Trivariate spline quasi-interpolants based on simplex splines and polar forms

Accepted Manuscript Trivariate spline quasi-interpolants based on simplex splines and polar forms A. Serghini, A. Tijini PII: DOI: Reference:

S0378-4754(14)00307-3 http://dx.doi.org/10.1016/j.matcom.2014.11.008 MATCOM 4120

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Mathematics and Computers in Simulation

Received date: 4 March 2014 Revised date: 10 November 2014 Accepted date: 21 November 2014 Please cite this article as: A. Serghini, A. Tijini, Trivariate spline quasi-interpolants based on simplex splines and polar forms, Math. Comput. Simulation (2014), http://dx.doi.org/10.1016/j.matcom.2014.11.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Trivariate spline quasi-interpolants based on simplex splines and polar forms. ⋆ A. Serghini a University

a,∗

, A. Tijini a,

Mohammed I, ESTO, MATSI Laboratory, Oujda, Morocco.

Abstract In this work we describe an approximating scheme based on simplex splines on a tetrahedral partition using volumetric data. We use the trivariate simplex B-spline representation of polynomials or piecewise polynomials in terms of their polar forms to construct several differential or discrete quasi interpolants which have an optimal approximation order. Key words: Polar form, Quasi-interpolation, Simplex B-spline. PACS: 41A05, 41A25, 41A50, 65D05.

1

Introduction

Multivariate simplex splines for approximation theory have been extensively investigated in mathematical science for many years. Motivated by an idea of Schoenberg for a geometric interpretation of univariate B-splines, de Boor [8] first presented a brief description of multivariate simplex splines. Since then, their theoretical aspects have been explored extensively. From the point of view of blossoming (see [12,13,27], for instance), Dahmen et al. [7] proposed triangular B-splines called also DMS-splines. Greiner and Seidel [14] demonstrated the practical feasibility of multivariate B-splines algorithms in graphics and shape design. Pfeifle and Seidel [25,26] proposed a fast evaluation technique for a quadratic bivariate DMS-splines surface and demonstrated the fitting of triangular B-spline surfaces to scattered data through the use ⋆ Research supported by URAC-05. ∗ Corresponding author. Email addresses: [email protected] (A. Serghini ), [email protected] (A. Tijini).

Preprint submitted to Elsevier

10 November 2014

of least squares and optimization techniques. Others bivariate and trivariate spline spaces are studied and analyzed in [16]. In recent years, the reconstruction of volumetric data became an active area of research since it is important for many applications such as scientific visualization, medical imaging and solid modeling. To model volumetric objects with high-order continuity, techniques based on splines are frequently used. Nonetheless, modeling with B-splines have several shortcomings. Its modeling scope is extremely constrained in terms of geometric and topological aspects. The several existing studies (see [1,20,22,24], for instance) use the tensor product splines in three variables. Standard examples are trilinear, continuous splines, where the total degree of the polynomial pieces is three, as well as triquadratic and tricubic, smooth splines, where the total degree of the polynomial pieces are six and nine, respectively. A local and smooth model of the reconstruction of volume data, close to tensor-product schemes, is represented by blending sums of univariate and bivariate C 1 quadratic spline quasi-interpolants in [29–31]. To overcome the difficulties due to the trivariate tensor product splines, N¨ urnberger et al. [23] and T. Sorokina et al. [35] constructed on type-6 tetrahedral partition a trivariate quadratic and cubic piecewise polynomials quasiinterpolants which are based on trivariate splines of lowest possible degree. In these methods, the Bernstein-B´ezier coefficients of the spline quasi-interpolants are immediately available from the given values by applying local averaging. Despite the elegance of these methods, the obtained quasi-interpolants do not reproduce the whole space P2 (R3 ) of trivariate polynomials of degree less than or equal to 2, and they are only second order accurate. In [28], S. Remogna constructed new quasi-interpolation schemes based on the trivariate C 2 quartic box spline defined on a type-6 tetrahedral partition with a fourth approximation order. Spline quasi-interpolants are very useful approximants in practice. In general, a spline quasi-interpolant for a given function f is obtained as linear combination of some elements of a suitable set of basis functions, i.e; Qf =

X

µj (f )Bj ,

j∈J

where {Bj , j ∈ J } is the B-spline basis of some spline space. If µj (f ) is a linear combination of values of derivatives of f , the associated quasi-interpolant is called a differential quasi-interpolant (abbr. DQI) and if µj (f ) is a linear combination of discrete values of f , the associated quasi-interpolant is called a discrete quasi-interpolant (abbr. dQI) (see [30], for instance). In this paper, we use the trivariate simplex B-splines which are piecewise polynomials of the lowest possible degree and the highest possible smoothness everywhere for constructing discrete and differential simplex spline quasi-interpolants which 2

have an optimal approximation order. We extend the approach given in [32,33] to R3 and to simplex splines of any degree. It is based on the polar form of a chosen local polynomial approximant like a local interpolant or another approximant having the optimal approximation order. The paper is organized as follows. In Section 2 we give some definitions and properties of trivariate simplex B-splines. In Section 3 we introduce the B-spline representation of all trivariate polynomials or DMS-splines over a tetrahedral partition ∆ of a bounded domain D ⊂ R3 , in terms of their polar forms. In Section 4 we apply the approach introduced in [32] to trivariate DMS B-splines. Then we describe some differential and discrete simplex spline quasi interpolants in three variables which reproduce trivariate polynomials and provide the full approximation order in the space of trivariate simplex splines. Some interesting results concerning the quadratic case are developed in Section 5. In Section 6 we give some upper bounds of the infinity norms of some families of trivariate discrete quasi-interpolants. Finally, some numerical examples are proposed in Section 7.

2

Trivariate DMS-splines

In this subsection, we review some basic definitions and properties of trivariate simplex splines. The following results can be found in [2,3,5–7,14,15,21,26,34]. For any ordered set of affinely independent points W = {w0 , w1 , w2 , w3} of R3 , we define    1 1 1 1  d(W ) = det  . w0 w1 w2 w3

Let V = {v0 , · · · , vm } be an arbitrary set in R3 , we denote by [V ] = [v0 , · · · , vm ] the convex hull of the set V . if we denote by ei , i = 1, 2, 3 the unit vectors of R3 , the half-open convex hull of V is defined as [V ) = {x ∈ [V ]/∃ǫ1 , ǫ2 , ǫ3 , ∀0 ≤ α1 ≤ α2 ≤ α3 ≤ 1,

(x + α1 ǫ1 e1 + α2 ǫ2 e2 + α3 ǫ3 e3 ) ∈ [V ]} ,

i.e., x belongs to the half open convex hull [V ) of V , if there exists a small tetrahedron that lies entirely within the convex hull [V ]. The simplex B-spline M(x|V ) = M(x|v0 , · · · , vm ) is defined recursively as follows: When m = 3 we set M(x|V ) =

X[v0 ,v1 ,v2 ,v3 ) (x) , |d(V )|

(2.1)

where X[v0 ,v1 ,v2 ,v3 ) (x) is the characteristic function of the half-open convex hull 3

[v0 , v1 , v2 , v3 ). When m > 3, select a subset W = {vi0 , vi1 , vi2 , vi3 } of four points from V such that W is affinely independent, then M(x|V ) =

3 X

j=0

λj (x|W )M(x|V \ {vij }),

(2.2)

where λj (x), j = 0, . . . , 3 are the barycentric coordinates of x with respect to W , i.e. x=

3 X

λj (x)vij and

3 X

λj (x) = 1.

j=0

j=0

The function M(x|V ) is a positive piecewise polynomial of degree n = m − 3, supported in the convex hull [V ], and is C n−1 continuous everywhere. Let T = {∆(I) = [ti0 , ti1 , ti2 , ti3 ], I = (i0 , i1 , i2 , i3 ) ∈ I ⊂ Z4+ } be a tetrahedral S partition of a bounded domain D ⊂ R3 . This means that D = I∈I ∆(I) and T for any two I, J ∈ I, ∆(I) ∆(J) is empty or is a common vertex or edge or face of ∆(I) and ∆(J). To each vertex ti = ti,0 of the tetrahedral partition, we assign a sequence of auxiliary knots ti,1 , · · · , ti,n in such a way that any subset of four knots of {ti,0 , · · · , ti,n } is affinely independent, i.e., forms a proper tetrahedron. Let β, γ ∈ Z4+ and define Γn = {β = (β0 , β1 , β2 , β3 ) ∈ Z4+ , |β| = β0 + β1 + β2 + β3 = n} and ∆Iγ = ∆I(γ0 ,γ1 ,γ2 ,γ3 ) = [ti0 ,γ0 , ti1 ,γ1 , ti2 ,γ2 , ti3 ,γ3 ]. Consider the regions ΩIβ = ∩γ≤β ∆Iγ

and ΩIn = int( ∩ ΩIβ ). β∈Γn

where int(A) denotes the interior of the set A. If ΩIn 6= ∅, then K = {ti0 ,0 , · · · , ti0 ,n , ti1 ,0 , · · · , ti1 ,n , ti2 ,0 , · · · , ti2 ,n , ti3 ,0 , · · · , ti3 ,n } is called a knot net associated with the tetrahedron ∆(I) = [ti0 ,0 , ti1 ,0 , ti2 ,0 , ti3 ,0 ]. The set Ci = {ti,0 , · · · , ti,n } is called the cloud of knots associated with the vertex ti = ti,0 . If we put VβI = {ti0 ,0 , · · · , ti0 ,β0 , ti1 ,0 , · · · , ti1 ,β1 , ti2 ,0 , · · · , ti2 ,β2 , ti3 ,0 , · · · , ti3 ,β3 },

β ∈ Γn ,

then the functions NβI (x) =| d(ti0 ,β0 , ti1 ,β1 , ti2 ,β2 , ti3 ,β3 ) | ·M(x | VβI ), 4

I ∈ I,

β ∈ Γn ,

called trivariate DMS-splines (or trivariate triangular B-splines) over K, are linearly independent and locally linearly independent over ΩIn . Like the univariate B-splines, the trivariate DMS-splines have a number of nice properties, such as the convex hull property, local support, affine n invariance. Furthermore o each simplex spline s in the space Sn := span NβI (x), I ∈ I, β ∈ Γn is a piecewise polynomial of degree n over the subtetrahedral partition induced by its knot net, which is C n−1 continuous everywhere. If we denote by Sen the space of the piecewise polynomials of degree n over the tetrahedral partition T that is C n−1 continuous everywhere in the domain D, then Sen ⊂ Sn . Further information can be found in [7,14,26,34] and references therein.

3

Polar form and trivariate simplex spline space

Definition 1 : Given a trivariate polynomial p ∈ Pn (R3 ), the polar form pˆ of p is defined as the unique function of n vector variables u1 , · · · , un ∈ R3 satisfying the following properties: • multiaffine: for any index i = 1, · · · , n and any real number λ

pˆ(u1 , · · · , ui−1 , λu + λv, ui+1, · · · , un ) = λˆ p(u1 , · · · , ui−1, u, ui+1, · · · , un ) p(u1 , · · · , ui−1, v, ui+1, · · · , un ), +λˆ

where λ = 1 − λ. • symmetry: for any permutation π of {1, 2, · · · , n}

pˆ(u1, · · · , un ) = pˆ(uπ(1) , · · · , uπ(n) ), • diagonal: pˆ reduces to p when evaluated on its diagonal, i.e., pˆ(u, · · · , u) = p(u). The following result gives a method for computing the polar form of trivariate polynomials in a special case. Corollary 2 Let A1 , · · · , An be n points of R3 and R1 , · · · , Rn be n polynomials in P1 (R3 ). If p(x, y, z) =

n Y

Ri (x, y, z),

i=1

then we have pˆ(A1 , · · · , An ) =

n 1 X Y Ri (Aπ(i) ), n! π∈Sn i=1

(3.1)

where Sn is the symmetric group of all permutations of the set {1, · · · , n}. 5

Proof. Let us consider the function q defined by: q:

(R3 )n → R 1 P Qn (A1 , · · · , An ) 7→ π∈Sn i=1 Ri (Aπ(i) ) = q(A1 , · · · , An ). n!

It is easy to see that the function q satisfies the properties given in Definition 1. Hence q is the polar form of p. Let A = (xA , yA , zA ) and B = (xB , yB , zB ) be two points of R3 . Using the above corollary, it is easy to compute the polar form of each monomial emn = xm y n , 0 ≤ m + n ≤ 2 at (A, B) (see Table 1). p(x, y, z)

pˆ(A, B)

1

x2

1 xA + xB 2 yA + yB 2 zA + zB 2 xA xB

y2

yA yB

z2

zA zB xA y B + xB y A 2 xA zB + xB zA 2 yA zB + yB zA 2

x y z

xy xz yz

Table 1 Polar forms of the monomials of P2 (R3 ).

An important property of the space Sn is that the simplex B-splines can represent any polynomial of degree less than or equal to n (see [7]) as well as any spline s ∈ Sen (see [34]) in terms of their polar forms. Theorem 3 Let s ∈ Sen , then for each x in D we have s(x) =

X X

cIβ NβI (x),

(3.2)

I∈I β∈Γn

where cIβ = sˆI (ti0 ,0 , · · · , ti0 ,β0−1 , ti1 ,0 , · · · , ti1 ,β1 −1 , ti2 ,0 , · · · , ti2 ,β2 −1 , ti3 ,0 , · · · , ti3 ,β3 −1 ) and sI is the restriction of s to the tetrahedron ∆(I). 6

A full proof of Theorem 3 is given in [34]. Corollary 4 For each polynomial p ∈ Pn (R3 ) and x ∈ D we have p(x) =

X X

cIβ NβI (x).

I∈I β∈Γn

where cIβ = pˆ(ti0 ,0 , · · · , ti0 ,β0−1 , ti1 ,0 , · · · , ti1 ,β1 −1 , ti2 ,0 , · · · , ti2 ,β2 −1 , ti3 ,0 , · · · , ti3 ,β3 −1 ). For p ≡ 1, we deduce that

X X

NβI = 1,

I∈I β∈Γn

i.e., the normalized simplex B-splines NβI form a partition of unity. The following theorem shows how each spline s in Sn can be explicitly represented as a linear combination of the normalized B-splines NβI , β ∈ Γn , I ∈ I. Theorem 5 For each spline s ∈ Sn and x ∈ D we have s(x) =

X X

cIβ NβI (x),

I∈I β∈Γn

where cIβ = sˆI (ti0 ,0 , · · · , ti0 ,β0−1 , ti1 ,0 , · · · , ti1 ,β1 −1 , ti2 ,0 , · · · , ti2 ,β2 −1 , ti3 ,0 , · · · , ti3 ,β3 −1 ) and sI is the restriction of s to the region ΩIn . Proof. Let s ∈ Sn , then we have s=

X X

dIβ NβI .

I∈I β∈Γn

If we denote by sI the restriction of s to the region ΩIn , then we have sI (x) =

X

dIβ NβI (x),

β∈Γn

∀x ∈ ΩIn ,

furthermore sI is a polynomial of degree n. By using Corollary 4, we get for each x ∈ ΩIn X cIβ NβI (x). sI (x) = β∈Γn

where

cIβ = sˆI (ti0 ,0 , · · · , ti0 ,β0−1 , ti1 ,0 , · · · , ti1 ,β1 −1 , ti2 ,0 , · · · , ti2 ,β2 −1 , ti3 ,0 , · · · , ti3 ,β3 −1 ) Since NβI , β ∈ Γn are linearly independent over ΩIn , it is easy to see that dIβ = cIβ . 7

4

Trivariate simplex spline quasi-interpolants

In this section, we propose the method used in [32,33] for constructing some quasi-interpolants using the trivariate simplex B-splines in order to approximate given values or derivatives of a function f . We are interested in quasi-interpolants of the form: Qf =

X X

µIβ (f )NβI ,

(4.1)

I∈I β∈Γn

where µIβ , I ∈ I, β ∈ Γn are linear functionals. In order to ensure the locality property of the quasi-interpolant, we suppose that the data-sites of scattered data used to construct µIβ (f ) belong to the support of NβI and allow us to construct a local linear polynomial operator JβI reproducing the space of polynomials of degree n, i.e., JβI (p) = p for all p ∈ Pn (R3 ), then we have the following results. Theorem 6 Let f be a function defined from R3 to R such that the values and (or) the derivatives of f are given at some discrete points in the support of NβI , I ∈ I, β ∈ Γn and allowing us to construct an operator JβI . Set pIβ = JβI (f ). Then, the quasi-interpolant defined by (4.1) with µIβ (f ) = pˆIβ (ti0 ,0 , · · · , ti0 ,β0 −1 , ti1 ,0 , · · · , ti1 ,β1 −1 , ti2 ,0 , · · · , ti2 ,β2 −1 , ti3 ,0 , · · · , ti3 ,β3−1 ), (4.2)

satisfies Qp = p,

∀p ∈ Pn (R3 ).

Proof. For f = p ∈ Pn (R3 ) we have pIβ = JβI (f ) = p for any I ∈ I, and β ∈ Γn . Then by using Corollary 4, it is easy, to see that Qp = p,

∀p ∈ Pn (R3 ).

Theorem 7 If the data sites corresponding to JβI , β ∈ Γn belong to the same region ΩIn , for all I ∈ I, then the quasi-interpolant Q defined by Theorem 6 satisfies Qs = s, ∀s ∈ Sn . Proof. Let s ∈ Sn and let sI be its restriction to the region ΩIn . We know that ΩIn contains the data sites which determine in a unique way the polynomials 8

pIβ , β ∈ Γn . Then we have pIβ = sI ,

for I ∈ I.

By using Theorem 5, it is easy to see that Qs = s, ∀s ∈ Sn . If s ∈ Sen , one can use Theorem 3 to obtain the following result.

Theorem 8 If the data sites corresponding to pIβ , β ∈ Γn belong to the same tetrahedron ∆(I), for all I ∈ I, then the quasi-interpolant Q defined by Theorem 6 satisfies Qs = s, ∀s ∈ Sen . 4.1 Differential trivariate simplex spline quasi interpolants

For constructing a differential simplex spline quasi-interpolant of degree n, we choose the local operator JβI as the Hermite interpolant at some fixed points MβI = (mIβ,1 , mIβ,2 , mIβ,3 ) ∈ supp(NβI ). For this, it suffices to have the values and derivatives of f at these points which give rise to a unisolvent scheme in Pn (R3 ). In this subsection we describe how the approach given in Theorems 6, 7, 8 can be used to construct some particular differential trivariate simplex spline quasi-interpolants. Let f be a function of class C n on D. For I ∈ I and β ∈ Γn , we denote by pIβ the Taylor expansion of order n of f at a point MβI in the support of NβI , i.e., for V = (x, y, z) ∈ R3 we have pIβ (V ) =

(x − mIβ,1 )i (y − mIβ,2 )j (z − mIβ,3 )k ∂ i+j+k f (MβI ). (4.3) i ∂y j ∂z k i!j!k! ∂x 0≤i+j+k≤n X

Theorem 9 Let Q be the quasi-interpolant defined by (4.1), (4.2) and (4.3). Then Qp = p, ∀p ∈ Pn (R3 ). Moreover, if each MβI , β ∈ Γn , belongs to the region ΩIn (resp. the tetrahedron ∆(I)), then Qs = s for each simplex spline s ∈ Sn (resp. spline s ∈ Sen ). Proof. The polynomial pIβ is the unique interpolant of f which satisfies:

pIβ (MβI ) = f (MβI ) and JβI (f )

Since and 8.

∂ i+j+k pIβ ∂ i+j+k f I (M ) = (M I ), ∂xi ∂y j ∂z k β ∂xi ∂y j ∂z k β

1 ≤ i+j +k ≤ n.

(4.4) = f for all f ∈ Pn (R ), the claim follows from Theorems 6, 7 3

9

4.2 Discrete trivariate simplex spline quasi interpolants

In order to construct a discrete trivariate spline quasi-interpolant, it suffices to take data points in the support of NβI , for I ∈ I and β ∈ Γn , which allow to interpolate f , in a unique way, in Pn (R3 ). The multivariate (in particular trivariate) Lagrange interpolation problem has been studied by many authors ( see [9,11,17], for instance). To interpolate with a unique function in the space of trivariate polynomials of degree less or equal to n at a set of points Xn ⊂ R3 (n + 3)(n + 2)(n + 1) with cardinal |Xn | = = αn , Liang [18] proved that it 6 is necessary and sufficient that Xn is not contained in any algebraic surface in Pn (R3 ). To simplify the solution of the Lagrange interpolation problem, Chung et al. [4] showed that if a set of nodes Xn (with |Xn | = αn ) satisfies the geometric characterization (GCn condition), i.e., if for each x ∈ Xn , there exist n hyperplanes R1,x , · · · , Rn,x of R3 such that n

n

i=1

i=1

Xn \{x} ⊂ ∪ Ri,x and x ∈ / ∪ Ri,x , then the Lagrange interpolation problem in Pn (R3 ) is uniquely solvable, and each element Lx of the Lagrange basis is given by the following expression Lx =

R1,x × · · · × Rn,x . R1,x (x) × · · · × Rn,x (x)

I Let Mβ,k , 1 ≤ k ≤ αn , be distinct points in the support of NβI satisfying the GCn condition. Then there exists a Lagrange basis {LIβ,k , k = 1, · · · , αn } I such that LIβ,r (Mβ,s ) = δr,s , r, s = 1, · · · , αn and the polynomial

JβI (f ) = pIβ =

αn X

I f (Mβ,k )LIβ,k

k=1

I interpolates f at the points Mβ,k , k = 1, · · · , αn . Therefore we have the following theorem.

Theorem 10 Let Q be the quasi-interpolant defined by (4.1), (4.2), where

pIβ

=

αn X

I f (Mβ,k )LIβ,k . Then

k=1

Qp = p,

∀p ∈ Pn (R3 ).

I Moreover, if the points Mβ,k , k = 1, · · · , αn , β ∈ Γn , belong to the region ΩIn (resp. the triangle ∆(I)), then Qs = s for each simplex spline s ∈ Sn (resp. spline s ∈ Sen ).

10

5

Quadratic simplex spline quasi interpolants

Now we give some differential quasi-interpolants constructed in the quadratic case. Consider WβI = {ti0 ,0 , · · · , ti0 ,β0 −1 , ti1 ,0 , · · · , ti1 ,β1 −1 , ti2 ,0 , · · · , ti2 ,β2−1 , ti3 ,0 , · · · , ti3 ,β3 −1 }. It is easy to see that for each β ∈ Γ2 there exist two points AIβ = (aIβ,1 , aIβ,2 , aIβ,3 ) and BβI = (bIβ,1 , bIβ,2 , bIβ,3 ) such that WβI = {AIβ , BβI }. If we denote by QIβ = (xIβ , yβI , zβI ) the midpoint of [AIβ BβI ], then by using the definition of the polar form and Corollary 2, we get the following results: • If MβI = AIβ for I ∈ I, β ∈ Γ2 , then µIβ (f ) = pˆIβ (AIβ , BβI ) = f (AIβ ) + ∇f (AIβ )(QIβ − AIβ )T .

(5.1)

• If MβI = QIβ for I ∈ I, β ∈ Γ2 , then 1 µIβ (f ) = pˆIβ (AIβ , BβI ) = f (QIβ ) − (AIβ − QIβ )∇2 f (QIβ )(AIβ − QIβ )T . 2

(5.2)

To construct a discrete quadratic simplex spline quasi-interpolant, the coefficients µIβ (f ) are expressed using α2 = 10 scattered data points which are in the support of NβI . The study of this case gives some interesting results, we can construct the linear functional µIβ using only 3 scattered data points. Then if we set WβI = {AIβ , BβI }, we have the following results. I Theorem 11 Let I ∈ I, β ∈ Γ2 and Mβ,k , k = 1, 2, 3, be three distinct I I points in the support of Nβ . The points Mβ,k , k = 1, 2, 3, are collinear with I ∈ R\{0}, k = 1, 2, 3 such that the AIβ and BβI if and only if there exist qβ,k quasi-interpolant X X Qf = µIβ (f )NβI I∈I β∈Γ2

with

µIβ (f ) =

3 X

I I qβ,k f (Mβ,k ),

k=1

satisfies

Qp = p,

I ∈ I,

β ∈ Γ2

∀p ∈ P2 (R3 ).

Proof. I We first prove the necessary condition. Indeed, let Mβ,1 = (mIβ,1,1 , mIβ,1,2 , mIβ,1,3 ), I I = (mIβ,3,1 , mIβ,3,2 , mIβ,3,3 ) be three points Mβ,2 = (mIβ,2,1 , mIβ,2,2 , mIβ,2,3 ) and Mβ,3 I I I I collinear with AIβ and BβI . Assume that Mβ,2 ∈ [Mβ,1 Mβ,3 ]. Let Mβ,k , k = I 4, · · · , 10 be seven auxiliary points in the support of Nβ (see Figure 1) such

11

that I I I Mβ,6 ∈ / (Mβ,1 Mβ,3 ), I I I Mβ,4 ∈ [Mβ,1 Mβ,6 ],

I I I Mβ,5 ∈ [Mβ,3 Mβ,6 ],

I I I I Mβ,7 ∈ / (Mβ,1 Mβ,3 Mβ,6 ), I I I Mβ,8 ∈ [Mβ,1 Mβ,7 ],

I I I Mβ,9 ∈ [Mβ,3 Mβ,7 ],

I I I Mβ,10 ∈ [Mβ,6 Mβ,7 ].

I We note that Mβ,k , k = 1, · · · , 10 satisfy the GC condition. Let LIi,k , k = I 1, · · · , 10 be the Lagrange basis corresponding respectively to Mβ,k , k = 1 · · · , 10. The GC2 condition implies that

R1,k (x, y, z)R2,k (x, y, z) , I I R1,k (Mβ,k )R2,k (Mβ,k )

LIβ,k (x, y, z) =

I where R1,k and R2,k are two planes containing the nodes Mβ,j , j = 1, · · · , 10, I I I j 6= k. Since the points Mβ,1 , Mβ,2 and Mβ,3 are collinear with AIβ and BβI , we deduce that for k = 4, · · · , 10, R1,k or R2,k contains the points AIβ and ˆ I (AI , B I ) = 0 for BβI . Hence, by using Corollary 2, it is easy to see that L β,k β β k = 4, · · · , 10. We set I ˆ Iβ,k (AIβ , BβI ) = qβ,k , L

k = 1, 2, 3,

then the quasi-interpolant Q defined by

X X

Qf =

µIβ (f )NβI

I∈I β∈Γ2

I

.A β I

I

Mβ,7.

Mβ,8

.. I

Mβ,10

.M B .

I

β,1

I β

I

Mβ,4

. I

.M

β,6

. . I

I

Mβ,9

. .

Mβ,2

I

Mβ,5

I

M β,3

Fig. 1. Position of auxiliary interpolation points.

12

with µIβ (f )

=

3 X

I I qβ,k f (Mβ,k ),

k=1

I ∈ I,

β ∈ Γ2

(5.3)

satisfies Qp = p,

∀p ∈ P2 (R3 ).

To prove the sufficient condition, we set AIβ = (0, 0, 0). If we write each monomial of P2 (R3 ) in terms of the simplex B-spline of S2 , firstly by using Corollary 4 and Table 1 and secondly by using Relation (5.3), we obtain the following system                                

I I I qβ,1 + qβ,2 + qβ,3 =1 I I I qβ,1 mIβ,1,1 + qβ,2 mIβ,2,1 + qβ,3 mIβ,3,1 = xIβ I I I qβ,1 mIβ,1,2 + qβ,2 mIβ,2,2 + qβ,3 mIβ,3,2 = yβI I I I qβ,1 mIβ,1,3 + qβ,2 mIβ,2,3 + qβ,3 mIβ,3,3 = zβI I I I qβ,1 (mIβ,1,1 )2 + qβ,2 (mIβ,2,1 )2 + qβ,3 (mIβ,3,1 )2 = 0

 I I I   (mIβ,3,2 )2 (mIβ,2,2 )2 + qβ,3 (mIβ,1,2 )2 + qβ,2 qβ,1      I I I   qβ,1 (mIβ,1,3 )2 + qβ,2 (mIβ,2,3 )2 + qβ,3 (mIβ,3,3 )2       I I I  qβ,1 mIβ,1,1 mIβ,1,2 + qβ,2 mIβ,2,1 mIβ,2,2 + qβ,3 mIβ,3,1 mIβ,3,2       I I I  qβ,1 mIβ,1,1 mIβ,1,3 + qβ,2 mIβ,2,1 mIβ,2,3 + qβ,3 mIβ,3,1 mIβ,3,3        q I mI mI + q I mI mI + q I mI mI β,1

β,1,2

β,1,3

β,2

β,2,2

β,2,3

β,3

β,3,2

β,3,3

(5.4)

=0 =0 =0 =0 =0

System (5.4) admits a solution only if the following systems                 

I I I qβ,1 + qβ,2 + qβ,3 =1 I I I qβ,1 mIβ,1,1 + qβ,2 mIβ,2,1 + qβ,3 mIβ,3,1 = xIβ I I I qβ,1 mIβ,1,2 + qβ,2 mIβ,2,2 + qβ,3 mIβ,3,2 = yβI

                qI

I I I qβ,1 (mIβ,1,1 )2 + qβ,2 (mIβ,2,1 )2 + qβ,3 (mIβ,3,1 )2 = 0 I I I qβ,1 (mIβ,1,2 )2 + qβ,2 (mIβ,2,2 )2 + qβ,3 (mIβ,3,2 )2 = 0

I I β,1 mβ,1,1 mβ,1,2

I I + qβ,2 mIβ,2,1 mIβ,2,2 + qβ,3 mIβ,3,1 mIβ,3,2 = 0

13

(5.5)

                

and

               

                

I I I qβ,1 + qβ,2 + qβ,3 =1 I I I qβ,1 mIβ,1,1 + qβ,2 mIβ,2,1 + qβ,3 mIβ,3,1 = xIβ I I I qβ,1 mIβ,1,3 + qβ,2 mIβ,2,3 + qβ,3 mIβ,3,3 = zβI I I I qβ,1 (mIβ,1,1 )2 + qβ,2 (mIβ,2,1 )2 + qβ,3 (mIβ,3,1 )2 = 0

(5.6)

I I I qβ,1 (mIβ,1,3 )2 + qβ,2 (mIβ,2,3 )2 + qβ,3 (mIβ,3,3 )2 = 0 I I I mIβ,3,1 mIβ,3,3 = 0 qβ,1 mIβ,1,1 mIβ,1,3 + qβ,2 mIβ,2,1 mIβ,2,3 + qβ,3 I I I qβ,1 + qβ,2 + qβ,3 =1 I I I qβ,1 mIβ,1,2 + qβ,2 mIβ,2,2 + qβ,3 mIβ,3,2 = yβI I I I qβ,1 mIβ,1,3 + qβ,2 mIβ,2,3 + qβ,3 mIβ,3,3 = zβI

                qI

I I I qβ,1 (mIβ,1,2 )2 + qβ,2 (mIβ,2,2 )2 + qβ,3 (mIβ,3,2 )2 = 0

(5.7)

I I I qβ,1 (mIβ,1,3 )2 + qβ,2 (mIβ,2,3 )2 + qβ,3 (mIβ,3,3 )2 = 0

I I β,1 mβ,1,2 mβ,1,3

I I mIβ,3,2 mIβ,3,3 = 0 + qβ,2 mIβ,2,2 mIβ,2,3 + qβ,3

admit a same solution. For each system, if we use the same proof given in [19] (Theorem 3 page 8), we conclude that: • System (5.5) admits a solution only if the points (mIβ,1,1 , mIβ,1,2 , 0), (mIβ,2,1 , mIβ,2,2 , 0) and (mIβ,3,1 , mIβ,3,2 , 0) are collinear with (aIβ,1 , aIβ,2, 0) and (xIβ , yβI , 0). • System (5.6) admits a solution only if the points (mIβ,1,1 , 0, mIβ,1,3), (mIβ,2,1 , 0, mIβ,2,3 ) and (mIβ,3,1 , 0, mIβ,3,3 ) are collinear with (aIβ,1 , 0, aIβ,3) and (xIβ , 0, zβI ). • System (5.7) admits a solution only if the points (0, mIβ,1,2, mIβ,1,3 ), (0, mIβ,2,2 , mIβ,2,3 ) and (0, mIβ,3,2 , mIβ,3,3 ) are collinear with (0, aIβ,2 , aIβ,3 ) and (0, yβI , zβI ). I I Hence the system (5.4) admits a solution only if the points Mβ,1 , Mβ,2 and I I I Mβ,3 are collinear with Aβ and Bβ . I We can also compute the coefficients qβ,k , k = 1, 2, 3. Indeed, let us consider I I I three distinct points Mβ,1 , Mβ,2 and Mβ,3 lying on the line joining the points AIβ and BβI , then there exist three real numbers νβI , ̺Iβ and εIβ such that I Mβ,1 = νβI AIβ +(1−νβI )BβI ,

I Mβ,2 = ̺Iβ AIβ +(1−̺Iβ )BβI ,

I Mβ,3 = εIβ AIβ +(1−εIβ )BβI . (5.8) Using these notations, we obtain the following theorem. I Theorem 12 The coefficients qβ,k , k = 1, 2, 3, defined in Theorem 11, are given by: εIβ (1 − ̺Iβ ) + ̺Iβ (1 − εIβ ) I qβ,1 = , 2(̺Iβ − νβI )(νβI − εIβ )

14

I qβ,2

εIβ (1 − νβI ) + νβI (1 − εIβ ) , = 2(νβI − ̺Iβ )(̺Iβ − εIβ )

I qβ,3 =

νβI (1 − ̺Iβ ) + ̺Iβ (1 − νβI ) . 2(̺Iβ − εIβ )(εIβ − νβI )

Proof. I I I I Assume that Mβ,2 ∈ [Mβ,1 Mβ,3 ]. We consider the seven auxiliary points Mβ,k , k = 4, · · · , 10, defined in the previous proof (see Fig. 1). Let Rr,s,t(x, y, z) = I I I ar,s,t x + br,s,ty + cr,s,tz + dr,s,t = 0 be the equation of the plane (Mβ,r Mβ,s Mβ,t ). Put ur,s,t = (ar,s,t , br,s,t, cr,s,t) and denote by hu, vi the scalar product of the vectors u and v. Then, Corollary 2 and the GC2 condition imply that I qβ,1

R2,4,8 (AIi )R3,6,7 (BiI ) + R3,6,7 (AIi )R2,4,8 (BiI ) = . I I 2R2,4,8 (Mβ,1 )R3,6,7 (Mβ,1 )

I I Since R2,4,8 (Mβ,2 ) = 0 and R3,6,7 (Mβ,3 ) = 0, we deduce that for each point M ∈ 3 I I R we have R2,4,8 (M) = hM − Mβ,2 , u2,4,8i and R3,6,7 (M) = hM − Mβ,3 , u3,6,7 i. Therefore, using Relation 5.8, it is easy to see that

I qβ,1 =

εIβ (1 − ̺Iβ ) + ̺Iβ (1 − εIβ ) . 2(̺Iβ − νβI )(νβI − εIβ )

I I By the same technique we obtain the expressions of qβ,2 and qβ,3 .

Remark 13 For a particular choice of the parameters µIβ , εIβ and ̺Iβ , the coefficients µIβ (f ) can be expressed using only values of f on two data sites εIβ which are collinear with AIβ and BβI . For example, if we set µIβ = 2εIβ − 1 1 where εIβ 6= 0, , 1, then it is easy to see that 2 I qβ,1

6

(2εIβ − 1)2 =− I , 4εβ (1 − εIβ )

I I qβ,2 = 0 and qβ,3 =

1 . − εIβ )

4εIβ (1

Upper bounds of the infinity norm of discrete quasi interpolants

In this section we provide upper bounds of the infinity norms of the discrete simplex spline quasi-interpolants of degree n. For this, we denote by C I the smallest closed ball containing the tetrahedron ∆(I) and κI its diameter. We 15

will consider the case where the interpolation points satisfy the GCn condition T in the region supp(NβI ) C I .

Denote by ||f ||∞,Υ = supx∈Υ |f (x)|, and by ||Q||∞,Υ the corresponding induced norm. It is well-known that if Q reproduces trivariate polynomials of degree less than or equal to n, then we have ||Qf − f ||∞,D ≤ (1 + ||Q||∞,D )

inf

p∈Pn (R3 )

||f − p||∞,D

(6.1)

Thus, bounding ||Q||∞,D implies that the quasi interpolant is (n + 1)th order accurate, that is it provides the optimal approximation order in Sn . T I Let Mβ,k , k = 1, · · · , αn , be distinct points in supp(NβI ) C I satisfying the GCn condition. We consider the discrete simplex spline quasi interpolant defined in Theorem 10. It is easy to see that ||Q||∞,D ≤ max max I∈I β∈Γn

αn X

ΛIβ,k

(6.2)

k=1

where ˆ I (ti ,0 , · · · , ti ,β −1 , ti ,0 , · · · , ti ,β −1 , ti ,0 , · · · , ti ,β −1 , ti ,0 , · · · , ti ,β −1 )|. ΛIβ,k = |L β,k 0 0 0 1 1 1 2 2 2 3 3 3 (6.3) We know that LIβ,k (x, y, z) =

I I (x, y, z) (x, y, z) × · · · × Rβ,n,k Rβ,1,k , I I I I Rβ,1,k (Mβ,k ) × · · · × Rβ,n,k (Mβ,k )

k = 1, · · · , αn

where for each s = 1, · · · , n, I I I Rβ,s,k (x, y, z) = aIβ,s,k x + bIβ,s,k y + cIβ,s,k z + dIβ,s,k and Rβ,s,k (Mβ,k ) 6= 0.

Put n

I I Zβ,s,k = M ∈ {Mβ,m ,

o

I m = 1, · · · , αn , } such that Rβ,s,k (M) = 0 .

I I From [10], we have |Zβ,s,k | ≥ 3 and Rβ,s,k is the unique plane which contains I I Zβ,s,k . This means that Rβ,s,k contains at least three non collinear points of I I I {Mβ,m , m = 1, · · · , αn , }. Then for each s = 1, · · · , n there exist Mβ,i , Mβ,j I I I I I and Mβ,l in Rβ,s,k such that area(Mβ,i , Mβ,j , Mβ,l ) 6= 0. If we put I Kβ,s,k =

q

(aIβ,s,k )2 + (bIβ,s,k )2 + (cIβ,s,k )2 ,

λI0 = max0<η≤β0 ||ti0 ,η − ti0 ,0 ||, λI1 = max0<η≤β1 ||ti1 ,η − ti1 ,0 ||,

λI2 = max0<η≤β2 ||ti2 ,η − ti2 ,0 ||, λI3 = max0<η≤β3 ||ti3 ,η − ti3 ,0 ||, λI = max{λI0 , λI1 , λI2 , λI3 }, 16

and ϑIβ =

min

I I ,M I ,M I )>0, {volR3 (Mβ,m ,Mβ,n β,p β,q

m,n,p,q=1,··· ,αn }

I I I I , Mβ,n , Mβ,p , Mβ,q ), volR3 (Mβ,m

then, it is easy to see that ∀AIβ ∈ WβI

I I I I |Rβ,s,k (AIβ )| ≤ Kβ,s,k ||AIβ − Mβ,i || ≤ Kβ,s,k (κI + λI ),

and I I I |Rβ,s,k (Mβ,k )| = 3Kβ,s,k

I I I I , Mβ,j , Mβ,l , Mβ,k ) volR3 (Mβ,i I I I area(Mβ,i , Mβ,j , Mβ,l )

I 3Kβ,s,k ϑIβ . ≥ I I I area(Mβ,i , Mβ,j , Mβ,l )

(6.4)

(6.5)

I I I I Since the radius of the circumsphere of the tetrahedron (Mβ,i , Mβ,j , Mβ,l , Mβ,k ) I 4 (κ ) 1 , then we have is less than or equal to × I I I I , Mβ,l , Mβ,k ) 4 volR3 (Mβ,i , Mβ,j I I I area(Mβ,i , Mβ,j , Mβ,l )≤

π (κI )8 × I 2. 16 (ϑβ )

Then Relation (6.5) becomes I I |Rβ,s,k (Mβ,k )|

I .(ϑIβ )3 48 Kβ,s,k ≥ × . π (κI )8

(6.6)

Using Relations (6.4), (6.6) and Corollary 2, we obtain ˆ I (ti0 ,0 , · · · , ti0 ,β0 −1 , ti1 ,0 , · · · , ti1 ,β1 −1 , ti2 ,0 , · · · , ti2 ,β2 −1 , ti3 ,0 , · · · , ti3 ,β3−1 )| |L β,k ≤

π (κI )8 (κI + λI ) 48 (ϑIβ )3

!n

∀k = 1 · · · , αn . (6.7)

If we put π (κI )8 (κI + λI ) = 48 (ϑIβ )3 then we get the following theorem. HβI

!n

(6.8)

Theorem 14 Let Q be the trivariate simplex spline quasi interpolant defined in Theorem 10. Then we have ||Q||∞,D ≤ αn max max HβI . I∈I β∈Γn

(6.9)

Remark 15 According to the results given in this section we observe that the upper bound of the infinity norm of any discrete quasi-interpolant depends on: 17

• the geometric characteristics of the tetrahedral partition T , • the maximum distance between each vertex ti = ti,0 of the tetrahedral partition and the auxiliary points ti,1 , · · · , ti,n , • the distribution of the interpolation points in the support of the B-spline NβI . In addition, if we choose an appropriate collection of simplex B-splines such that the quantity λI decreases, then the upper bound of ||Q||∞,D also decreases. Therefore, for suitable tetrahedral partition T one can construct simplex Bsplines such that the distance between each vertex and its auxiliary points is minimal. In this case, the obtained simplex B-splines may be more successful for building bounded discrete quasi-interpolants.

7

Numerical examples

In this section, we show the results of some numerical tests on the method proposed in this paper. To do this, we performed experiments with some quadratic differential and discrete quasi-interpolants given in Section 4. Let ∆(m) , m ≥ 1, be a tetrahedral partition of the bounded domain D = [0, 1] × [0, 1] × [0, 1] obtained as follows: • Use m+1 parallel planes in each of the three space dimensions and subdivide D into m3 subcubes [i, i + 1]h × [j, j + 1]h × [k, k + 1]h where i, j, k = 0, · · · , m − 1 and h = 1/m. • Subdivide each subcube into six tetrahedra (see Figure 2)

Fig. 2. A tetrahedral partition of the unit cube into six tetrahedra.

18

7.1 Example 1: We consider the trivariate test function defined as follows: f (x, y, z) =

1 −10((x−1/4)2 +(y−1/4)2 ) 3 −16((x−1/4)2 +(y−1/4)2 +(z−1/4)2 ) e + e 2 4 1 −10((x−3/4)2 +(y−3/8)2 )+(z−1/2)2 ) 1 −20((x−3/4)2 +(y−3/4)2 ) + e − e 2 4

where (x, y, z) ∈ D. Using the method defined in this paper, we construct two quadratic differential quasi-interpolants DQI1 and DQI2 (see Theorem 9). The interpolation points MβI are given respectively by 1) MβI = AIβ . 2) MβI = QIβ , and at the boundary where QIβ ∈ / D we put MβI = AIβ . We also construct two quadratic discrete quasi-interpolants dQI1 and dQI2, such that the interpolation points are given respectively as follows 1) the vertices of the tetrahedron ∆(I) and the midpoints of its edges. 2) AIβ , QIβ , and BβI , We note that for each s ∈ Sen the quasi-interpolants DQI1 and dQI1 satisfy Qs = s. Using Theorem 12, one can easily see that the upper bound of the infinity norm of dQI2 is given by ||dQI2||∞,D ≤ max max I∈I β∈Γ2

where I qβ,1

with

k=1

I |qβ,k |,

εIβ (1 − ̺Iβ ) + ̺Iβ (1 − εIβ ) = , 2(̺Iβ − νβI )(νβI − εIβ )

I qβ,2 =

I qβ,3

3 X

εIβ (1 − νβI ) + νβI (1 − ̺Iβ ) , 2(νβI − ̺Iβ )(̺Iβ − εIβ )

νβI (1 − ̺Iβ ) + ̺Iβ (1 − νβI ) = , 2(̺Iβ − εIβ )(εIβ − νβI )

νβI = 1, ̺Iβ =

1 and εIβ = 0. 2

Then ||dQI2||∞,D ≤ 3. We define the error between f and each quasi-interpolant defined above by max

r,s,t=0,··· ,200

|f (xr , ys , zt ) − Qf (xr , ys , zt )|, 19

where

r s t , ys = and zt = . 200 200 200 Table 2 gives the errors between f and the quadratic quasi-interpolants DQI1, DQI2, dQI1 and dQI2, for the tetrahedral partitions ∆(m) , m = 16, 32, 64. Tables 2 and 3 allow us to compare the maximum errors of the quasi-interpolants DQI1, DQI2, dQI1 and dQI2 constructed by our approach and the quadratic and the cubic quasi-interpolants defined in [23,35]. Numerical results confirm the theoretical results and show that the approximation order of our method is better than the ones developed in [23] and [35]. xr =

m

DQI1

order

DQI2

order

dQI1

order

dQI2

order

16

6.51 × 10−2

2.83

3.49 × 10−2

2.96

1.91 × 10−2

2.96

2.69 × 10−2

2.73

32

9.11 ×

10−3

2.96

10−3

2.91

10−3

3.03

10−3

3.07

64

1.17 × 10−3

4.46 ×

2.44 ×

5.95 × 10−4

−

2.97 × 10−4

−

m

QI [23]

order

QI [35]

order

16

4.29 × 10−2

1.97

4.26 × 10−2

1.96

32

1.09 ×

10−2

1.98

1.09 ×

10−2

1.98

2.76 ×

10−3

2.76 ×

10−3

−

−

4.05 ×

4.81 × 10−4

−

Table 2 Error between the function f and the quadratic quasi-interpolants DQI1, DQI2, dQI1 and dQI2 for the tetrahedral partitions ∆(m) , m = 16, 32, 64.

64

−

Table 3 Error between the function f and the quadratic and cubic quasi-interpolant devoloped in [23,35] for the tetrahedral partitions ∆(m) , m = 16, 32, 64.

In Figure 3 we show, in two different positions, the visualization of the isosurface of the function f with an isovalue equal to 0.3 (f (x, y, z) = 0.3) and 1 the isosurfaces of the quadratic quasi-interpolants DQI1 and dQI2 (h = ) 64 with the same isovalue.

7.2 Example 2: Now, we consider another function frequently used as a test in volume visualization. This is the Marschner-Lobb test function defined in [−1, 1] × [−1, 1] × [−1, 1] by g(x, y, z) =

π 1 πq 2 1 2 − sin( z) + cos(12π cos( x + y 2)) 2 5 2 10 2

This function is extremely oscillating and therefore it is a difficult test for any efficient three dimensional reconstruction method (see [23]). Since this function 20

f

f

Dqi1

dqi1

Dqi1

dqi1

Fig. 3. Left to right: isosurfaces of f , DQI1 and dQI1 with an isovalue equal to

3 . 10

is not differentiable at (0, 0, z) we use the two discrete quasi-interpolants dQI1 and dQI2 for reconstructing the function g. Table 4 gives the errors between g and the discrete quasi-interpolants dQI1 and dQI2 defined in Example 1, for the tetrahedral partitions ∆(m) , m = 16, 32, 64. Table 4 also gives a comparison of the maximum errors between the two quasi-interpolants dQI1 and dQI2 constructed by our method and the quadratic and the cubic quasi-interpolants developed in [23] and [35]. m

dQI1

order

dQI2

order

QI [23]

order

QI [35]

order

16

5.12 × 10−2

2.36

5.27 × 10−2

2.31

1.79 × 10−1

0.57

1.84 × 10−1

0.59

32

9.96 × 10−3

2.48

1.06 × 10−2

2.40

1.20 × 10−1

1.60

1.22 × 10−1

1.62

64

1.78 × 10−3

−

2.01 × 10−3

−

3.95 × 10−2

−

3.95 × 10−2

−

Table 4 Error between the function g and the quadratic quasi-interpolants dQI1 and dQI2 for the tetrahedral partitions ∆(m) , m = 16, 32, 64.

Figure 4 present the isosurfaces of the Marschner-Lobb test function g and 1 the quasi-interpolants dQI1 and dQI2 (h = ) with an isovalue equal to 0.3 64 (g(x, y, z) = 0.3, dQI1(x, y, z) = 0.3 and dQI2(x, y, z) = 0.3). 21

Fig. 4. Left to right: isosurfaces of g, dQI1 and dQI2 with an isovalue equal to 0.3.

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