Interpolatory Wavelets on the Sphere x0. Introduction - CiteSeerX

Interpolatory Wavelets on the Sphere Daniel Potts and Manfred Tasche Dedicated to Prof. E. W. Cheney on the occasion of his 65th birthday

Abstract. In this paper,3we construct an interpolatory wavelet basis

on the unit sphere S  lR . Using spherical coordinates, we apply the tensor product of interpolatory trigonometric and algebraic polynomial wavelets. The described decomposition and reconstruction algorithms work in the frequency domain.

x0. Introduction The wavelet theory is closely related to shift{invariant subspaces of

L2(lRd). Therefore, wavelets are naturally adapted to problems on the

whole space lRd. Very often, one has to deal with functions de ned on a bounded domain such that the construction of wavelets on a domain is desirable. This problem is solved for the torus T := lR=2Z [5], and for compact intervals [1, 4]. In the sequel, we will apply 2{periodic trigonometric polynomial wavelets [6] and algebraic polynomial wavelets on I := [?1; 1] [3, 9] in order to construct interpolatory wavelets on the unit sphere S  lR3 . Other approachs to wavelets on the sphere can be found in [2, 7]. Using modi ed spherical coordinates x := (x1 ; x2 ) 2 H := T  I , we can identify the sphere S with H by the mapping  : H ! S with (z1 ; z2 ; z3) = (x) 2 S ,

z1 := cos x1 cos x2 2 ; z2 := sin x1 cos x2 2 ; z3 := sin x2 2 ;

where T f1g corresponds to the north/south pole of S . Let F : S ! lR be given and let f := F   : H ! lR be its coordinate representation. Approximation Theory VIII Charles K. Chui and Larry L. Schumaker (eds.), pp. 335{342. Copyright oc 1995 by World Scienti c Publishing Co., Inc. All rights of reproduction in any form reserved. ISBN 981-02-2972-0

335

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D. Potts and M. Tasche

Then F 2 C 1(S ) if and only if f 2 C 1(H ) ful ls the following conditions (see [8]) f (x1 ; 1) = c; (0:1) (@x2 f )(x1 ; 1) = a cos x1 + b sin x1 ; (0:2) (@x1 @x2 f )(x1 ; 1) = ?a sin x1 + b cos x1 ; (0:3) where the mixed derivative exists for all x1 2 T. Here a ; b ; c denote some constants. Note that (0.1) corresponds to the continuity of F at both poles of S . The conditions (0.2) { (0.3) assure that the tangent plane of F varies continuously at the poles of S . The aim of this paper is to construct an almost smooth wavelet decomposition of functions de ned on the unit sphere S . Roughly spoken, a function F is almost smooth on S , if f := F   2 C 1(H ) with (0.1) satis es the conditions (0.2) { (0.3) approximately. This paper is organized as follows. In Sections 1 and 2, we brie y recall the construction of trigonometric polynomial wavelets on T and of algebraic polynomial wavelets on I . Using tensor product approach, we introduce a multiresolution of the weighted Hilbert space L2w (H ) in Section 3. Finally, we describe decomposition and reconstruction algorithms.

x1. Trigonometric polynomial wavelets on T In the following, we sketch the construction of interpolatory trigonometric polynomial wavelets on T. For f 2 L2 (T), let Z 2 1 1 cu(f ) := 2 f (t) e?iut dt (u 2 Z ): 0 With the Dirichlet kernel Xl Dl1 := 21 + cos(k
) (l 2 lN) k=1

and Mj := 2j+3 (j 2 lN0), we consider the de la Vallee Poussin kernel as scaling function of level j j ?1 5
2X 1 Mj '1j := 2j Dl1 : l=3
2j

Hence we have c1u('1j ) = c1?u('1j ) (u 2 Z ) and for u 2 lN0

8 < 1?1 u = 0; : : : ; 3
2j , 1 1 Mj cu('j ) = : 2 (5 ? 2?j u) u = 3
2j + 1; : : : ; 5
2j ? 1, 0 u = 5
2j ; 5
2j + 1; : : : .

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Note that the transformed two{scale relation of '1j reads as follows

c1u('1j ) = A1j+1;u c1u('1j+1) (u 2 Z ) (1:1) with A1j+1;u = A1j+1;?u = Aj+1;u+Mj+1 (u 2 Z ) and 82 u = 0; : : : ; 3
2j , < A1j+1;u := : 5 ? 2?j u u = 3
2j + 1; : : : ; 5
2j ? 1, 0 u = 5
2j ; : : : ; Mj . 1 : L2 (T) ! L2 (T) With h1j := 2=Mj we consider the shift operators j;k 1 f := f (
? kh1j ) (k 2 Ij1 ), where Ij1 denotes the index set de ned by j;k 1 '1j . f0; : : : ; Mj ? 1g. Introduce the shifted scaling functions '1j;k := j;k Then '1j;k are Lagrange fundamental functions with respect to the equidistant grid G1j := h1j Ij1, since we have '1j;k (lh1j ) = k;l (k; l 2 Ij1). As sample space of level j we use Vj1 := span f'1j;k : k 2 Ij1g. Obviously, the operator L1j : C (T) ! Vj1 de ned by L1j f :=

X

k2Ij1

f (kh1j ) '1j;k (f 2 C (T))

is the interpolation projector onto Vj1 with respect to the grid G1j . Then 2 fVj1g1 j =0 forms a multiresolution of L (T) (see [5, 6]). Setting j1 := (2'1j+1 ?'1j )(
?h1j+1), then we obtain c1u( j1 ) = c1?u( j1 ) (u 2 Z ) and for u 2 lN0 8 (2?j u ? 3) !u u = 3
2j + 1; : : : ; 5
2j ? 1, > j +1 < u j ; : : : ; 3
2j +1 , Mj+1 c1u( j1 ) = > 2 !j+1?j?1 u u = 5
2j+1 u = 3
2 + 1; : : : ; 5
2j+1 ? 1, : (50 ? 2 u) !j+1 otherwise with !j := exp(?2i=Mj ). Hence the transformed two-scale relation of j1 reads as follows (1:2) c1u( j1 ) = Bj1+1;u c1u('1j+1) (u 2 Z ) with Bj1+1;u = Bj1+1;?u = Bj+1;u+Mj+1 and

8 0 u = 0; : : : ; 3
2j , < Bj1+1;u := : (2?j u ? 3) !ju+1 u = 3
2j + 1; : : : ; 5
2j ? 1, 2 !ju+1 u = 5
2j ; : : : ; Mj .

Let Wj1 := Vj1+1 Vj1 be the wavelet space of level j . Then we obtain that 1 := j1 (
? kh1j ) : k 2 Ij1 g. Wj1 = span f j;k

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x2. Algebraic polynomial wavelets on I Now we brie y describe interpolatory polynomial wavelets on the interval I . Let w(x) := (1 ? x2)?1=2 (x 2 (?1; 1)). For f 2 L2w (I ), let

c2 (f ) := 2 n



Z

I

w(x) f (x) Tn (x) dx (n 2 lN0 );

where Tn denotes the n-th Chebyshev polynomial. With the Chebyshev{ Dirichlet kernel Xl 1 2 Dl := 2 + Tk (l 2 lN) k=1

and Nj := 2j+2 (j 2 lN0); we consider the corresponding de la Vallee Poussin mean as scaling function '2j of level j : j ?1 5
2X 1 2 Dl2 : Nj 'j := 2j l=3
2j

Hence we have

8 < 2 n = 0; : : : ; 3
2j , Nj c2n('2j ) = : 5 ? 2?j n n = 3
2j + 1; : : : ; 5
2j ? 1, 0 n = 5
2j ; 5
2j + 1; : : : .

The transformed two{scale relation of '2j reads as follows

c2n('2j ) = A2j+1;n c2n('2j+1 ) (n 2 lN0)

(2:1)

with A2j+1;n = A2j+1;n+Nj+2 , A2j+1;Nj+2?l = A2j+1;l (l = 0; : : : ; Nj+1) and

82 < ?j n = 0; : :j: ; 3
2j , 2 Aj+1;n := : 5 ? 2 n n = 3
2 + 1; : : : ; 5
2j ? 1, 0 n = 5
2j ; : : : ; Nj+1 .

Setting h2j;v := cos Nvj (v 2 Z ), we introduce the Chebyshev{shift operators 2 : L2w (I ) ! L2w (I ) by c2n ( 2 f ) = h2 c2n (f ) (f 2 L2w (I )). Put '2 := j;v j;v j;nv j;v 2 '2 (v 2 Z ). Then '2 2 5
2j ?1 ful ls the interpolation properties j;v j j;n '2j;n(h2j;l) = "?1  with "j;0 = "j;Nj := 1=2, "j;l := 1 (l = 1; : : : ; Nj ? 1). n;l j;l 2 Hence 'j;n are modi ed Lagrange fundamental polynomials related to the Chebyshev grid G2j := fh2j;k : k 2 Ij2g with Ij2 := f0; : : : ; Nj g.

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As sample space of level j (j 2 lN0) we use Vj2 := span f'2j;k : k 2 Ij2g. Obviously, the operator L2j : C (I ) ! Vj2 de ned by

L2j f :=

X

k2Ij2

"j;k f (h2j;k ) '2j;k (f 2 C (I ))

is the interpolation projector onto Vj2 with respect to the grid G2j . Then 2 fVj2g1 j =0 forms a multiresolution of Lw (I ) (see [3, 4, 9]). Introducing j2+1 2 Vj2+1 by

8 2?j n ? 3 n = 3
2j + 1; : : : ; 5
2j ? 1, > < 5
2j ; : : : ; 3
2j+1, Nj c2n( j2+1 ) := > 25 ? 2?j?1 n nn = = 3
2j+1 + 1; : : : ; 5
2j+1 ? 1, : 0

otherwise,

then we obtain the transformed two{scale relation of j2+1

c2n( j2+1 ) = Bj2+1;n c2n('2j+1) (n 2 lN0)

(2:2)

with Bj2+1;n = Bj2+1;n+Nj+2 (n 2 lN0), Bj2+1;Nj+2?l = Bj2+1;l (l 2 Ij2+1) and

80 n = 0; : : : ; 3
2j , < ?j 2 Bj+1;n := : 2 n ? 3 n = 3
2j + 1; : : : ; 5
2j ? 1, 2 n = 5
2j ; : : : ; Nj+1 .

Let Wj2 := Vj2+1 Vj2 be the wavelet space of level j . Then we have Wj2 = span f j2+1;2k+1 := j2+1;2k+1 j2+1 : k 2 Kj2 g with Kj2 := f0; : : : ; Nj ? 1g.

x3. Multiresolution of L2w (H ) Let L2w (H ) be the Hilbert space of all functions f : H ! lR with 22 kf k2 :=

Z

D

w(x2 ) jf (x)j2 dx < 1

with D := [0; 2]  I and x := (x1 ; x2 ). Applying the tensor product method, we introduce a multiresolution of L2w (H ). For f 2 L2w (H ), let

2 ck (f ) :=

Z

D

w(x2 ) f (x) e?ik1 x1 Tk2 (x2 ) dx

with k := (k1 ; k2) 2 Z  lN0. As sample space of level j we use Vj := Vj1 Vj2 . Hence we have Vj = span f'j;k : k 2 Ij g with Ij := Ij1  Ij2

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D. Potts and M. Tasche

and 'j;k := '1j;k1 '2j;k2 . By (1.1) and (2.1), the transformed two{scale relation of the j -th scaling function 'j := '1j '2j reads as follows

ck ('j ) = Aj+1;k ck ('j+1) (k 2 Z  lN0)

(3:1)

with Aj+1;k := A1j+1;k1 A2j+1;k2 . Since we have j;l2 'j;k (hj;l ) = k1 ;l1 k2 ;l2 (k; l 2 Ij ) for hj;l := (l1h1j ; h2j;l2 ), the shifted scaling functions 'j;k are modi ed Lagrange functions with respect to the grid Gj := G1j  G2j . Therefore, the operator Lj := L1j L2j : C (H ) ! Vj given by

Lj f :=

X

k2Ij

"j;k2 f (hj;k ) 'j;k (f 2 C (H ))

is the interpolation projector onto Vj with respect to the grid Gj . The points of (Gj ) are lying more dense at the poles of S . If f 2 C 1(H ) with (0.1) { (0.3) is given, then Lj f ful ls (0.1) exactly and (0.2) { (0.3) approximately in the following sense that the partial derivatives in (0.2) { (0.3) must be replaced by corresponding dierence quotients on Gj . From Lj f jGj = f jGj it follows that Lj f is almost smooth. Then fVj g1 j =0 forms 2 sense: a multiresolution of Lw (H ) in theSfollowing 1 (M 1) Vj  Vj+1; clos( j=0 Vj ) = L2w (H ): (M 2) For all j 2 lN0 and for any j;k 2 lR (k 2 Ij ), we have 1 X " j j2  M N k X "  ' k2  X " j j2: j j j;k2 j;k j;k j;k2 j;k 4 k2Ij j;k2 j;k k2Ij k2Ij As known, the orthogonal complement Wj of Vj in Vj+1 is generated by Vj1, Vj2 , Wj1 and Wj2. We obtain that Wj = WjI  WjII  WjIII with the wavelet spaces WjI := Vj1 Wj2, WjII := Wj1 Vj2 and WjIII := Wj1 Wj2. Then we have I; III : k 2 Kj g; W II = span f II : k 2 Ij g WjI; III = span f j;k j j;k I := '1 2 II 1 2 with Kj := Ij1 Kj2 and with j;k j;k1 j +1;2k2 +1 , j;k := j;k1 'j;k2 III := 1 2 and j;k j;k1 j +1;2k2 +1 . By (1.1) { (1.2) and (2.1) { (2.2), the transformed two{scale relations of the wavelets jI := '1j j2+1 , jII := 1 2 III 1 2 j 'j and j := j j +1 read as follows

cl( jI?III ) = BjI?III +1;l cl('j +1 ) (l = (l1 ; l2 ) 2 Z  lN0 )

(3:2)

with BjI+1;l := A1j+1;l1 Bj2+1;l2 , BjII+1;l := Bj1+1;l1 A2j+1;l2 and BjIII+1;l := Bj1+1;l1 Bj2+1;l2 .

341

Wavelets on the Sphere

x4. Decomposition and reconstruction algorithms Now we derive ecient decomposition and reconstruction algorithms. In order to decompose a given function fj+1 2 Vj+1 of the form X fj+1 = "j+1;k2 j+1;k 'j+1;k ; (4:1) k2Ij+1

uniquely determined functions fj 2 Vj and gjI?III 2 WjI?III have to be found such that fj+1 = fj + gjI + gjII + gjIII . Assume that the coecients j+1;k 2 lR of fj+1 or their transformed data

^j+1;l :=

X

1 l1 cos k2 l2  "j+1;k2 j+1;k !jk+1 Nj+1 k2Ij+1

(4:2)

are known. The wanted functions fj 2 Vj and gjI?III 2 WjI?III can be uniquely represented in the form (4.1) and X I;III I;III X II II j;k j+1;k ; gjII = "j;k2 j;k gjI;III = j +1;k k2Kj

k2Ij

I?III 2 lR. Let ^ j;l ; ^I?III 2 C be transwith unknown coecients j;k ; j;k j;l formed data given in the form (4.2) or by X ^I;III := I;III !k1l1 cos (2k2 + 1)l2 ; (4:3) j

Nj+1 II := X "j;k II ! k1 l1 cos k2 l2  : ^j;l 2 j;k j Nj k2Kj

j;l

k2Ij

j;k

j

(4:4)

Note that the discrete transforms (4.2) { (4.4) and their inverses can be realized by the row{column method using fast Fourier transform and fast algorithms of discrete cosine transforms. In order to reconstruct fj+1 2 Vj+1 (j 2 lN0), we have to compute the sum fj+1 = fj + gjI + gjII + gjIII with given functions fj 2 Vj and I?III 2 lR or their transformed data gjI?III 2 WjI?III . Assume that j;k ; j;k are known. Then fj+1 2 Vj+1 can be uniquely represented in the form (4.1). The decomposition and reconstruction algorithms are based on the I?III . Introducing for l = following connection between ^j+1;l and ^j;l ; ^j;l (l1 ; l2) 2 Ij the two-scale symbol matrices

 A1 Bj1+1;l1  j +1 ;l 1 1 ; Sj+1;l1 := A1 1 j +1;Mj +l1 Bj +1;Mj +l1  A2  2 B j +1 ;l j +1 ;l 2 2 2 Sj+1;l2 := A2 2 j +1;Nj+1 ?l2 ?Bj +1;Nj+1?l2

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D. Potts and M. Tasche

and the Kronecker product Sj+1;l := Sj1+1;l1 Sj2+1;l2 , then we obtain by (3.1) { (3.2) that for l 2 Kj

0 ^j;l 1 0 ^j+1;l1;l2 1 I C B ^j;l  ^ j+1;l1;N +1?l2 C B B @ ^j+1;M +l1;l2 A = Sj+1;l @ ^j;lII C A j

j

^j+1;Mj +l1;Nj+1?l2

and for l1 2 Ij1

III ^j;l

 ^j;l1;N    ^ j +1;l1;N 1 : ^j+1;M +l1;N = Sj+1;l1 ^II j

j

j

j

j;l1 ;Nj

Note that Sj+1;l (l 2 Kj ) and Sj1+1;l1 (l1 2 Ij1) are regular matrices.

Acknowledgments. This research was supported by the Deutsche For-

schungsgemeinschaft.

References

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