Reconstruction Of Bivariate Cardinal Splines Of Polynomial Growth From Their Local Average Samples

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1 Applied Mathematics E-Notes, 17017), c ISSN Available free at mirror sites of amen/ Reconstruction Of Bivariate Cardinal Splines Of Polynomial Growth From Their Local Average Samples Yugesh Shanmugam, Devaraj Ponnaian Received June Abstract We analyse the following local average sampling problem for two variables: Let h be a nonnegative function supported in the rectangle [ 1, ] [ 1 1, ] 1. Given a sequence of samples {y ij} i,j Z, find a bivariate spline fx, y) such that f h)i, j) = y ij. It is shown that this problem has infinitely many solutions. Further, under some realistic conditions on h it is shown that the above said problem has unique solution when both samples {y ij} and the spline f are of polynomial growth. 1 Introduction The extension of classical Whittaker-Shannon-Kotel nikov sampling formula for k- dimension may be stated as follows [8, 6, 7]: Any function f bandlimited to the k- dimensional cube [ 1, 1 ]k can be reconstructed from its sequence of samples {fn)} n Z k using the formula ft 1, t,..., t k ) = α Z k fα)sinct 1 α 1 )sinct α )... sinct k α k ), where the sinc function is defined by sincx) = sinx). Although the bandlimited condition is eminently useful, it is not always realistic, since a bandlimited signal is of x infinite duration. It is natural to investigate other signal classes for which a sampling theorem holds. The reconstruction has been investigated for non-bandlimited functions in [1,, 5, 6, 7, 8, 9, 10]. In this paper, we consider the class of bivariate splines of polynomial growth. The B-splines with equally spaced knots in two variables is defined [3, 4] as β d1d x, y) = β d1 x)β d y), Mathematics Subject Classifications: 4A15, 94A0. Department of Mathematics, SSN College of Engineering, Kalavakkam , Tamil Nadu, India, mathsyugesh@gmail.com, yugeshs@ssn.edu.in School of Mathematics, Indian Institute of Science Education and Research Thiruvananthapuram, Kerala, India, devarajp@iisertvm.ac.in 47

2 48 Reconstruction of Bivariate Cardinal Splines where β d is the cardinal central B-spline of degree d in single variable which is given by β d := χ [ 1, 1 ]... χ [ 1, 1 ], d 1 terms) and denotes the convolution. β d1d is a tensor-product of two B-splines and its piecewise pieces are separated by a rectangular partition. Let S d1,d be the class of functions fx, y) satisfying the following properties: uv 1. The d 1 d partial derivatives x u y v fx, y), 0 u d 1 1, 0 v d 1 are continuous in the entire plane R.. Let Π x,y denote set of all polynomials in x and y of degree d 1 d ), i.e., { d1 } d Π x,y = a uv x u y v : a uv R. u=0 v=0 In each square [i 1, i] [j 1, j], fx, y) Π x,y for both d 1 and d odd. If d 1 is odd and d is even, then in each square [i, i] [j 1, j 1 ], fx, y) Π x,y. In each square [i 1 1, i 1 ] [j 1, j], fx, y) Π x,y for d 1 even and d odd. Also, when d 1 and d are even, then in each square [i 1, i 1 ] [j 1, j 1 ], fx, y) Π x,y. We note that Π x,y depends on d 1 1)d 1) parameters and we can write S d1,d = fx, y) = a ij β d1 x i)β d y j) : a ij R. Bivariate Cardinal Spline Interpolation The bivariate cardinal spline interpolation problem defined in [10] is as follows: Given a double sequence {y ij } i,j Z of real numbers, find a bivariate spline f S d1,d such that fi, j) = y ij, i, j Z. 1) It can be easily checked that for d 1 = d = 1, this problem has a unique solution. LEMMA 1. Let d 1, d > 1. Then given a double sequence {y ij } i,j Z of real numbers, there are infinitely many bivariate splines f S d1,d such that fi, j) = y ij, for i, j Z. Moreover, the set of all such solutions in S d1,d form a linear manifold of dimension d 1 1)d 1) 4 when both d 1, d are odd or d 1 is odd and d is even or d 1 is even and d is odd and of dimension d 1 1)d 1) 3 when both d 1, d are even. PROOF. Case i): d 1, d are odd. In this case every fx, y) S d1,d can be uniquely represented in the form fx, y) = P x, y) a uv x u) d1 y v) d a u v x u) d1 y v) d, u>0 v>0 u 0 v 0

3 S. Yugesh and P. Devaraj 49 where P x, y) Π x,y, a uv are constants, and x := max0, x). Since fx, y) = P x, y) in [0, 1] [0, 1], we have the relations, P 0, 0) = y 00, P 1, 0) = y 10, P 0, 1) = y 01 and P 1, 1) = y 11. The coeffi cients a uv are uniquely determined by the interpolation conditions fi, j) = y ij, i, j Z. Therefore fx, y) linearly depends on d 1 1)d 1) 4 parameters in P x, y). Caseii): d 1 is odd and d is even. Every fx, y) S d1,d has a unique representation of the form fx, y) = P x, y) u>0 u 0 v 0 v>0 a uv x u) d1 a u v x u) d1 y 1 ) ) d v y 1 ) v Therefore if fx, y) satisfies equation 1), then P 0, 0) = y 00, P 1, 0) = y 10, P 0, 1) = y 01 and P 1, 1) = y 11. As in the previous case the coeffi cients a uv are calculated from the interpolation conditions 1. Hence fx, y) linearly depends on d 1 1)d 1) 4 parameters. Caseiii): d 1 is even and d is odd. In this case every fx, y) S d1,d can be uniquely written in the form fx, y) = P x, y) u>0 v>0 a u v u 0 v 0 ) d a uv x 1 ) ) d1 u y v) d x 1 ) ) d1 u y v) d. In this case also P 0, 0) = y 00, P 1, 0) = y 10, P 0, 1) = y 01 and P 1, 1) = y 11. As in the previous cases, fx, y) linearly depends on d 1 1)d 1) 4 parameters. Caseiv): Both d 1 and d are even. Every fx, y) S d1,d has unique representation of the form fx, y) = P x, y) b uv x 1 ) d1 y ) u 1 ) d ) v u>0 v>0 b u v x 1 ) ) d1 u y 1 ) ) d v, u 0 v 0 where P x, y) Π x,y. Then P x, y) satisfies the relations P 0, 0) = y 00, P 1, 0) = y 10 and P 0, 1) = y 01. Thus three coeffi cients in P x, y) can be uniquely found from these relations. The coeffi cients b uv are uniquely determined using the conditions fi, j) = y ij, i, j Z. Therefore, in this case fx, y) linearly depends on d 1 1)d 1) 3 parameters. In order to obtain uniqueness of solution Schoenberg [10] has applied the following power growth condition on the bivariate cardinal spline and the bi-infinite samples: S γ = {fx, y) S d1,d : fx, y) = O x y 1) γ },.

4 50 Reconstruction of Bivariate Cardinal Splines D γ = {{y ij } i,j Z : y ij = O i j 1) γ }. Further, he has shown in [10] that for γ 0 and for a given double sequence of real numbers {y ij } i,j Z D γ, there exists a unique bivariate spline f S γ such that fi, j) = y ij, i, j Z. In practice, the available samples are not always exact. The samples of f are the local averages of the function f at m, n). i.e., n 1 n 1 m 1 m 1 fx, y)hn x, m y)dxdy, where h is suitable weight function which reflects the characteristic of the measurement process. Average sampling problem: Given a sequence of real numbers{y ij } i,j Z, find a bivariate spline f S d1,d 3 such that f hi, j) = y ij, i, j Z, where the averaging function h satisfies supph) [ 1, 1 ] [ 1, 1 ] and hx, y) 0, ) 0 < hx, y)dxdy < and 0 < 0 0 hx, y)dxdy <. 3) We show that this average sampling problem has infinitely many solutions for every d. Further, by applying the polynomial growth conditions as that of [10], the uniqueness is obtained. LEMMA. If the averaging function h satisfies ) and 3), then for a given double sequence of real numbers {y ij } i,j Z, there are infinitely many bivariate splines f S d1,d such that f hi, j) = y ij, i, j Z. 4) The set of such solutions in S d1,d form a linear manifold of dimension d 1 1)d 1) if d 1, d are odd and of dimension d 1 1)d 1) 1 for the other three cases. PROOF. Casei): Both d 1 and d are odd. In this case, the functions f S d1,d can be uniquely represented in the form fx, y) = P x, y) u>0 v>0 a uv x u) d1 y v) d u 0 v 0 a u v x u) d1 y v) d with appropriate coeffi cients a uv and P x, y) = fx, y) in [0, 1] [0, 1]. If fx, y) satisfies 4), then f h1, 1) = y 1,1. i.e., h f1, 1) = h P 1, 1) a h1 x, 1 y)x 1) d y 1) d dxdy

5 S. Yugesh and P. Devaraj 51 and h1 x, 1 y)x 1) d1 y 1) d dxdy > 0. From this the coeffi cient a 11 can be uniquely determined such that f h1, 1) = y 1,1. Similarly the other coeffi cients a ij can be uniquely determined using the other conditions of f hi, j) = y i,j. Thus the solutions linearly depends on d 1 1)d 1) parameters. Caseii): d 1 is odd and d is even. Then every fx, y) S d1,d can be uniquely written in the form fx, y) = P x, y) u>0 u 0 v 0 v>0 a uv x u) d1 a u v x u) d1 If fx, y) satisfies equation 4), then in this case h f0, 0) = y 1 ) v ) d y 1 ) d ) v. h x, y)p x, y)dxdy. The coeffi cients a uv can be found from the conditions 4). Hence fx, y) linearly depends on d 1 1)d 1) 1 parameters. Caseiii): d 1 is even and d is odd. In this case every function f S d1,d has a unique representation of the form fx, y) = P x, y) u>0 u 0 v 0 v>0 a uv x 1 ) u ) d1 a u v x 1 ) u ) d1 In this case the condition 4) implies P x, y) satisfies h f0, 0) = h x, y)p x, y)dxdy. y v) d y v) d. Therefore fx, y) linearly depends on d 1 1)d 1) 1 coeffi cients. Caseiv): Suppose that both d 1 and d are even. Then every function f S d1,d can be uniquely represented in the form fx, y) = P x, y) u>0 u 0 v 0 v>0 b uv x 1 ) u ) d1 b u v x 1 ) u ) d1 y 1 ) v ) d y 1 ) v ) d 5)

6 5 Reconstruction of Bivariate Cardinal Splines with appropriate coeffi cients b uv. Since h f0, 0) = = h x, y)fx, y)dxdy h x, y)p x, y)dxdy and fx, y) satisfies h f0, 0) = y 00, the solutions fx, y) linearly depends on d 1 1)d 1) 1 parameters. 3 Average Sampling Theorem THEOREM 1. [Main Theorem] Let d 1, d N and hx, y) = h 1 x)h y) satisfy the conditions ) and 3). Then for a given double sequence of real numbers {y ij } i,j Z D γ, there exists a unique bivariate spline f S γ such that f hi, j) = y ij i, j Z. In order to prove this theorem, we introduce the vector G h z, w) = G h1,d 1 z), G h,d w)), where G h1,d 1 z) and G h,d w) are Laurent polynomials defined by G h1,d 1 z) = G h,d w) = Υ z,d1 x) = i Z h 1 x)υ z,d1 x)dx, h y)υ w,d y)dy, z i β d1 i x), Υ w,d y) = j Z w j β d j y). For the proof we also need the following properties [9]: LEMMA 3 [9]). For d 1 N, n Z and z C \ {0} we have i) Υ z,d 1 x) = Υ z,d1 x). ii) Υ z,d1 x n) = z n Υ z,d1 x). iii) Υ z,d 11 x) = 1 z)υ z,d 1 x 1 ). iv) Υ,d1 x) is even, Υ,d1 1 ) = 0 and Υ,d 1 x) > 0 for x, 1 ).

7 S. Yugesh and P. Devaraj 53 THEOREM. Consider the linear space := {fx, y) Sd1,d : h fi, j) = 0 i, j Z}. If z 1, z,..., z p are the simple roots of G h1,d 1 z) and w 1, w,..., w q are the simple roots of G h,d w) then the set of functions forms a basis of. {Υ z Υ w s,d : 1 r p, 1 s q} PROOF. We have to find l linearly independent functions in related to the roots of G h z, w). The dimension of is d 1 1)d 1) if both d 1 and d are odd, d l := 1 1)d ) if d 1 is odd and d are even, d 1 d 1) if d 1 is even and d are odd, d 1 d if both d 1 and d are even, Now Υ z Υ w s,d hi, j) = Υ z r = = z i r = z i r 1 1,d 1 h 1 i) Υ w,d h j) Υ z i u)h 1 u)du 1 s 1 Υ z u)h 1 u)du ws j Υ z u)h 1 u)du ws j = z i rg h1,d 1 z r ) w j sg h,d w s ) Υ w s,d i v)h v)dv 1 1 = 0 for r = 1,,..., p and s = 1,,..., q. Υ w s,d v)h v)dv Υ w s,d v)h v)dv Now we prove Υ z Υ w s,d are linearly independent. For, suppose that Then i.e., p r=1 s=1 p q r=1 s=1 q c rs [ p c rs [Υ z r ],d 1 x) Υ w,d y) = 0. zrβ i d1 i x) wsβ j d j y) = 0. i Z j Z r=1 s=1 s ] q c rs zrw i s j β d1 i x)β d j y) = 0.

8 54 Reconstruction of Bivariate Cardinal Splines As β d1 i x)β d j y) are linearly independent, p q r=1 s=1 c rszrw i s j = 0 i, j Z. The above system is a set of linear equations in c ij with coeffi cient matrix, the Vandermonde s matrix. Since the Vandermonde determinant is not zero, c rs = 0. Therefore the functions Υ z Υ w s,d form a basis of. THEOREM 3. Suppose that d 1, d N, γ 0 and h is in the separable form satisfying conditions ) and 3). If the roots of G h1,d 1 z),g h,d w) are simple and no roots on the unit circles z = 1, w = 1 respectively, then for a given double sequence of real numbers {y ij } i,j Z D γ the problem, of finding a bivariate spline f S γ satisfying f hi, j) = y ij, i, j Z has a unique solution. The solution is of the form fx, y) = i Z y ij L h1,d 1 x i)l h,d y j), j Z where L h1,d 1 x) = i Z c iβ d1 x i), L h,d y) = j Z d jβ d y j), c i and d j are coeffi cients of the Laurent expansion of G z) and G h,d w) respectively. The spline L h1,d 1 and L h,d have exponential decay. PROOF. The coeffi cients c i, d j are given by Cz) = G z) = i Z c iz i and Dw) = G h,d w) = j Z d jw j. These coeffi cients have exponential decay. Therefore c i Oφ x ) and L h,d = Oφ i ), d j = Oφ j h,d ), where φ h1,d 1, φ h,d 0, 1). Hence L h1,d 1 = = Oφ y h,d ). For x >, y > consider, i Z j Z i j 1)γ φ x i φ y j h,d x y 1) γ i Z j Z x i 1 y j 1 1)γ φ i φ j h,d x y 1) γ 1 i j ) γ φ i Therefore from the order of y ij we get φ j h,d <. fx, y) = y ij L h1,d 1 x i)l h,d y j) = i j 1) γ φ x i φ y j h,d K x y 1) γ x, y) R. Hence fx, y) = O x y 1) γ x, y) R.

9 S. Yugesh and P. Devaraj 55 Now fx, y) = y ij L h1,d 1 x i)l h,d y j) = [ ] [ ] y ij c u β d1 x u i) d v β d y v j) u Z v Z = y ij c u i d v j β d1 x u)β d y v). u Z v Z From this we conclude that f S γ. As Cz)G h1,d 1 z) = 1 and Dw)G h,d w) = 1 we obtain L h1,d 1 h 1 i) = u Z c u [h 1 β d1 ]i u) = δ i, L h,d h j) = v Z d v [h β d ]j v) = δ j. Hence we get f hi, j) = y ij L h1,d 1 x i)l h,d y j) hi, j) = y ij [[L h1,d 1 x i) h 1 i)][l h,d y j) h j)]]. Clearly f hi, j) = y ij i, j Z. We conclude that fx, y) is a solution. Now we shall show the uniqueness. If the bivariate spline f, g S γ are two solutions, then by the Theorem we have fx, y) gx, y) = p q r=1 s=1 c rs [Υ z r ],d 1 x) Υ w,d y), for some constants c rs. Using the behaviour of Υ z x) Υ w s,d y) at x ± and y ± we get that c rs = 0. Therefore f = g. PROOF OF THEOREM 1. In view of Theorem 3, it is suffi cient to prove that the roots of G h1,d 1 z) and G h,d w) are simple and none of them is on the unit circles z = 1 and w = 1 respectively. We can write P z) = z p Gh1,d 1 z) p ) p ) p ) = h 1 β d1 h 1 β d1 1 z h 1 β d1... h 1 β d1 p ) z p, s z where { d1 1 if d p := 1 is odd, if d 1 is even. d 1

10 56 Reconstruction of Bivariate Cardinal Splines Also where Qw) = w q Gh,d w) q ) q ) q ) = h β d h β d 1 w h β d... h β d q ) w q, { d 1 if d q := is odd, if d is odd. d It is shown in [5] that the roots of G h1,d 1 z) and G h,d w) are simple and none of them is on the unit circles z = 1 and w = 1 respectively. Acknowledgment. The first author would like to thank the Management of SSN College of Engineering, Kalavakkam , Tamil Nadu, India. w References [1] A. Aldroubi and M. Unser, Sampling procedure in function spaces and asymptotic equivalence with Shannon s sampling theory, Numer. Funct. Anal. Optim., ), 1 1. [] A. Aldroubi and K.Gröchenig, Nonuniform sampling and reconstruction in shiftinvariant spaces, SIAM Rev., 43001), [3] C. K. Chui, Multivariate Splines, SIAM Regional Conference series in Applied Mathematics,1988. [4] C. de Boor, K. Höllig and S. Riemenschneider, Box Splines, Applied Mathematical Sciences, 98. Springer-Verlag, New York, [5] P. Devaraj and S. Yugesh, On the zeros of the generalized Euler-Frobenius Laurent polynomial and reconstruction of cardinal splines of polynomial growth from local average samples, J. Math. Anal. Appl., 43015), [6] A. G. Garcia and G. Perez-Villalon, Multivariate generalized sampling in shift-invariant spaces and its approximation properties, J. Math. Anal. Appl., ), [7] A. G. Garcia, M. J. Munoz-Bouzo and G. Perez-Villalon, Regular multivariate sampling and approximation in L p shift-invariant spaces, J. Math. Anal. Appl., ), [8] J. Xian and W. Sun, Local sampling and reconstruction in shift-invariant spaces and their applications in spline subspaces, Numer. Funct. Anal. Optim., 31010),

11 S. Yugesh and P. Devaraj 57 [9] G. Perez-Villalon and A. Portal, Reconstruction of splines from local average samples, Appl. Math. Lett., 501), [10] I. J. Schoenberg, Cardinal Spline Interpolation, SIAM Regional Conference series in Applied Mathematics, 1973.