positive real parameters. We consider the bilinear forms Dm,s, Rm,s

and Ma,bm,s defined on Xm,s(Rn; Rn) as follows, for all u = (u1, . . . , un) and v = (v1, . . . , vn) in

Xm,s(Rn; Rn),Dm,s(u, v) = (div u|div v)m−1,s,

Rm,s(u, v) =n∑

i=1

((rot u)i|(rot v)i)m−1,s,

Ma,bm,s(u, v) = aDm,s(u, v) + bRm,s(u, v).

(3.7)


The associated quadratic forms denoted by Dm,s(u) = Dm,s(u, u), Rm,s(u) = Rm,s(u, u) and

Ma,bm,s(u) = Ma,b

m,s(u, u) are called the div energy, the curl energy and the div-curl energy, respec-tively. The semi-norm (. | .)m−1,s is given in (2.2).

Let us consider the following matrix-polynomials defined by

Pdiv(ξ) = ξ.ξT = (ξkξl)1≤k,l≤n,Pcurl(ξ) = |ξ|2In − ξ.ξT = (δk,l|ξ|

2 − ξkξl)1≤k,l≤n,Pa,b(ξ) = aPdiv(ξ) + bPcurl(ξ),

(3.8)

for all ξ = (ξ1, · · · , ξn)T ∈ Rn, where In denotes the matrix identity of size n × n. It is easy tocheck that the matrix-polynomials Pdiv and Pcurl satisfy the following relations

Pdiv(ξ)Pcurl(ξ) = 0n,Pdiv(ξ)Pdiv(ξ) = |ξ|2 Pdiv(ξ),

Pcurl(ξ)Pcurl(ξ) = |ξ|2 Pcurl(ξ),

where 0n denotes the zero matrix of size n × n. An immediate consequence is that, for all realnumbers a 6= 0 and b 6= 0, we have

Pa,b(ξ)P 1

a, 1b(ξ) = |ξ|4 In. (3.9)

The associated differential matrix-operators are defined as following

Pdiv(i∇) = (i∇).(i∇)T = −∇.∇T = (−∂2k,l)1≤k,l≤n,

Pcurl(i∇) = (i∇)T .(i∇)In − (i∇).(i∇)T = −∆In + ∇.∇T = (−δk,l∆ + ∂2k,l)1≤k,l≤n,

Pa,b(i∇) = aPdiv(i∇) + bPcurl(i∇),(3.10)

and satisfy the following relations

Pdiv(i∇)Pcurl(i∇) = Pcurl(i∇)Pdiv(i∇) = 0n,Pdiv(i∇)Pdiv(i∇) = ∆ Pdiv(i∇) = Pdiv(i∇)∆,Pcurl(i∇)Pcurl(i∇) = ∆ Pcurl(i∇) = Pcurl(i∇)∆,Pa,b(i∇)P 1

a, 1b(i∇) = ∆2 In, for ab 6= 0.

For a vector tempered distribution u ∈ S ′(Rn; Rn), we have the following relations

F [Pdiv(i∇)u] = Pdiv(ξ)u,F [Pcurl(i∇)u] = Pcurl(ξ)u,F [Pa,b(i∇)u] = Pa,b(ξ)u,

(3.11)

We have the following proposition

Proposition 3.2 For all u ∈ Xm,s(Rn; Rn) and ϕ ∈ D(Rn; Rn) we have

Dm,s(u, ϕ) = 〈| ξ |2(m−1)+2s Pdiv(ξ)u, ϕ〉,

Rm,s(u, ϕ) = 〈| ξ |2(m−1)+2s Pcurl(ξ)u, ϕ〉,

Ma,bm,s(u, ϕ) = 〈| ξ |2(m−1)+2s Pa,b(ξ)u, ϕ〉.

(3.12)


Proof. For all ϕ, φ ∈ D(Rn; Rn) and for all p ∈ Πm−1(Rn; Rn) we have Dm,s(φ + p, ϕ) =

Dm,s(φ, ϕ). By using the definition of the bilinear form Dm,s, we have

Dm,s(φ, ϕ) =∑

|α|=m−1

(m − 1)!

α!

∫

Rn

| ξ |2s div (∂αφ)(ξ) div( ∂αϕ)(ξ)dξ

=∑

|α|=m−1

(m − 1)!

α!

∫

Rn

| ξ |2s (iξ)α(div φ)(ξ)(iξ)α(div ϕ)(ξ)dξ

=

∫

Rn

| ξ |2s (∑

|α|=m−1

(m − 1)!

α!ξ2α)(div ϕ)(ξ)(div ϕ)(ξ)dξ

=

∫

Rn

| ξ |2(m−1+s) F(div φ)(ξ)F(div ϕ)(ξ)dξ.

Since

(div φ)(ξ)(div ϕ)(ξ) =( n∑

k=1

iξkφk(ξ))( n∑

l=1

iξlϕl(ξ))

=n∑

l=1

( n∑

k=1

ξlξkφk(ξ))ϕl(ξ) = Pdiv(ξ)φ(ξ)ϕ(ξ).

We obtain

Dm,s(φ + p, ϕ) =

∫

Rn

| ξ |2(m−1+s) Pdiv(ξ)φ(ξ)ϕ(ξ)dξ

= 〈| ξ |2(m−1+s) Pdiv(ξ)φ, ϕ〉.

But a polynomial p ∈ Πm−1(Rn; Rn) satisfies the following relation

| ξ |2(m−1+s) Pdiv(ξ)p =| ξ |2s F [Pdiv(i∇)∆m−1p] = 0.

Thus the following relation holds

Dm,s(φ + p, ϕ) = 〈| ξ |2(m−1+s) Pdiv(ξ) (φ + p), ϕ〉.

According to Proposition 3.2, the density of D(Rn; Rn)+Πm−1(Rn; Rn) in Xm,s(Rn; Rn) together

with the continuous imbedding of Xm,s(Rn; Rn) in S ′(Rn; Rn) prove that

Dm,s(u, ϕ) = 〈| ξ |2(m−1+s) Pdiv(ξ)u, ϕ〉,

for all u ∈ Xm,s(Rn; Rn) and for all ϕ ∈ D(Rn; Rn).The other relations may be obtained by similar arguments. 2

The following proposition will be useful in Section 4.

Proposition 3.3 For all u, v ∈ Xm,s(Rn; Rn), we have

M1,1m,s(u, v) = (u | v)m,s (3.13)

and for all positive real numbers a and b

inf (a, b) | u |2m,s≤ Ma,bm,s(u) ≤ sup (a, b) | u |2m,s (3.14)


Proof. For all u ∈ Xm,s(Rn; Rn), for all ϕ ∈ D(Rn; Rn) and for all p ∈ Πm−1(Rn; Rn) we have

M1,1m,s(u, ϕ + p) = M1,1

m,s(u, ϕ). According to (3.12) we obtain

M1,1m,s(u, ϕ) = 〈| ξ |2(m−1)+2s P1,1(ξ)u, ϕ〉.

Since P1,1(ξ) = Pdiv(ξ) + Pcurl(ξ) = |ξ|2In, it follows that

M1,1m,s(u, ϕ) = 〈| ξ |2m+2s u, ϕ〉 = (u|ϕ)m,s = (u|ϕ + p)m,s,

which together with the density property give the equation (3.13).Since

inf (a, b)M1,1m,s(u) ≤ Ma,b

m,s(u) ≤ sup (a, b)M1,1m,s(u)

we obtain the inequality (3.14). 2

Let us now introduce the matrix-function F a,bm,s defined by

F a,bm,s(t) = P 1

a, 1b(i∇)Km,s(x) = −P 1

a, 1b(∇)Km,s(x)

=(−

1

bδk,j∆Km,s(x) + (

1

b−

1

a)∂2

k,jKm,s(x))

1≤k,j≤n

(3.15)

where Km,s is the function defined in (2.6) and P 1

a, 1b(i∇) is the differential matrix-operator given

by (3.10).A vector-measure ω = (ω1, · · · , ωn) with compact support is said to be orthogonal to the

space Πm−1(Rn; Rn) if it satisfies the following condition

〈ω, q〉 :=n∑

k=1

〈ωk, qk〉 =n∑

k=1

∫

Rn

qk(x)dωk(x) = 0, (3.16)

for all q = (q1, · · · , qn) ∈ Πm−1(Rn; Rn).

Proposition 3.4 For any vector-measure ω = (ω1, · · · , ωn) with compact support and orthogo-

nal to the space Πm−1(Rn; Rn) the convolution matrix-product F a,b

m,s ∗ ω has the following prop-erties

1- F a,bm,s ∗ ω ∈ Xm,s(Rn; Rn).

2- For all u = (u1, · · · , un) ∈ Xm,s(Rn; Rn)

Ma,bm,s(F

a,bm,s ∗ ω, u) = 〈ω, u〉 =

n∑

k=1

〈ωk, uk〉. (3.17)

Proof.

1- The expression (3.15) shows that F a,bm,s ∗ ω is obtained by using the second derivatives of

ωk ∗ Km,s with ω = (ω1, · · · , ωn). Thus Proposition 2.4 implies the result.


2- For all u ∈ Xm,s(Rn; Rn), for all ϕ ∈ D(Rn; Rn) and for all polynomial p in Πm−1(Rn; Rn)

we have Ma,bm,s(F

a,bm,s ∗ ω, ϕ + p) = Ma,b

m,s(Fa,bm,s ∗ ω, ϕ). According to Proposition 3.2, we get

Ma,bm,s(F

a,bm,s ∗ ω, ϕ + p) = 〈| ξ |2(m−1)+2s Pa,b(ξ)F(F a,b

m,s ∗ ω), ϕ〉.

By using the relation (3.11) together with the Fourier transform of a convolution product

we obtain that the Fourier transform of F a,bm,s ∗ ω is given by

F(F a,bm,s ∗ ω) = P 1

a, 1b(ξ)Km,sω.

Furthermore, by the relation (3.9), we get

Ma,bm,s(F

a,bm,s ∗ ω, ϕ + p) = 〈| ξ |2(m−1)+2s| ξ |4 InKm,sω, ϕ〉

= 〈| ξ |2(m+1)+2s Km,sω, ϕ〉.

By taking the relation (2.7) into account and according to the relation 〈ω, ϕ〉 = 〈ω, ϕ〉 weobtain that

Ma,bm,s(F

a,bm,s ∗ ω, ϕ + p) = 〈ω, ϕ〉.

Since ω is orthogonal to Πm−1(Rn; Rn), we have 〈ω, p〉 = 0. Thus

Ma,bm,s(F

a,bm,s ∗ ω, ϕ + p) = 〈ω, ϕ + p〉.

The density of D(Rn; Rn) + Πm−1(Rn; Rn) in Xm,s(Rn; Rn) gives the desired result.

2

4 Pseudo-polyharmonic div-curl splines

Let ΩN = x1, · · · , xN be a subset of distinct points in Rn satisfying the Πm−1(Rn)-unisolvance

condition, which means that[p(xi) = 0, and p ∈ Πm−1(R

n)]⇒ p ≡ 0.

Let us consider the operator A : Xm,s(Rn; Rn) → RN×n given by

Au =

u1(x1) · · · un(x1)...

...u1(xN ) · · · un(xN )

.

We give the following definition

Definition 4.1 For all Z ∈ RN×n, a > 0, b > 0 and ε ≥ 0 we define a div-curl spline functionas a solution σa,b,ε of the following approximation problem

(DC − Pa,b,ε)(Z) : minv∈Im,s

ε (Z)

(Ma,b

m,s(v) + ε||Av − Z||2RN×n

), (4.1)

where

Im,sε (Z) =

A−1(Z) for ε = 0 (Interpolating Problem)Xm,s(Rn; Rn) for ε > 0 (Smoothing Problem)

(4.2)

A−1(Z) = v ∈ Xm,s(Rn; Rn) : A v = Z (4.3)


Remark 4.1 Depending on the approximating problem we want to consider, we can choose theenergy Ma,b

m,s in different ways

1- In order to obtain an energy which depends only on the ratio ρ = ab , it is sufficient to

divide the energy Ma,bm,s(u) by b. The obtained energy M1,ρ

m,s(u) allows the control of thedivergence and the curl by the parameter ρ.

2- By dividing the energy Ma,bm,s(u) by a + b and by setting ρ = a

a+b ∈]0, 1[ one obtains the

energy Mρ,1−ρm,s (u) which is a convex tradeoff between the div-approximation and the curl-

approximation.

We have the following theorem

Theorem 4.1 For all Z ∈ RN×n, a > 0, b > 0 and ε ≥ 0, there is a unique solution σa,b,ε ofthe problem (4.1). The solution σa,b,ε is the unique element of Xm,s(Rn; Rn) characterized by

Ma,bm,s(σ

a,b,0, v) = 0, ∀v ∈ ker(A), (4.4)

for the interpolating problem (ε = 0), and

Ma,bm,s(σ

a,b,ε, v) + ε〈Aσa,b,ε − Z | Av〉RN×n = 0, ∀v ∈ Xm,s(Rn; Rn), (4.5)

for the smoothing problem (ε > 0). Furthermore, there exists a unique vector-measure ωa,b,ε of

the form ωa,b,ε = (∑N

i=1 λa,b,εi1 δxi

, . . . ,∑N

i=1 λa,b,εin δxi

) orthogonal to the space Πm−1(Rn; Rn) and

satisfying

Ma,bm,s(σ

a,b,ε, v) = 〈ωa,b,ε, v〉 =

n∑

j=1

N∑

i=1

λa,b,εij vj(xi), ∀v ∈ Xm,s(Rn; Rn). (4.6)

The coefficients λa,b,εij are real numbers.

Proof. According to the inequality (3.14) in Proposition 3.3, the symmetric positive bilinear

form Ma,bm,s is continuous. Then, there exists a positive and symmetric continuous linear operator

S : Xm,s(Rn; Rn) → Xm,s(Rn; Rn) such that

Ma,bm,s(u, v) = (Su | v)m,s,

for all u, v ∈ Xm,s(Rn; Rn). The operator S admits a symmetric positive square-root. Namely,there exists a symmetric and positive continuous linear operator T : Xm,s(Rn; Rn) → Xm,s(Rn; Rn)such that S = T 2. In consequence, we have

Mα,βm,s(u) = Mα,β

m,s(u, u) = (T 2u | u)m,s = (Tu | Tu)m,s =| Tu |2m,s,

for all u ∈ Xm,s(Rn; Rn).The operators A and T satisfy the following properties:


1- A is continuous and surjective: The continuity of A is a consequence of the hypothesis(2.3) and the continuous imbedding Sobolev’s Theorem (see Proposition 3.1). We haveXm,s(Rn; Rn) → C0(Rn; Rn) for n

2 < m+s. Since the points x1, · · · , xN are distincts, thenit is possible to find functions φ1, . . . , φN in D(Rn) such that the following condition

φj(xi) = δij , i, j = 1, · · · , N (4.7)

holds. For all Z = (Zi,j) 1≤i≤N1≤j≤n

∈ RN×n, we define the functions Φj ∈ D(Rn) given by

Φj(x) =N∑

i=1

zi,jφi for 1 ≤ j ≤ n where the functions φi are given by (4.7). Then, the

function ΦZ = (Φ1, · · · , Φn) is an element of D(Rn; Rn) satisfying AΦZ = Z.

2- ker(T ) = Πm−1(Rn; Rn) and T (Xm,s(Rn; Rn)) is closed: It is a consequence of Proposition

2.2 and the inequality (3.14).

3- ker(T )+ker(A) is closed: It is a consequence of the fact that ker(A) is closed and ker(T ) =Πm−1(R

n; Rn) is a finite dimensional space.

4- ker(T )∩ker(A) = 0: Let u = (ui)1≤i≤n ∈ ker(T )∩ker(A), then u ∈ Πm−1(Rn; Rn) and

ui(xj) = 0 for i = 1, · · · , n and j = 1, · · · , N . The Πm−1(Rn)-unisolvence of x1, · · · , xN

implies that ui = 0 for i = 1, · · · , n.

According to the general spline theory (see [4, 9, 22]) the variational problem (4.1) has a

unique solution σa,b,ε. Since (Tσa,b,ε|Tv)m,s = Ma,bm,s(σa,b,ε, v), the solution σa,b,ε satisfies the

characterization given by (4.4) and (4.5). The characterization (4.6) is obtained by similararguments as in the proof of Proposition 3.2 in [7]. 2

Theorem 4.2 There are unique vectors Λa,b,εi ∈ Rn for i = 1, ..., N , and a unique polynomial

pa,b,ε ∈ Πm−1(Rn; Rn) such that the unique solution σa,b,ε of the problem (4.1) is explicitly given

by

σa,b,ε(x) =N∑

i=1

F a,bm,s(x − xi)Λ

a,b,εi + pa,b,ε(x), ∀x ∈ Rn. (4.8)

Furthermore, the solution σa,b,ε has the following properties

1- σa,b,ε ∈ Cη(Rn; Rn) where

η =

[2m + 2s − n] for 2m + 2s − n 6∈ 2N?

2m + 2s − n − 1 otherwise,

where the notation [r] stands for the integer part of the real r.

2- limε→+∞

σa,b,ε = σa,b,0 and limε→0

σa,b,ε = p0 in Xm,s(Rn; Rn), where p0 ∈ Πm−1(Rn; Rn) is the

unique solution of the mean-square problem

P0(Z) : minp∈Πm−1(Rn;Rn)

||Ap − Z||2RN×n .


Proof. Using Theorem 4.1, there exists a unique vector-measure ωa,b,ε of the form ωa,b,ε =(∑N

j=1 λa,b,ε1j δxj

, · · · ,∑N

j=1 λa,b,εnj δxj

) which is orthogonal to the space Πm−1(Rn; Rn) and which

satisfies the condition (4.6).

Let %a,b,ε = F a,bm,s ∗ ωa,b,ε. Proposition 3.4 implies that %a,b,ε ∈ Xm,s(Rn; Rn) and satisfies

Ma,bm,s(%

a,b,ε, v) = Ma,bm,s(F

a,bm,s ∗ ωa,b,ε, v) = 〈ωa,b,ε, v〉,

for all v ∈ Xm,s(Rn; Rn). Hence, with respect to the condition (4.6), we obtain

Ma,bm,s(%

a,b,ε, v) = Ma,bm,s(F

a,bm,s ∗ ωa,b,ε, v) = Ma,b

m,s(σa,b,ε, v),

for all v ∈ Xm,s(Rn; Rn). Thus

Ma,bm,s(σ

a,b,ε − %a,b,ε, v) = 0,

for all v ∈ Xm,s(Rn; Rn). In particular, by putting v = σa,b,ε − %a,b,ε , we get

Mα,βm,s(σ

a,b,ε − %a,b,ε) = 0.

It follows thatσa,b,ε − %a,b,ε = pa,b,ε ∈ Πm−1(R

n; Rn).

It is easy to see that

%a,b,ε(x) = F a,bm,s ∗ ωa,b,ε(x) =

N∑

i=1


a,b,εi ,

where the vectors Λa,b,εi = (λa,b,ε

i1 , · · · , λa,b,εin )T are in Rn, for i = 1, · · · , N , and the coefficients

λij are given in Theorem 4.1.For the properties 1) and 2) we have

1- It is clear that the function Km,s is C∞(Rn \ 0). It was shown in [3] that ∂αKm,s(0) = 0for | α |≤ η + 2, then Km,s ∈ Cη+2(Rn; Rn). Since σa,b,ε is obtained using the secondderivatives of Km,s (see (3.15) and (4.9)), we get the result.

2- See [4].

2

The computation method of the solution of the problem (4.1) is given in the followingtheorem.

Theorem 4.3 Let Z = (zij) 1≤i≤N1≤j≤n

∈ RN×n. Let d = (n+m−1)!n!(m−1)! be the dimension and (q1, · · · , qd)

be a basis of the space Πm−1(Rn), respectively. The solution σa,b,ε = (σa,b,ε

1 , . . . , σa,b,εn ) of the

problem (4.1) is explicitly given by

σa,b,εk (x) =

n∑

j=1

N∑

i=1

λa,b,εi,j

[−

1

bδj,k∆Km,s(x − xi) + (

1

b−

1

a)∂2

j,kKm,s(x − xi)]

+

d∑

l=1

αa,b,εl,k ql(x) for k = 1, . . . , n.

(4.9)


The coefficients λa,b,εi,j and αa,b,ε

l,k are computed by solving the following nonsingular linear systemof size n(N + d) × n(N + d)

(K + cεInN M

MT O

)(Λa,b,ε

αa,b,ε

)=

(z0

)with cε =

0 if ε = 0,1

εif ε > 0,

(4.10)

where Λa,b,ε and αa,b,ε are the vectors given by

Λa,b,ε = (λa,b,ε1,1 , . . . , λa,b,ε

N,1 , . . . , λa,b,ε1,n , . . . , λa,b,ε

N,n )T ∈ RnN ,

αa,b,ε = (αa,b,ε1,1 , . . . , αa,b,ε

d,1 , . . . , αa,b,ε1,n , . . . , αa,b,ε

d,n )T ∈ Rnd.

The vector z = (z1,1, . . . , zN,1, . . . , z1,n, . . . , zN,n)T ∈ RnN is obtained by stacking the columnsof the matrix Z and InN is nN -unit matrix. The matrices K = (Kl,k)1≤l,k≤n of size nN × nNand M = (Ml,k)1≤l,k≤n of size nN × nd are given by the blocks Kl,k and Ml,k, respectively. Theblocks Kl,k and Ml,k are given by

Kl,k =

[−

1

bδl,k∆Km,s(xi − xj) + (

1

b−

1

a)∂2

l,kKm,s(xi − xj)

]

1≤i≤N1≤j≤N

Ml,k = [δl,kqj(xi)] 1≤j≤d1≤i≤N

.

Proof. From Theorem 4.2 we have the explicit expression (4.9). With respect to the interpolating(resp. smoothing) conditions together with the orthogonality conditions we obtain the linearsystem (4.10). 2

5 Elastic splines with linear elasticity

In this Section, we will show how the div-curl splines can arise in the linear elasticity problem.In elasticity theory the strain tensor is given by

Eu = (Ei,ju)1≤i,j≤n =1

2(∇u + (∇u)T ) =

1

2(∂iuj + ∂jui)1≤i,j≤n.

We consider the bilinear forms Sm,s and Eµ,λm,s defined on Xm,s(Rn; Rn) as follows

Sm,s(u, v) =

n∑

i,j=1

(Ei,ju | Ei,jv)m−1,s,

Eµ,λm,s(u, v) = 2µSm,s(u, v) + λDm,s(u, v),

(5.1)

for all u = (u1, . . . , un) and v = (v1, . . . , vn) in Xm,s(Rn; Rn). The associated quadratic forms

are denoted by Sm,s(u) = Sm,s(u, u) and Eµ,λm,s(u) = Eµ,λ

m,s(u, u). The form Eµ,λm,s is the energy

stored in the body and the real numbers µ and λ are called the Lame constants of the isotropicbody (see [8, 14]).

We give the following definition

Definition 5.1 For all Z ∈ RN×n, µ, λ and ε ≥ 0 we define an elastic spline function as asolution θµ,λ,ε of the following approximation problem

(E − Pµ,λ,ε)(Z) : minv∈Im,s

ε (Z)

(Eµ,λ

m,s(v) + ε||Av − Z||2RN×n

), (5.2)

where Im,sε (Z) is given by (4.2).


The problem of elastic splines corresponding to the particular case s = 0 was studied in[5, 6]. The next theorem gives, on the one hand, the connection between the div-curl energyand the elastic energy and in the other hand, it gives the connection between the elastic splineand the div-curl spline.

Theorem 5.1 1- For all u in Xm,s(Rn; Rn), and for all real numbers a, b, λ and µ we have

Sm,s(u) = Dm,s(u) +1

2Rm,s(u), (5.3)

Eµ,λm,s(u) = M2µ+λ,µ

m,s (u) (5.4)

andMa,b

m,s(u) = E2b,a−2bm,s (u). (5.5)

2- The elastic spline θµ,λ,ε (rep. the div-curl spline σa,b,ε) relatively to the parameters µ andλ (resp. a and b) is the div-curl spline σ2µ+λ,µ,ε (rep. the elastic spline θ2b,a−2b,ε) relativelyto the parameters a = 2µ + λ and b = µ(resp. µ = 2b and λ = a − 2b), namely

θµ,λ,ε = σ2µ+λ,µ,ε (resp. σa,b,ε = θ2b,a−2b,ε).

Proof.

1- For all ϕ ∈ D(Rn; Rn) and for all p ∈ Πm−1(Rn; Rn) we have Sm,s(ϕ + p) = Sm,s(ϕ). By

using the definition of the quadratic form Sm,s, we have

Sm,s(ϕ) =∑

|α|=m−1

(m − 1)!

α!

∫

Rn

| ξ |2sn∑

k,l=1

Ek,lϕ(ξ)Ek,lϕ(ξ)dξ.

Thanks to the properties of the Fourier transform, we can reformulate the following sumas

n∑

k,l=1

Ek,lϕEk,lϕ =1

4

n∑

k,l=1

(iξkϕl + iξlϕk)(−iξkϕl − iξlϕk)

=1

2

n∑

k=1

| ξ |2 ϕkϕk +1

2

n∑

k=1

(n∑

l=1

ξkξlϕl)ϕk.

It follows thatn∑

k,l=1

Ek,lϕEk,lϕ =1

2(| ξ |2 Inϕ) · ϕ +

1

2(Pdiv(ξ)ϕ) · ϕ

=1

2

[| ξ |2 In + Pdiv(ξ)

]ϕ · ϕ = (Pdiv(ξ) +

1

2Pcurl(ξ))ϕ · ϕ.

Then

Sm,s(φ) =∑

|α|=m−1

(m − 1)!

α!

∫

Rn

| ξ |2s (Pdiv(ξ) +1

2Pcurl(ξ))ϕ(ξ) · ϕ(ξ)dξ

=

∫

Rn

| ξ |2(m−1)+2s (Pdiv(ξ) +1

2Pcurl(ξ))ϕ(ξ) · ϕ(ξ)dξ

= 〈| ξ |2(m−1)+2s Pdiv(ξ)ϕ, ϕ〉 +1

2〈| ξ |2(m−1)+2s Pcurl(ξ)ϕ, ϕ〉

= Dm,s(φ) +1

2Rm,s(φ).


Thus, we have

Sm,s(φ + p) = Sm,s(φ) = Dm,s(φ) +1

2Rm,s(φ)

= Dm,s(φ + p) +1

2Rm,s(φ + p)

According to Proposition 2.3 and by using the density property, we obtain the relation(5.3). Hence, for all u in Xm,s(Rn; Rn), we have

Eµ,λm,s(u) = 2µSm,s(u) + λDm,s(u)

= (2µ + λ)Dm,s(u) + µRm,s(u) = M2µ+λ,µm,s (u).

and

Ma,bm,s(u) = aDm,s(u) + bRm,s(u) = aDm,s(u) + 2b(Sm,s(u) − Dm,s(u))

= 2bSm,s(u) + (a − 2b)Dm,s(u) = Eb,a−2bm,s (u).

2- It is a direct consequence of Definition 5.1 (resp. Definition 4.1) and the relation (5.4)(resp. (5.5)).

2

An immediate corollary of Theorem 4.1, Theorem 4.2 and the above theorem follows

Corollary 5.1 1- For all Z ∈ RN×n and all real numbers µ and λ such that µ > 0 and2µ + λ > 0, the problem (5.2) admits a unique solution θµ,λ,ε in Xm,s(Rn; Rn).

2- There are unique vectors Λµ,λ,εi ∈ Rn for i = 1, ..., N , and a unique polynomial pµ,λ,ε ∈

Πm−1(Rn; Rn) such that the unique solution θµ,λ,ε of the problem (5.2) is explicitly given

by

θµ,λ,ε(x) =N∑

i=1

F 2µ+λ,µm,s (x − xi)Λ

µ,λ,εi + pµ,λ,ε(x), ∀x ∈ Rn. (5.6)

Furthermore, the solution θµ,λ,ε has the following properties

2-1 θµ,λ,ε ∈ Cη(Rn; Rn) where

η =

[2m + 2s − n] for 2m + 2s − n 6∈ 2N?

2m + 2s − n − 1 otherwise

2-2 limε→+∞

θµ,λ,ε = θµ,λ,0 and limε→0

θµ,λ,ε = p0 in Xm,s(Rn; Rn), where p0 ∈ Πm−1(Rn; Rn)

is the unique solution of the mean-square problem

P0(Z) : minp∈Πm−1(Rn;Rn)

||Ap − Z||2RN×n .

It is clear that the computation of the elastic spline θµ,λ,ε solution of the problem (5.2) canbe formulated in a similar theorem to Theorem 4.3 relative to the computation of the div-curlspline σa,b,ε by setting a = µ and b = 2µ + λ.


6 Extensions

6.1 Hermite approximation

Under the assumption −m+1+ n2 < s < n

2 , the continuous imbedding inclusion Xm,s(Rn; Rn) →C1(Rn; Rn) holds. In this case, one can use the Hermite approximation as following. Let usconsider the operator A : Xm,s(Rn; Rn) → RN×n(n+1) given by

Au =

u1(x1) · · · un(x1) ∇u1(x1) · · · ∇un(x1)...

......

...u1(xN ) · · · un(xN ) ∇u1(xN ) · · · ∇un(xN )

, (6.1)

where∇uk(xi) = (∂1uk(xi), · · · , ∂nuk(xi)) ∈ Rn.

We can establish that the problem (4.1) with A is now given by (6.1) has also a unique solutionσa,b,ε with the expression

σa,b,ε(x) =

N∑

i=1


a,b,εi +

n∑

j=1

N∑

i=1

∂jFa,bm,s(x − xi)Γ

a,b,εi,j + P a,b,ε(x),

where Λi, Γi,j for i = 1, · · · , N and j = 1, · · · , n are vectors in Rn and P a,b,ε is a polynomial in

Πm−1(Rn; Rn). The notation ∂jF

a,bm,s stands for the matrix-function whose components are the

partial derivative ∂j of each components of the matrix-function F a,bm,s.

6.2 Div-curl approximation by radial basis function

In the relation (4.9), in stead of the function Km,s we can use any other function Φ which isradial and conditionally positive definite function on Rn of order m + 1 and of class at least C2

on Rn. In this case, we also obtain a system of the form (4.10) which is also nonsingular forthe both interpolating and smoothing cases. It is also possible to replace the function Km,s bythe shift-function φm,s(x) = Km,s(

√| x |2 +c2). But it is important to notice that in this case

the interpolating (or smoothing) resulted function is not derived from a minimization problemof certain energy in a some Hilbert space. For more details about the radial basis functions, wecan see for instance [11, 24].

7 Numerical results

In this section, we preset an example illustrating the approximation by pseudo-polyharmonicdiv-curl splines. We restrict ourselves to the case of interpolation problem. We consider a simplecase where the vector field to be interpolated is defined by the function f : R3 → R3 given by

f(x, y, z) = (2ye−(x2+y2+z2)/49,−2xe−(x2+y2+z2)/49, 1)

and restricted to the domain Ω = [−7, 7]× [−7, 7]× [−7, 7]. We interpolate the function f at Nrandom scattered data points ti = (xi, yi, zi) ∈ Ω, i = 1, · · · , N . We choose s = 1

2 and m = 2and we have n = 3. The condition −m + n

2 < s < n2 is satisfied and 2ν = 2(m + 1) + 2s−n = 4,


Figure 1: Stream of ribbons in the wind represented by the original function f

Figure 2: Stream of ribbons in the wind represented by σa,b,0 with a = 2.5, b = 1 and N = 50


Figure 3: Stream of ribbons in the wind represented by σa,b,0 with a = 1, b = 2.5 and N = 50

this imposes the use of the kernel Km,s(x, y, z) = r4 log(r) where r =√

x2 + y2 + z2. We recallthat in this case, the div-curl energy appearing in the problem (4.1) is more explicitly given asfollowing

Ma,b

2, 12

(u) = a

∫

R3

|∇(div u)(ξ)|2|ξ|dξ + b3∑

i=1

∫

R3

| ∇(rot u)i(ξ)|2|ξ|dξ.

The interpolating div-curl spline σa,b,0 is the vectorial function in X2, 12 (R3; R3) solution of the

interpolating problemmin

u∈I(f)Ma,b

2, 12

(u),

where I(f) = u ∈ X2, 12 (R3; R3) : u(ti) = f(ti), i = 1, · · · , N. Now we explain how to

represent a function from R3 to R3 in a quite significative way. We suppose that the value(u, v, w) = f(x, y, z) represent the velocity of a field of wind at the position (x, y, z). In orderto represent the vectorial function f we give some planer vector-values of f and we representsome streams of ribbons in the field of the wind represented by f . To compare a vector fieldg to f we compare only the behavior of the streams of ribbons in the wind represented by fand g, respectively. Figure 1 represents streams of ribbons in a wind represented by the originalfunction f .

Figure 2 and Figure 3 represent the div-curl spline σa,b,0 interpolating f at N = 50 pointswith a = 2.5, b = 1 and a = 1, b = 2.5, respectively.

Figure 4 and Figure 5 represent the div-curl spline σa,b,0 interpolating f at N = 800 pointswith a = 2.5, b = 1 and a = 1, b = 2.5, respectively.


Figure 4: Stream of ribbons in the wind represented by σa,b,0 with a = 2.5, b = 1 and N = 800

Figure 5: Stream of ribbons in the wind represented by σa,b,0 with a = 1, b = 2.5 and N = 800


Table 1 gives the relatives error R(a, b) = ||f − σa,b,0||1/||f ||1 between the original functionf and the interpolating div-curl spline σa,b,0 for different values of a, b and N .

N R(2.5,1) R(1,2.5)

50 1.45640e-1 2.43915e-1100 9.16660e-2 1.88718e-1200 7.25162e-2 1.68431e-1400 6.92226e-2 1.66165e-1800 6.88343e-2 1.65687e-1

Table 1. Relative Error

References

[1] L. Amodei and M. N. Benbourhim, A vector spline approximation, J. Approx. Th, vol 67,(1991), 51–79.

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Documents

Pseudo-polyharmonic div-curl splines and elastic splines