Fourier Transform of Bernstein-Bézier Polynomials

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1 Fourier Transform of Bernstein-Bézier Polynomials Charles. K. Chui, Tian-Xiao He 2, and Qingtang Jiang 3 Department of Mathematics and Computer Science University of Missouri-St. Louis, St. Louis, MO 632, USA and Department of Statistics, Stanford University, Stanford, CA 9435, USA 2 Department of Mathematics and Computer Science Illinois Wesleyan University, Bloomington, IL , USA 3 Department of Mathematics and Computer Science University of Missouri-St. Louis, St. Louis, MO 632, USA Abstract Explicit formulae, in terms of Bernstein-Bézier coefficients, of the Fourier transform of bivariate polynomials on a triangle and univariate polynomials on an interval are derived in this paper. Examples are given and discussed to illustrate the general theory. Finally, this consideration is related to the study of refinement masks of spline function vectors. AMS Subject Classification: 42C4, 42C5, 4A5. Key Words and Phrases: Fourier transform, Bernstein-Bézier polynomials, minimal support, quasi-minimal support, refinement Introduction The objective of this paper is to present a compact formula of the Fourier transform of bivariate polynomials on a triangle and univariate polynomials on a bounded interval in terms of their Bernstein-Bézier BB coefficients. Of course the BB coefficients are formulated, as usual, in terms of the Barycentric coordinates, as opposed to the Cartesian coordinates x = x, x 2 for x R 2 or x R. We will focus on the bivariate setting and only consider the univariate formulae as simple consequences. In this regard, although our method of derivation can be Research supported by ARO Grant #W9NF

2 2 Fourier transform of BB polynomials on triangles extended to multivariate polynomials on simplexes, we have decided to present the detailed derivation only for bivariate polynomials, since the motivation of this research is the study of subdivision masks for parametric spline curves and surfaces. Let P n x, x R 2, x = x, x 2, be a Bernstein-Bézier polynomial on a triangle A A 2 A 3 with vertices A i = a i, b i i =, 2, 3. We write P n x in terms of the Barycentric coordinates u, v, w of A A 2 A 3 as follows. P n x P n x, x 2 = j,k,l n, j+k+l=n a j,k,l n! j!k!l! uj v k w l,. where u, v, w is the Barycentric coordinates of x = x, x 2 A A 2 A 3 ; i.e., x, x 2 = a u + a 2 v + a 3 w, b u + b 2 v + b 3 w, u, v, w, u + v + w =. We need the following notation. For m, s {, 2,, 3, 2, 3}, the forwardbackward operator m,s and backward-forward operator m,s defined on sequences {a j,k,l } j,k,l of multi-indices j, k, l are given by: 2 a j,k,l := a j+,k,l a j,k,l, 2 a j,k,l := a j,k,l a j,k+,l, 3 a j,k,l := a j+,k,l a j,k,l, 3 a j,k,l := a j,k,l a j,k,l+,.2 23 a j,k,l := a j,k+,l a j,k,l, 23 a j,k,l := a j,k,l a j,k,l+. In addition, we set k m,s := m,s k m,s and k m,s := m,s k m,s for k =, 2,. For a triangle T = A A 2 A 3, we use V T to denote its area, given by V T := 2 det a a 2 a 3 b b 2 b 3. The main result of this paper can be stated as follows. Theorem. The Fourier transform of P n x as in. over a triangle A A 2 A 3 has the explicit formulation: P n ξ := P n xe iξ x dx A A 2 A 3 n l = 2V T k+l n! n k l! γ l+.3 l= k= { α k+ k 2 l 23a n l,l, e ia ξ +b ξ 2 k 2 l 23a,n, e ia 2ξ +b 2 ξ 2 β k+ k 3 l 23a n l,,l e ia ξ +b ξ 2 k 3 l 23a,,n e ia 3ξ +b 3 ξ 2 },

3 C. K. Chui, T. X. He, and Q. T. Jiang 3 where ξ = ξ, ξ 2, and α := i a 2 a ξ + b 2 b ξ 2, β := i a3 a ξ + b 3 b ξ 2, γ := i a3 a 2 ξ + b 3 b 2 ξ 2. The above formulation is valid for univariate polynomials by setting w =, namely: n! p n x = b j,k j!k! uj v k,.4 j,k n,j+k=n with u = x b/a b, v = x a/b a, and x [a, b. By using and to denote the forward-backward and backward-forward operators: b j,k := b j+,k b j,k, b j,k := b j,k b j,k+,.5 and k := k and k := k, for k =, 2,. Then the formulation of the Fourier transform of univariate polynomials is an immediate consequence of Theorem., as follows. Corollary.2 The Fourier transform of p n x in.4 over a bounded interval [a, b has the explicit formulation: ˆp n ξ := b a p n xe ixξ dx = b ae ibξ k n! n k! ib aξ k+ k b n, k b,n..6 k= We will present the proof of Theorem. in the next section. In Section 3, we will compute the Fourier transforms of certain minimal supported bivariate splines to illustrate the general theory, and discuss the relation to subdivision refinement masks for parametric spline surface rendering. 2 Proof of Theorem The integral of the Fourier transform P n of a bivariate polynomial P n restricted to the triangle is related to the hypergeometric function F, called a confluent hypergeometric function see e.g. [9, defined by F α, β ; z := + α z β! + α α + β β + z 2 2! + α α + α + 2 β β + β + 2 where α, β C with β / {,, 2, }. Clearly, F, β ; z =, F β, β ; z = e z. z 3 +, z C, 3! We need the following two properties of F for any nonnegative integers α, β :

4 4 Fourier transform of BB polynomials on triangles i For integers k, m >, z m F k +, m + ; z = F k +, m; z F k, m; z, z C; 2. ii For integers k, m, u and ρ C, u ν k u ν m e ρν dν = k!m! k + m +! F k +, k + m + 2; uρ. 2.2 See [9, p.3 and p.343. To prove Theorem., we need the following result. Lemma 2. For any integer s, real numbers b m,j, and ρ, z C, s s +! zs+ F m +, s + 2; ρz m= s l l s l l = ρ l+ l b s, s l! eρz ρ l+ l b,s s l!. 2.3 l= Proof. By applying 2., we see that Left-hand side of 2.3 = = s m= s m= ρ s! l= ρ s! F m +, s + ; ρz = b s, ρ s! F s +, s + ; ρz + ρ s b,s ρ s! F, s + ; ρz = b s, ρ s! eρz + ρ b,s s m= s m= s m= F m +, s + ; ρz F m, s + ; ρz s m= s! F m +, s + ; ρz ρ s! F m, s + ; ρz s! F m +, s + ; ρz ρ s! F m, s + ; ρz ρ s! b m+,s m ρ s! F m +, s + ; ρz m= = ρ b s, s! eρz ρ b,s s! + ρ s m= s! F m +, s + ; ρz.

5 C. K. Chui, T. X. He, and Q. T. Jiang 5 Now, repeating this process, we may conclude that the left-hand side of 2.3 is given by ρ b s, s! + ρ 2 b s, s! + + l l ρ l+ l b s l,l s l! + + s ρ s+ s b,s ρ b,s s! + ρ 2 b,s s! + + l l ρ l+ l b,s s l! + + s ρ s+ s b,s s l l s l l = ρ l+ l b s, s l! eρz ρ l+ l b,s s l!, l= l= e ρz where the last equality follows from the identity l b s l,l = l b s,. Hence we obtain 2.3. We are now ready to prove the main result of the paper. Proof of Theorem.. By simple calculations, we have P n ξ = e iξ x n! P n xdx = 2V T a j,k,n j k A A 2 A 3 j!k!n j k! j= k= u e iξ a 3 +a a 3 u+a 2 a 3 v e iξ 2 b 3 +b b 3 u+b 2 b 3 v u j v k u v n j k dvdu = 2V T e ia 3ξ +b 3 ξ 2 = 2V T e ia 3ξ +b 3 ξ 2 k= j= k= j= k= a j,k,n j k n! j!k!n j k! a j,k,n j k n! j!n j +! e βu u j u n j+ F k +, n j + 2; uγdu u e βu u j e γv v k u v n j k dvdu = 2V T n!e ia 3ξ +b 3 ξ 2 βu uj e j= j! { n j } a j,k,n j k n j +! un j+ F k +, n j + 2; uγ du, where the third equality follows from 2.2. Applying Lemma 2. to the sum in the curly brackets of the rightmost equality with s = n j, b m,l = a j,m,l so that

6 6 Fourier transform of BB polynomials on triangles b m,l = 23 a j,m,l, b m,l = 23 a j,m,l, z = u, ρ = γ, we see that P n ξ = 2V T n!e ia 3ξ +b 3 ξ 2 { n j l= j= βu uj e j! l u n j l n j γ l+ l 23a j,n j, n j l! e uγ = 2V T n!e ia 2ξ +b 2 ξ 2 2V T n!e ia 3ξ +b 3 ξ 2 = 2V T n!e ia 2ξ +b 2 ξ 2 2V T n!e ia 3ξ +b 3 ξ 2 = 2V T n!e ia 2ξ +b 2 ξ 2 2V T n!e ia 3ξ +b 3 ξ 2 j= l= j= l= j= l= j= l= l= l= l γ l+ l= l γ l+ l 23a j,n j, j!n j l! l γ l+ l 23a j,,n j j!n j l! } l u n j l γ l+ l 23a j,,n j du n j l! u j u n j l e αu du u j u n j l e βu du l γ l+ l 23a j,n j, n l +! F j +, n l + 2; α l γ l+ l γ l+ l 23a j,,n j n l +! F j +, n l + 2; β n l j= n l j= l 23a j,n j, n l +! F j +, n l + 2; α l 23a j,,n j n l +! F j +, n l + 2; β. Finally, applying Lemma 2. to the first term the second term resp. in the above equation with s = n l, b m,k = l 23a m,k+l,, z =, ρ = α with s = n l, b m,k = l 23a m,,k+l, z =, ρ = β resp., we have P n ξ = 2V T n!e ia 2ξ +b 2 ξ 2 { n l k= k l= l γ l+ α k+ k 2 l e α n l 23a n l,l, n l k! k= 2V T n!e ia 3ξ +b 3 ξ 2 { n l k= k l= l γ l+ β k+ k 3 l e β n l 23a n l,,l n l k! k= } k α k+ k 2 l 23a,n, n l k! } k β k+ k 3 l 23a,,n n l k!

7 C. K. Chui, T. X. He, and Q. T. Jiang 7 n l = 2V T k+l n! n k l! γ l+ l= k= { α k+ k 2 l 23a n l,l, e ia ξ +b ξ 2 k 2 l 23a,n, e ia 2ξ +b 2 ξ 2 β k+ k 3 l 23a n l,,l e ia ξ +b ξ 2 k 3 l 23a,,n e ia 3ξ +b 3 ξ 2 }, as desired. Remark. By applying Lemma 2. and following the above procedure, the main result in this paper can be extended to higher dimensions. That is, explicit formulae of the Fourier transform of Bernstein-Bézier representations of multivariate polynomials on simplexes can be derived in a similar way. 3 Application to refinable bivariate splines on triangles Refinable spline functions are instrumental to surface subdivisions. For example, the bi-cubic B-spline is used in the Catmull-Clark scheme [ and the threedirection box-spline B 222 is used in the Loop scheme [. The simple reasons are that firstly, the refinement masks of such spline functions immediately give the so-called local averaging rules for the subdivision schemes; and secondly, the parametric spline representations are precisely the subdivision surfaces. While the refinement masks of the bi-cubic B-spline and the box-spline B 222, being defined by convolutions of the characteristic function of the unit square along the appropriate directions, are readily computable, those for others, such as basis functions with minimum and quasi-minimum supports, are not as easy to compute. Examples of the recent development in this direction are the refinable bivariate C 2 -cubic, C 2 -quartic and C 2 -quintic spline functions in [6, 7, 8, introduced for matrix-valued surface subdivisions to gain such desirable properties as surface geometric shape control parameters, smaller subdivision template size to better address the often unavoidable extraordinary vertices, and interpolation of the position components of the initial control vertices. Computations of the refinement masks of these bivariate spline functions are very tedious, requiring formulating and solving large linear systems in terms of Bernstein-Bézier coefficients. For this reason, the original motivation for this research was to extend the standard Fourier approach to computing the scalar-valued refinement masks of refinable spline functions to computing the matrix-valued refinement masks of refinable spline function vectors. As an application, let us consider the Fourier transform of the minimally supported ms and quasi-minimally supported qms bivariate splines in S k mk i i =, 2, where and 2 denote respectively 3 and 4-direction meshes in

8 8 Fourier transform of BB polynomials on triangles R 2 with integer grid points, and S k mk i i =, 2 the spaces of functions in C k whose restrictions on the triangular cells are polynomials of degree mk. Here, mk denotes the smallest nonnegative integer such that S k mk i contains at least one locally supported function f. It is well known that for m2k = 3k and m2k = 3k + ; and for 2 we have m3k = 4k, m3k = 4k +, and m3k + = 4k + 2 see [2, 3. Minimally ms and qms bivariate splines, considered in [2, 3, 4, are defined as follows. The support of a locally supported function f in a spline space is the closure of the set on which f does not vanish and is denoted by suppf. A set S is called a minimal support of a spline space if there is some f, called an ms spline, in the space with suppf = S, but there does not exist a nontrivial g in the space with suppg properly contained in S. A function f in a spline space is called a qms spline if i f cannot be written as a finite linear combination of ms splines in the space, and ii for any h in the space properly contained in suppf, h is some finite linear combination of ms spines in the space. For example, for the 3-direction mesh, while the spline space S has only one ms spline g, which is the box spline B with direction set {,,,,, }, the space S has two ms splines g2 and g 3 which are the characteristic functions of χ A and χ B, where A is the triangle bounded by the 3 lines: x =, y =, and y = x, and B the triangle bounded by the 3 lines: x =, y = and y = x. On the other hand, for the 4-direction mesh 2, the space S2 2 has only one ms spline f = B, the box spline with direction set {,,,,,,, }, but there are two ms splines in the space S 2 by f2 and f 3, where f2 is the Courant hat function having a diamond support with the vertices,,,,,, and,, and having value of at the center, of the support, and f3 the other Courant hat function supported on the unit square [, 2 with value at its center /2, /2. Furthermore, there are two ms splines f4 and f 5 and one qms spline f 6 in S2 4 2 see [2, 3, 4. In the following, let us consider the k-fold 2-dimensional convolutions: g k = g g g }{{} k+ and f k = f f f }{{} k+ spline function vectors of interest are: [ G k g k 2 g k 3 [ g where k and f h [ := g k g 2 g3 := [, F k f k 2 f3 k [ f g f h. of g and f, respectively. Now, the [ := f k f 2 f3, 3. We remark that among the functions gi k, f i k, i =, 2, 3, only gk and f k are box splines, while all of them are the unique ms and qms splines in the corresponding

9 C. K. Chui, T. X. He, and Q. T. Jiang 9 spline spaces where the notion of uniqueness is according to the statement of Theorem 3.2 in [2. The Fourier transforms of the initial ms splines gi, f i, i = 2, 3, can be evaluated by using the formula provided in this paper, and the Fourier transforms of the other splines are given by the corresponding products with those of g k or f k. Finally, the refinement masks of G k, F k can be easily computed by making use of 3. from the refinements of the initial G and F. Example. The Fourier transform of G k is given by Ĝk ξ, ξ 2 = [ĝk 2, ĝk 3 T ξ, ξ 2, where ĝk 2 = ĝ 2 ĝ k and ĝk 3 = ĝ 3 ĝ k. Here, we have ĝ ξ, ξ 2 = B ξ, ξ 2 = e iξ e iξ2 iξ iξ 2 e iξ+ξ2 iξ + ξ 2, and ĝ 2 ξ, ξ 2 = e iξ +ξ 2 ξ ξ + ξ 2 ĝ 3 ξ, ξ 2 = e iξ +ξ 2 ξ 2 ξ + ξ 2 e iξ 2 ξ ξ 2, 3.2 e iξ ξ ξ Next, let us compute the Fourier transform of F k. For this purpose, recall that φa k ξ = deta e iξt A k φ A T ξ, 3.4 for any invertible matrix A of dimension s. Example 2. Let f2, f 3 be the two Courant hat functions in S 2 as introduced previously. To compute the Fourier transform of f2, we use the x-y axes to partition its support into four triangles:, 2, 3, and 4 in the the first, second, third, and fourth quadrants, respectively. Then f2 can be written as the sum of four functions: φ, φ 2, φ 3, and φ 4, with supports given by, 2, 3, and 4, respectively. By the formula.3, the Fourier transform of φ is given by φ ξ, ξ 2 = ξ ξ 2 + i e iξ ξ 2 ξ ξ 2 + i e iξ2 ξ 2 2 ξ 2 ξ. 3.5 Since φ 2 x, y = φ x, y, it follows from 3.5 and 3.4 that φ 2 ξ, ξ 2 = φ ξ, ξ 2 = e iξ i ξ ξ 2 ξ 2ξ + ξ 2 + i e iξ2 ξ2 2ξ + ξ 2.

10 Fourier transform of BB polynomials on triangles Similarly, since φ 3 x, y = φ x, y and φ 4 x, y = φ x, y, we have Consequently, we arrive at f 2 ξ, ξ 2 = φ 3 ξ, ξ 2 = e iξ + i ξ ξ 2 ξ 2ξ 2 ξ + i e iξ2 ξ2 2ξ ξ 2 ; φ 4 ξ, ξ 2 = + i e iξ ξ ξ 2 ξ 2ξ + ξ 2 i e iξ2 ξ2 2ξ + ξ 2. 4 e φ iξ 2 e iξ 2 j ξ, ξ 2 = 2i ξ 2 ξ + ξ 2 ξ 2 ξ + 2i e iξ e iξ ξ ξ + ξ 2 ξ ξ 2. j= [ x Next, observe that the linear transformation B y B = [, [, with 3.6 maps the suppf3 to suppf 2, with the vertices,,,,,, and, of suppf3 corresponding to the vertices,,,,,, and, of suppf2, respectively. Hence, we may write [ [ f3 x, y = f2 x, B y and apply 3.4 to obtain f 3 ξ, ξ 2 = 2 e iξ 2 B[,T f 2 2 Bξ = 2 e i ξ +ξ 2 = 2i e iξ 2 e iξ ξ ξ 2 ξ 2 ξ + e iξ+ξ2 ξ + ξ 2 2 f 2 ξ + ξ 2 2, ξ ξ Finally, to compute the refinement masks of G k and F k in 3., we observe that the convolution of two finitely refinable functions remains to be finitely refinable, and that G k is the convolution of G with the refinable box spline g k, and F k the convolution of F with the refinable box spline f k. Hence, we only need to compute the refinement masks of the initial ms splines. Precisely, from ĝ2 and ĝ 3 given by 3.2 and 3.3, we have Ĝ 2ξ, 2ξ 2 = Q ξ, ξ 2 Ĝ ξ, ξ 2 where [ + z2 + z Q ξ, ξ 2 = z 2 z 2 z + z + z z 2, z = e iξ, z 2 = e iξ 2 ;

11 C. K. Chui, T. X. He, and Q. T. Jiang and from f 2 and f 3 given by 3.6 and 3.7, we have F 2ξ, 2ξ 2 = R ξ, ξ 2 F ξ, ξ 2 where R ξ, ξ 2 = 4 [ + 2 z + z 2 z + z z + z 2 z z z + z 2, z = e iξ, z 2 = e iξ 2. The interested reader is referred to [5 for computing the refinement mask for F x by calculating the BB coefficients of F B x directly. Therefore, we conclude, from the definitions in 3., that with Ĝ k 2ξ, 2ξ 2 = Q k ξ, ξ 2 Ĝk ξ, ξ 2, F k 2ξ, 2ξ 2 = R k ξ, ξ 2 F k ξ, ξ 2 Q k ξ, ξ 2 = qξ, ξ 2 k Q ξ, ξ 2, R k ξ, ξ 2 = rξ, ξ 2 k R ξ, ξ where qξ, ξ 2 = 8 + e iξ + e iξ 2 + e iξ +ξ 2 is the mask or two-scale symbol of g, and rξ, ξ 2 = 6 +e iξ +e iξ 2 +e iξ +ξ 2 +e iξ ξ 2 the mask or two-scale symbol of f. That is, we have the following result. Theorem 3. The vector-valued functions G k and F k are finitely refinable with refinement masks given by 3.8. Similarly, f k f 4 f 5 f 6 is also finitely refinable, with refinement mask given by where S is the refinement mask of rξ, ξ 2 k S ξ, ξ 2, f 4 f 5 f 6. Computation of S as well as the refinement masks of other initial ms and qms bivariate splines in general is usually nontrivial and requires further investigation.

12 2 Fourier transform of BB polynomials on triangles References [ E. Catmull and J. Clark, Recursively generated B-splines surfaces on arbitrary topological meshes, Comput. Aided Design 978, [2 C.K. Chui, Multivariate Splines, NSF-CBMS Series #54, SIAM Publ., Philadelphia, 988. [3 C.K. Chui and T.X. He, Computation of minimally and quasi-minimally supported bivariate splines, J. Comput. Math 8 99, 8 7. [4 C.K. Chui and T.X. He, On minimal and quasi-minimal supported bivariate splines, J. Approx. Theory , [5 C.K. Chui and Q.T. Jiang, Multivariate balanced vector-valued refinable functions, In: Modern Development in Multivariate Approximation, ISNM 45, V.W. Haussmann, K. Jetter, M. Reimer, and J. Stöckler eds., Birhhäuser Verlag, Basel, 23, pp [6 C.K. Chui and Q.T. Jiang, Surface subdivision schemes generated by refinable bivariate spline function vectors, Appl. Comput. Harmon. Anal. 5 23, [7 C.K. Chui and Q.T. Jiang, Refinable bivariate C 2 -splines for multi-level data representation and surface display, Math Comp , [8 C.K. Chui and Q.T. Jiang, Refinable bivariate quartic and quintic C 2 -splines for quadrilateral subdivisions, J. Comp. Appl. Math., to appear. Available: jiang [9 I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, 6th edition, Academic Press, New York, 2. [ C. Loop, Smooth subdivision surfaces based on triangles, Master s Thesis, University of Utah, Department of Mathematics, Salt Lake City, 987.