Barysymmetric Multiwavelets on Triangle

Thomas P.Y. Yu�, , Krasmir Kolarovy, William Lynch1

Interval Research Corporation

IRC Technical Report: 1997-006

March 11, 1997

Abstract

In this paper, we give explicit construction of multiwavelets on polygonal region inR

2 that is associated with a nested triangular tessellation. Two di�erent constructionswill be presented. The �rst construction is very similar to Alpert's construction in[3], but unlike the latter 1-D construction, in which case symmetry of basis functionscomes in almost automatically, the multiwavelets from our �rst construction possessno symmetry of any sort. We de�ne a form of symmetry for functions that \live" ontriangles, which we call barysymmetry, and establish various results about it. We thenshow that by sacri�cing some vanishing moments in the �rst constructions, we canconstruct multiwavelets which possess barysymmetry.

Typical members of the bases from both constructions have at least M > 0 vanish-ing moments, but are discontinuous. We discuss how to apply moment-interpolationschemes to improve these orthonormal bases, which gives rise to their smooth biorthog-onal counterparts.

1 Introduction

Wavelet is typically designed on simple domains like Rn or rectangular subset of Rn. Mo-

tivated by applications on more general domains, for example, compression of data de�ned

on the sphere or more general surfaces [12], we tackle the problem of designing multires-

olution analysis on general manifolds. In this paper, we make some preliminary attempts

by constructing multiwavelets on triangulated domains in R2. Wavelets are invariably de-

signed with the aid of Fourier analysis, which is easily accessible in simple domain like R.

On complicated domains Fourier analysis becomes less accessible and we need to explore

design methodologies which are less dependent on Fourier means.

Alpert's design of multiwavelets in [3] and their smoother biorthogonal counterparts [10]

are only dependent on simple polynomial operations, e.g. Gram-Schmidt orthogonalization

and polynomial interpolation, and hence possess high potential to be generalized to more

complicated domains. In section 2 we revise the construction of Alpert multiwavelets in

1-D [3] and their smooth biorthogonal counterparts by Donoho, Dyn, Levin and Yu [10]. In

Section 3 we describe how to generalize Alpert's construction to triangles. Unlike the 1-D

case where symmetry of the resulted basis functions come in automatically, the resulted ba-

sis functions from the direct generalization possess no symmetry of any kind. In Section 4,

�Interval Research Corporation and SCCM, Department of Computer Science, Stanford Universityy1 Interval Research Corporation

1

we de�ne a form of symmetry for functions de�ned on triangles, which we will call barysym-

metry. We then show that one can construct multiwavelets which possess barysymmetry,

with the cost of sacri�cing some of the vanishing moments gotten in the �rst construction.

In section 5, we derive the �lter banks associated with the multiwavelets constructed in this

paper. We end this paper by presenting some of the possible generalizations in Section 6.

2 Alpert Bases in R1 and their smooth bi-orthogonal duals

Let �`(x) denote the l-th Legendre polynomial for the interval [0; 1], mutilated to have

support [0; 1]. Thus

�0(x) = 1[0;1](x) (2.1)

�1(x) =p12(x� 1=2)1[0;1](x) (2.2)

�2(x) =p180((x� 1=2)2 � 1=12)1[0;1](x) (2.3)

etc. For j and k integers, let Ij;k = [k=2j ; (k + 1)=2j) denote a typical dyadic interval, and

de�ne

�`j;k = 2j=2�`(2jx� k); (2.4)

the translate and dilate of �` that \lives" on Ij;k and has L2(R)-norm 1. Any two such

functions �`j;k, �l0

j;k0 with the same scale index j and di�erent k 6= k0 or l 6= l0 are orthogonal.

Fix m > 0 and set

V j = V(m)

j = Span (�`j;k : k 2 Z; 0 � ` < m)

V j is the collection of all piecewise polynomials which are of degree < m on intervals

Ij;k; k 2 Z.The operator P jf =

Pk2Z

Pm�1l=0 hf; �`j;ki�`j;k gives orthogonal projection onto V j . It is

convenient below to adopt the convention that �j;k denotes the vector of functions (�`j;k)

m�1`=0 ,

and that hf; �j;ki is a vector of inner products. Then we can write P jf =P

khf; �j;kiT�j;k,avoiding the summation over the ` index.

The spaces V j are nested: V j � V j+1; indeed a piecewise polynomial with pieces (Ij;k)kis also a piecewise polynomial on the �ner collection of pieces (Ij+1;k)k. It follows that the

basis elements (�`j;k)k;l all have a representation in terms of elements (�`j+1;k)k;l for V j+1.

In vector notation we write this as

�j;k = H0�j+1;2k +H1�j+1;2k+1 (2.5)

where the Hi are m bym matrices with entries (Hi)`;`0 = h�`0;0; �`0

1;ii. 0 � `; `0 < m, i = 0; 1.

In the scalar case m = 1, the Hi are scalars and H0 = H1 = 1=p2. In the simplest

vector case m = 2

H0 =

"� 0

�� �

#H1 =

"� 0

� �

#

where � = 1=p2, � =

p3=8, � = 1=

p8. (software is available for the general case m > 2.

We consider now the detail space W j � V j and associated projection Qjf = P j+1f �P jf . W j consists of piecewise polynomials with pieces (Ij+1;k)k which are orthogonal to

2

piecewise polynomials with coarser pieces (Ij;k)k. An orthogonal basis (h`j;k)k;l for W j can

be constructed as in Alpert [1]. As W j � V j+1, each h`j;k can be expressed in terms of

(�`0

j+1;k0)k0;`0 :

hj;k = G0�j+1;2k + G1�j+1;2k+1 (2.6)

(in vector notation). In the case m = 1, the Gi are scalars: G0 = 1=p2; G1 = �1=

p2. In

the case m = 2, the Gi obey

G0 =

"� �

0 �

#G1 =

"�� �

0 ��

#:

The functions h`j;k are piecewise polynomials supported in Ij;k with knots at the endpoints

and midpoint of Ij;k. For pictures see [3] or [2]. We mention that each basis functions from

Alpert's construction is either symmetric or antisymmetric.

A key fact about the two-scale matrices we have just de�ned is that the 2m�2m matrix

U =

"H0 H1

G0 G1

#

is orthogonal. This is equivalent to saying we have two di�erent orthogonal bases for

V 1 : f(�`1;k)k;`g and f(�`0;k)k;`; (h`0;k)k;`g.Members of Alpert bases are discontinuous, [10] provided an improvement of the Alpert

bases via moment-interpolation, which gave rise to their smooth bi-orthogonal counterparts.

Moreover, symmetry of the Alpert bases is preserved in this process.

3 Nonseparable construction in R2

Assume we start from a polygon R � R2 that can be triangulated by fT0;kjk 2 Ig, i.e.R = [disjointT0;k:

R could be a �nite or in�nite subset of R2. Consider successive re�nement fTj;kjj > 0; k 2Ijg where each triangle Tj;k in a �ner scale is constructed from one in a coarser level by

midpoint subdivision, denoted by

Tj;k = Tj+1;k0 [ Tj+1;k1 [ Tj+1;k2 [ Tj+1;k3 :

Note that when jIj < 1, jIj j = 4j jIj. For consistency, we let I0 = I. By convention, we

index the center and the three peripheral subtriangles in the mid-point subdivision of Tj;kby Tj+1;k0, Tj+1;k1 , Tj+1;k2 and Tj+1;k3 respectively; and when no confusion occurs, simply

by T0, T1, T2, T3. We also denote the edges of Tj+1;k0 by e1j;k, e

2j;k, e

3j;k or simply by e1, e2,

e3, so that ei represents the common boundary of T0 and Ti.

De�ne

Pd = fxiyj j i+ j � dg andPd(T ) = ff j f jT 2 Pd ; f jR2nT = 0g

We have dim(Pd) = dim(Pd(T )) = M =(d+1)(d+2)

2. We will also denote by h�; �i the usual

innerproduct in L2(R2). De�ne h�; �iT by hf; giT =R R

T f g dx dy. (Of course, if either f

or g is compactly supported at T , then hf; gi = hf; giT .)

3

Let T = T0[T1[T2[T3 be the midpoint subdivision of an arbitrary triangle T . Denote

V = Pd(T0)� Pd(T1)� Pd(T2)� Pd(T3)

and

W = V Pd(T ):

Note that dim(V ) = 4M and dim(W ) = 3M . Our goal in this section is to develop

orthonormal basis for Pd(T ) and W . An orthonormal basis for Pd(T ) will generally be

denoted by f�lT j0 � l < Mg or simply by �T . The latter notation is understood as a vector

of functions. An orthonormal basis of W will generally be denoted by fhlei j0 � l < M ; i =

1; 2; 3g or, simply by the vector notation h1; h2 and h3.

Once such orthonormal basis is constructed, one has orthonormal basis for L2(R) of the

form

f�T0;k ; heij;kji = 1; 2; 3; j � 0; k 2 Ijg; (3.7)

such that arbitrary member of L2(R2) can be decomposed as

f =Xk2Ihf; �T0;kiT�T0;k +

3Xi=1

Xj�0

Xk2Ijhf; hej;kiT hei

j;k(3.8)

where hf; �T0;ki is understood as

hf; �T0;ki = [hf; �0T0;ki; � � � ; hf; �M�1

T0;ki]T :

and similar interpretation applies to hf; heij;ki.

Homogeneous expansion (counterparts of Equation 3.8 where the �rst term of right

hand side disappears and the second term has j runs from �1 to 1) is also possible, but

will enforce impractical uniformity of the triangular tessellation. Therefore we will only

consider inhomogeneous expansion in this paper.

MRA. Given a nested triangular tessellation fTjg of R � R2, Let

Vj =Mk2Ij

Pd(Tj;k):

be the space of piecewise polynomials of degree less than d on each of Tj;k. We have

� V j � V j+1

� limj!1 V j = L2(R2)

� f�lTj;k j0 � l < M; k 2 T jg form an orthonormal basis for V j

� Each of �lTj;k is compactly supported at Tj;k.

� For each Tj;k, there exists four M by M matrices Hi such that

�Tj;k =3X

i=0

Hi �Tj+1;ki(3.9)

The two-scale relation in Equation 3.9 holds regardless of the choice of f�lTj;kg, because anyfunction which is piecewise polynomial on Tj+1;k is also piecewise polynomial on Tj;k.

4

3.1 Barycentric coordinates

Given any (nondegenerate) triangle T � R2 speci�ced by an ordered list of vertices P1 =

fx1; y1g; P2 = fx2; y2g; P3 = fx3; y3g, one can express any point P = fx; yg in R2 in terms

of its barycentric coordinates � = (�1; �2; �3)T with respect to T .

P =3X

i=1

�iTi:

The �i are conventionally made unique by the normalization requirement

j� j :=3X

i=1

�i = 1:

If P is inside T , one gets

�i � 0 for i = 1; 2; 3:

An important property of barycentric coordinates is their a�ne invariance: if the triangle

T together with P is transformed by an a�ne transformation �, then the barycentric

coordinates of �(P ) with respect to �(T ) is exactly the same as that of P with respect

to T . If we think of � and �i as functionals �(P; T ) and �i(P; T ) , then the above can be

written compactly as �(P; T ) = �(�(P );�(T )).

3.1.1 Change of coordinates

Change of coordinates from Cartesian to barycentric coordinates is very easy, one has the

transformation rule: x(�1; �2; �3)

y(�1; �2; �3)

!=

x(�1; �2)

y(�1; �2)

!=

x1 � x3 x2 � x3y1 � y3 y2 � y3

! �1�2

!+

x3y3

!: (3.10)

Hence the Jacobian for the change of coordinates is

jdet(

x1 � x3 x2 � x3y1 � y3 y2 � y3

!j = 2jT j:

Hence, integral of functions de�ned on triangles can be calculated easily by

Z ZTf(x; y) dx dy = 2jT j

Z 1

0

Z 1��1

0

f(x(�1; �2; 1� �1 � �2) ; y(�1; �2; 1� �1 � �2)) d�2 d�1

3.1.2 Mutilation of orthonormal basis to arbitrary triangles

Given an orthonormal basis of Pd(T ), expressed in terms of barycentric coordinates with

respect to T : �lT = �l(�1; �2; �3), one can immediately write down an orthonormal basis

for arbitrary P (T 0) by:

�lT 0(�1; �2; �3) =

sjT jjT 0j�

l(�1; �2; �3) (3.11)

where (�1; �2; �3) is interpreted with respect to T 0 on the left hand side and with respect to

T on the right hand side. It may sound confusing at the �rst glance, but the true power

5

of barycentric coordinates is that it allows us to think of �l(�1; �2; �3) as a \symbol" in

terms of the symbol triplet (�1; �2; �3), rather than as a function. For example, when we say

\�l(�1; �2; �3) = cos(�1)��2=�3" without specifying the reference triangle for (�1; �2; �3), thenall we can say is that �l(�1; �2; �3) is the symbol \cos(�1)��2=�3", rather than a well de�ned

function �l : R2 ! R. The potential ambiguity turns out to be a useful property, because

when a user request for an orthonormal basis of an arbitrary P (T 0), we can then interpret

each triplet of symbols (�1; �2; �3) as a physical point in R2 by letting T 0 be the reference

triangle, and then the symbols �l(�1; �2; �3) become well-de�ned functions. Followed by

proper scalings, they become to form an orthonormal basis for Pd(T0).

One can think of \translation" and \dilation" in the 1-D setting be naturally captured

by the a�ne invariance of barycentric coordinates in the 2-D triangular setting. In fact,

one can think of Equation 3.11 as just a 2-D version of Equation 2.4.

As a convention (to avoid future confusion, especially in Section 3.2.2 and 3.2.2), when

we subscript a function variable by a triangle variable (which in turn must be speci�ed as

an ordered list of three points in R2), e.g. \fT (�1; �2; �3)", it is understood that (�1; �2; �3)

are barycentric coordinates relative to triangle T .

3.1.3 Change of reference triangle

If a point P has barycentric coordinates � = (�1; �2; �3) relative to triangle T (de�ned by

f(xi; yi)ji = 1; 2; 3g. Then the barycentric coordinates � 0 of P relative to triangle T 0 (de�ned

by f(xi; yi)ji = 1; 2; 3g) can be calculated by:

(� 01; �02; �

03) = (�1; �2; �3)MT!T 0

where

M 0T!T 0 =

264 x01 x02 x03y01 y02 y031 1 1

375�1 264 x1 x2 x3

y1 y2 y31 1 1

375 (3.12)

3.1.4 Midpoint subdivision

Midpoint subdivision can also be easily described using barycentric coordinates. Let T =

T0 [ T1 [ T2 [ T3 be the midpoint subdivision of an arbitrary triangle T . Let (�1; �2; �3) be

de�ned with respect to T , then

T0 = f(�1; �2; �3)j0 � �1; �2; �3 � 1=2gT1 = f(�1; �2; �3)j�1 � 1=2; �2; �3 � 0gT2 = f(�1; �2; �3)j�2 � 1=2; �1; �3 � 0gT3 = f(�1; �2; �3)j�3 � 1=2; �1; �2 � 0g

Later we will study symmetry of functions de�ned on triangles, and the relative ordering

of vertices in the speci�cation of Ti become relevant. We adopt the convention that if T

is speci�ed by the ordered list T = fP1; P2; P3g, then Ti will be speci�ed by the following

ordered lists:

T0 = fP1 + P2

2;P2 + P3

2;P3 + P1

2g; T1 = fP1;

P1 + P2

2;P3 + P1

2g;

T2 = fP2;P2 + P3

2;P1 + P2

2g; T3 = fP3;

P3 + P1

2;P2 + P3

2g: (3.13)

6

3.2 Construction I

We now present a construction of multiwavelets for our setting, which is similar to the

construction by Alpert in 1-D.

3.2.1 Scaling Functions

We �rst seek for scaling functions �lT0;k which are orthogonal polynomials on T0;k and vanish

outside so that

h�lT0;k�l0

T0;kiR2 = h�lT0;k�l

0

T0;kiT0;k = �ll0 :

Virtually any choice of orthogonal polynomial systems (with respect to the h�; �iT0;k inner-

product) is valid. Due to our experience in 1-D, we will consider Legendre-typed polynomial

basis. Legendre polynomials on triangles can easily be constructed as follows. First of all,

due to the comment at the end of last section, we can restrict our attention to the base trian-

gle T = TB with vertices (0; 0); (0; 1); (1; 0). Consider the power basis f1; x; y; x2; xy; y2; : : :gwhich is equal to f1; �2; 1� �1 � �2; �

22 ; (1� �1 � �2)�2; (1� �1 � �2)

2; : : :g in barycentric co-

ordinates with respect to TB . Applying the Gram-Schimitt process to the latter sequence

(in the listed ordering) with respect to the inner product h�; �iTB will give us Legendre

polynomials f�ljl � 0g on TB .

De�ne, for l � 0, �lTB = �l1TB . The resulted sequence S = f�lTBg becomes a triangular

sequence of orthogonal polynomials in R2: for any d � 0, the �rst M = (d + 1)(d + 2)=2

elements of S form an orthonormal basis for both Pd(TB) and Pd(R2). Using the procedure

described above, we obtain the �rst few members of S as

�0TB(�1; �2) =

p2 1TB (�1; �2)

�1TB(�1; �2) = (�2 + 6�2) 1TB (�1; �2)

�2TB(�1; �2) = 2

p3(1� 2�1 � �2) 1TB (�1; �2)

�3TB(�1; �2) =

p6(1� 8�2 + 10�22 ) 1TB (�1; �2)

�4TB(�1; �2) = 3

p2(�1 + 2�1 + 6�2 � 10�1�2 � 5�22 ) 1TB (�1; �2)

�5TB(�1; �2) =

p30(1� 6�1 + 6�21 � 2�2 + 6�1�2 + �22 ) 1TB (�1; �2)

etc.. 1 See Figure 1 for a graphical display. By Equation 3.11, we can choose the scaling

functions as

�lTj;k =

s1

2jTj;kj�lTB ; 0 � l < M: (3.14)

3.2.2 Mother Multiwavelets

Having de�ned an orthonormal basis for Pd(T ), we now present a construction for an

orthonormal basis for

W = (Pd(T0)� Pd(T1)� Pd(T2)� Pd(T3)) Pd(T ):

Before we proceed, the authors would like to mention that the following construction is

strongly motivated by the 1-D construction in [3].

1When a function f(�1; �2; �3) is de�ned in terms of barycentric coordinates, we abuse the notation by

writing f(�1; �2) = f(�1; �2; 1� �1 � �2).

7

0

0.5

1

0

−2

−1

0

1

2

31

−2

0

2

2

0

2

4

63

−4

−2

0

2

44

−2

0

2

45

Figure 1: Legendre Polynomials on a triangle, d � 2

Recall that dim(W ) = 3M . We will generally denote by

fhliji = 1; 2; 3; 0 � l < Mg

any set of 3M functions that can be combined with a basis of P (d) to span Pd(T0)�Pd(T1)�Pd(T2)� Pd(T3).

Let f�lT g be the legendre polynomials mutilated to the triangle T (c.f. Equation 3.11.)

Consider

hli(�1; �2; �3) =

8><>:

�lTi((�1; �2; �3)MT!Ti) on Ti;

��lTi((�1; �2; �3)MT!Ti) on TnTi0 on R2nT

(3.15)

i = 1; 2; 3; l = 0 : : : (M � 1) (Note: (�1; �2; �3) in Equation 3.15 is de�ned relative to the

triangle T .) Notice that fhlig [ f�lT g spans Pd(T0)� Pd(T1)� Pd(T2)� Pd(T3).

We will modify fhlig successively to make an orthonormal basis for W .

We �rst orthogonalize fhlig against Pd(T ) by

hli � hli �M�1Xl=0

hhli; �lT i�lT ;

then span(fhlig) = Pd(T )? \L3

i=0 Pd(Ti). Note also that each member of fhlig now has

M = (d+ 1)(d + 2)=2 vanishing moments:

hhli; xpyqi = 0 for p+ q � d; p; q � 0:

8

In principle, all we have to do next is to orthogonalize span(fhlig) by Gram-Schmidt to

get an orthonormal basis for W . It is, however, possible to arm fhlig with more vanishing

moments before orthogonalization among themselves.

To do so, we order fhlig as

L = [h01; h02; h

03; h

11; h

12; h

13; : : : ; h

(M�1)

1 ; h(M�1)

2 ; h(M�1)

3 ]: (3.16)

In the following, we will generally denote the members of L by L = [h1; h2; : : :].

Recall that every member in L is orthogonal to �lT for l < M . Initialize t = M ,

�target = �tT and L0 = [ ] (empty list.) We apply to L the following operations:

1. Find the �rst element, h, in the list L which is non-orthogonal to �target, or if no

such h exists, repeat: set t � t+ 1 and �target = �tT until we �nd such a h.

2. reorder L as

L � [h; [Lnh]]

3. Orthogonalize all but the �rst function in L against �target by

hi � hi �hhi; �targetihh1; �targeti

h1:

4. L � [h2; h3; : : :], L0 � [L0; h1]. If L is now an empty list, t � t+ 1 goto 5, else

goto 1.

5. L � L0

Note that span(L) is una�ected by the above operations, but that each hj now has at

least M + j � 1 vanishing moments.

Finally we orthogonalize L among themselves by the Gram-Schmidt process. In order

to retain the vanishing moments acquired in the last steps, we must Gram-Schmidt L in

the reverse order.

Again, by Equation 3.11, we can de�ne

hleij;k

=

s1

2jTj;kjhli ; 0 � l < M;

assuming fhlig is derived on the triangle T = TB . We have completed our �rst construction

for multiwavelets on L2(R).

Figure 2 and 3 display the multiwavelets from construction I for d = 0 and d = 1

respectivelty.

4 Restoring Symmetry

While the �rst construction is a natural generation of the 1-D construction in [3], the

resulted multiwavelets from construction I lack a nice property that come in e�ortlessly in

the 1-D case, namely, symmetry. As mentioned in Section 2, every member of the Alpert

basis is either symmetric or anti-symmetric about the midpoint of its supports. Symmetry

in wavelets is desirable in applications such as image coding [4] and curve and surfaces

9

0

0.5

1

χ T

−3

−2

−1

0h 1

0

−3

−2

−1

0h 2

0

−3

−2

−1

0h 3

0

Figure 2: Multiwavelets on triangle from construction I, d = 0

editing [9]. In 1-D, it is well known that compactly supported orthonormal dyadic wavelets

based on translation and dilation of a single function cannot possess symmetry, with the

Haar basis as a (trivial) exception. See, for example, Chapter 8 of [8] for a proof of this

fact. Various solutions have been proposed by wavelet designers to cure this problem, e.g.

by trading o� between vanishing moments with the amount of symmetry [8] in order to

acquire \more symmetry" in wavelets, by bi-orthogonal wavelets[7], by triadic dilation [6],

by complex-valued wavelets [13], and, of course, by multiwavelets.

The multiwavelets considered in Section 3.2.2 was constructed with the goal of opti-

mizing the number of vanishing moments, and symmetry was completely disregarded. In

the next section we will discuss a form of symmetry on triangles, and show in subsequent

sections that multiwavelets can be constructed that possess such kind of symmetry, with

the cost of sacri�cing some of the vanishing moments acquired in construction I.

4.1 BarySymmetry

For functions that \live" on a triangles, it is less clear what symmetry means. Alternatively,

it is easier to think of a set of functions being mutually symmetric to each other in some

sense, and aim at designing multiwavelet systems with that property. Apparently it is a

weak form of symmetry we are considering. Multiwavelets with this kind of symmetry have

been proposed in 1-D by Geronimo-Hardin-Massopust [11], in which a basis element may

not be symmetric or antisymmetric, but that a \mirror" of the element is always presented

in the basis. Precisely,

De�nition 4.1 A set of function S = fpi(�1; �2; �3) j i = 1 : : : ng is barysymmetric with

respect to T if pi(��(1); ��(2); ��(3)) 2 S for all � 2 CyclicGroup(f1; 2; 3g) and 1 � i � n.

10

0

0.5

1

χ 1

−2

0

2

χ 2

−2

0

2

χ 3

−4

−2

0

2

h 11

−5

0

5

h 21

−6

−4

−2

0

2

h 31

−2

0

2

4h 1

2

−2

0

2

4

6h 2

2

−3

−2

−1

0

h 32

−5

0

5h 1

3

−5

0

5

10

15h 2

3

−10

−5

0

5

10

h 33

Figure 3: Multiwavelets on triangle from construction I, d = 1

11

(Note: CyclicGroup(f1; 2; 3g) := f(1; 2; 3); (2; 3; 1); (3; 1; 2)g.)We could potentially de�ne a stronger type of barysymmetry if the cyclic groupCyclicGroup(f1; 2; 3g)

is replaced by the full permutation group

PermuGroup(f1; 2; 3g) = f(1; 2; 3); (2; 3; 1); (3; 1; 2); (1; 3; 2); (3; 2; 1); (2; 1; 3)g

in the above de�nition. We will simply use the term \strong-barysymmetry" for this mod-

i�ed de�nition.

De�nition 4.2 Given a function p : R2 ! R de�ned in terms of barycentric coordinates

with respect to T , the set S = fp(��(1); ��(2); ��(3)) j � 2 CyclicGroup(f1; 2; 3g) is called the

barysymmetric orbit of p with respect to T , denoted generally by Orbit(p).

Lemma 4.3 For any p, jOrbit(p)j = 1 or 3.

Proof. Assume jOrbit(p)j = 2. Without lost of generality, we can assume (i) p(�1; �2; �3) =

p(�2; �3; �1) for all (�1; �2; �3) and (ii) there exist a point (�1; �2; �3) such that p(�2; �3; �1) 6=p(�3; �1; �2). But these immediately lead to a contradiction because (i) implies p(�2; �3; �1) =

p(�3; �1; �2). Therefore, jOrbit(p)j = 1 or 3. 2

One can de�ne p be symmetric with respect to T if jorbit(p)j = 1, but we will focus on

barysymmetry in the following construction.

4.2 Symmetric Orthogonalization

In this section, we will prove that under some mild conditions, an arbitrary set of indepen-

dent and barysymmetric functions S 0 can always be orthogonalized into another barysym-

metric set S 0. The proof of this result is a constructive one, and we will use the ideas in

this section extensively in our next construction of multiwavelets.

Given p 2 L2(T ) with jjpjjT :=php; piT = 1, we de�ne

P(p) := hp(�1; �2; �3); p(�2; �3; �1)iT : (4.17)

The normalization condition jjpjjT = 1 implies�1 � P(p) � 1 (Cauchy-Schwarz inequality.)

Moreover, a simple change of variable argument shows that

hp(�1; �2; �3); p(�2; �3; �1)iT = hp(�2; �3; �1); p(�3; �1; �2)iT = hp(�3; �1; �2); p(�1; �2; �3)iT :(4.18)

In other words, the number P(p) completely speci�es the Gram matrix of orbit(p).

In this paper, we are mainly interested in functions such that

jorbit(p)j = dim(span(orbit(p))): (4.19)

Notice that not every function de�ned on a given triangle satis�es Equation 4.19. Consider

p(�1; �2; �3) =

8>>><>>>:

1 if �1 > 1=2;

�1 if �2 > 1=2;

0 if �3 > 1=2;

0 if �1; �2; �3 � 1=2:

(4.20)

Then jorbit(p)j = 3, but p(�1; �2; �3)+p(�2; �3; �1)+p(�3; �1; �2) � 0, which implies dim(span(orbit(p))) <

3. (In fact, dim(span(orbit(p))) = 2.)

12

Lemma 4.4 For any p 2 L2(T ) with jjpjjT = 1 and jorbit(p)j = dim(span(orbit(p))), we

have

�1=2 < P(p) � 1: (4.21)

Moreover, jOrbit(p)j = 1 if and only if P(p) = 1 (Equivalently, jOrbit(p)j = 3 if and only

if �1=2 < P(p) < 1.)

Proof.

If jOrbit(p)j = 1, then clearly P(p) = 1.

It is well known that if a set of vectors fv1; : : : ; vng in an inner-product space is linearly

independent, then the Gram matrix G = [Gij ], Gi;j = hvi; vji is symmetric positive de�nite.

Therefore if jOrbit(p)j = dim(span(orbit(p))) = 3, then the Gram matrix of orbit(p) is

G =

0B@ 1 P(p) P(p)P(p) 1 P(p)P(p) P(p) 1

1CA

and

det(G) = (1 + 2P(p))(1 �P(p))2 > 0

which happens when and only when �1=2 < P(p) < 1.

To conclude, jOrbit(p)j = 3 if and only if �1=2 < P(p) < 1 and jOrbit(p)j = 1 if and

only if P(p) = 1. 2

For the function de�ned by Equation 4.20, P(p) = �1=2. Therefore, the dimensionality

condition Equation 4.19 is necessary.

Lemma 4.5 Given p 2 L2(T ) that satis�es Equation 4.19, there exists q 2 L2(T ) such that

orbit(q) is an orthonormal set with respect to h�; �iT and span(orbit(p)) = span(orbit(q)).

Proof. We can assume without loss of generality that jjpjjT = 1 (if not, then replace p by

p=jjpjjT .) If jorbit(p)j = 1, then we just set q = p. Otherwise, jorbit(p)j = 3. In order to

orthogonalize jorbit(p)j in a barysymmetric fashion, de�ne q by

q(�1; �2; �3) = r1q(�1; �2; �3) + r2q(�2; �3; �1) + r3q(�3; �1; �2):

So

orbit(q) = fq(�1; �2; �3); q(�2; �3; �1); q(�3; �1; �2)g:It remains to show the existence of r1; r2; r3 2 R such that orbit(q) is orthonormal. By

Equation 4.18, orthonormality implies only 2 constraints on (r1; r2; r3):

jjqjjT = 1 and P(q) = 0: (4.22)

Denote P = P(p), Equation 4.22 can easily be turned into explicit algebraic form:

1 = jjqjjT = r21 + r22 + r23 + 2(r1r2 + r2r3 + r3r1)P0 = P(q) = (r21 + r22 + r23)P + (r1r2 + r2r3 + r3r1)(P + 1):

One set of solution in C is

r1 = (

s1

1 + 2P � r3 +pD)=2; r2 = (

s1

1 + 2P � r3 �pD)=2

13

where

D =2 + 3P � 3r23 � 3Pr23 + 6P2r23

(1�P)(1 + 2P) + 2

s1

1 + 2P r3:

Note thatq

1

1+2P is well-de�ned because P > �1=2. It remains to show that for every

P > �1=2, we can always �nd r3 such that D � 0. A careful calculation shows that D > 0

if and only ifs1

1 + 2P �2

3

p(1�P)(1 + 2P)2(1�P)(1 + 2P) < r3 <

s1

1 + 2P +2

3

p(1�P)(1 + 2P)2(1�P)(1 + 2P) (4.23)

and the above range is non-empty since �1=2 < P < 1. (In fact, one can verify that r3 = 1

is always a valid choice as long as �0:47 < P < 1.) 2

Lemma 4.6 Given P;Q be two barysymmetric sets in L2(T ) such that P [ Q is linearly

independent. Assume P is orthonormal. Then there exist barysymmetric Q0 � L2(T ) such

that span(Q) = span(Q0), and P ? Q0.

Proof. Let P = fp1; : : : ; png and Q = fq1; : : : ; qmg. De�ne Q0 = fq01; : : : ; q0mg.

q0j = qj �nXi=1

hqj; piiT pi (4.24)

for 1 � j � m. Then

1. Q0 is a barysymmetric set and

2. Q0 ? P , which implies also span(Q0) = span(Q)

as desired. 2

Finally, we have,

Theorem 4.7 (S-orthogonalization) Given a linearly independent and barysymmetric

set S, it is always possible to generate another barysymmetric set S 0 which is orthonormal

and such that span(S) = span(S 0).Proof. De�ne the following equivalence relation

p �S q () q 2 orbit(p):

Then �S induces a partition on S:

S = [Nk=1;disjointorbit(pk) (4.25)

We proceed by induction on N . If N=1, then the result follows from Lemma 4.5. As-

sume the theorem is true for N = N 0. When N = N 0 + 1, we can S-orthogonalize

[N 0+1k=2 orbit(pk) into [N 0+1

k=2 orbit(p0k) by the induction hypothesis. Then, by Lemma 4.6,

we can S-orthogonalize orbit(p1) against [N 0+1k=2 orbit(p0k) and obtain p001 such that

orbit(p001) ? [N0+1

k=2orbit(p0k):

14

Finally, we use again Lemma 4.5 to obtain p01 such that orbit(p01) is orthonormal and

span(orbit(p01)) = span(orbit(p001)) = span(orbit(p1)):

Then [N 0+1k=1 orbit(p0k) is an orthonormal set that has the same span as S. 2

The proof of Theorem 4.7 actually gives the following algorithm for S-orthogonalizing

any given independent and barysymmetric set S.

ALGORITHM: S-ORTHO

Description:

Symmetric orthogonalization of a given barysymmetric set.

Inputs:

S , a linearly independent, barysymmetric set of functions

Outputs:

S 0 , an orthonormal, barysymmetric set such that span(S0) = span(S)

Algorithm

1. Initialize i = 1

2. Partition S into sets of barysymmetric orbits �! G = fG1; G2; : : : ; Gkg3. S-orthogonalize Gi �! G0

i

4. S-orthogonalize Gi+1; : : : ; Gk with respect to Gi

5. G � fGi+1; : : : ; Gkg.6. If G is empty, goto step 6, else set i � i+ 1 and goto step 3.

7. Output S 0 = fG01; G

02; : : : ; G

0kg

We state and prove an additional result that will also be used in our second construction.

We �rst need a new notation.

Let P = fp1; : : : ; pmg; Q = fq1; : : : ; qng � L2(T ), we will denote by hP; QiT the m by

n matrix

(hP; QiT )i;j = hpi; qjiT :De�nition 4.8 We say Q is non-orthogonal to P if hP; QiT has full rank. We say Q is

orthogonal to P if P � Q?, or equivalently, hP; QiT = [0]jP j�jQj.

Perhaps annoying, when P is non-orthogonal to Q in the above de�nition, it does not in

general imply that P is \not orthogonal" to Q. Neither the reverse in true in general. The

latter requires only hP; QiT has rank > 0, whereas the former is much more restrictive.

Theorem 4.9 Given barysymmetric sets P = orbit(p); Q = orbit(q) and R where (P [Q [ R) is linearly independent (i.e. dim(span(P [ Q [ R)) = jP j + jQj + jRj), Q is non-

orthogonal to P and jP j � jQj. We can always \S-orthogonalize R against P via Q", i.e.

there exists barysymmetric set R0 such that

R0 � span(R [Q);span(R [Q) = span(R0 [Q);

R0 � Q?: (4.26)

15

Proof. jP j � jQj implies we have three distinct cases: (i) jP j = jQj = 1, (ii) jP j = 1; jQj =3, (iii) jP j = jQj = 3.

Partition R as in Equation 4.25, R = [Ni=1orbit(ri).

In case (i), let

r0i = ri � hri; pihq; pi q;

and in case (ii), let P = fp1; p2; p3g and

r0i = ri �1

3

3Xj=1

hri; pihqj; pi

qj:

then it is clear that in both cases, R0 = [Ni=1orbit(r0i) are as desired.

Now consider case(iii), we can think of P;Q and Ri = orbit(ri) as column 3-vectors. If

jorbit(ri)j = 1, simply make the three entries of Ri all equal to ri. Notice that both hQ;P iTand hR;P iT are 3 by 3 circulant matrices. By assumption, hQ;P iT is invertible. Therefore

we can de�ne

R0i = Ri � hRi; P iT hQ; P i�1T Q: (4.27)

Observe

1. hQ; P i�1T is also a circulant matrix, so both hQ; P i�1

T Q and hRi; P iT hQ; P i�1T Q

constitute barysymmetric sets, hence so is R0i.

2. if jorbit(ri)j = 1, then all the nine entries of hRi; P iT are identically equal to hri; piT ,and all the three entries of R0i are identical.

It should be clear now that R0 = [Ni=1R0i is as desired by the statement of the theorem. 2

In fact the following more general statement is true, but we omit the proof because

the underlying idea is similar to that of Theorem 4.9 but requires more clumsy notations.

(After all, we don't need it for latter purposes.)

Theorem 4.10 Given barysymmetric sets P;Q and R where (P [Q[R) is linearly inde-

pendent (i.e. dim(span(P [Q[R)) = jP j+jQj+jRj), hQ ; P iT has full rank and jP j � jQj.We can always \S-orthogonalize R against P via Q", i.e. there exists barysymmetric set

R0 such that

1. R0 � span(R [Q),2. span(R [Q) = span(R0 [Q),3. R0 � Q?.

4.3 Construction II

We now construct multiwavelet bases for L2(R)

f�T0;k ; geij;kji = 1; 2; 3; j � 0; k 2 Ijg; (4.28)

such that each �T0;k or geij;k

is a M -vector of functions that form a barysymmetric set

with respect to the members' common support triangle. We will use extensively the tools

established in the last section.

16

4.3.1 Scaling Functions

Given a triangle T , there is a basis for Pd or Pd(T ) which is barysymmetric (in fact strong-

barysymmetric) with respect to T , namely, the Bernstein basis 2

fBd�(�1; �2; �3) = ��11 ��22 ��33 j j�j =

3Xi=1

�i = dg: (4.29)

The Bernstein basis is not an orthonormal basis for Pd(T ), and hence cannot be used directly

as the scaling function of an orthonormal multiwavelet system. Theorem 4.7 says that there

exists an orthogonalization of the Bernstein basis without destroying its barysymmetry

property.

It is enough to say that the application of algorithm S-ORTHO to the Bernstein bases

would generate the barysymmetric scaling functions for arbitrary d. There are however some

speci�c knowledges of the Bernstein bases that can be used to facilitate the implementation.

First notice that jorbit(Bd�)j = 1 if and only if �1 = �2 = �3. Therefore, when d is

indivisible by 3, the Bernstein basis of degree d is made up of orbits all with cardinality

3; and when d is divisible by 3, is made up of orbits all but one with cardinality 3, and

that the singleton orbit is exactly fBdd=3;d=3;d=3g. Second, we have the following formula for

P(Bd�) [15]:

hBd(�1;�2;�3)

; Bd(�0

1;�02;�03)iT =

2jT j (�1 + �01)! (�2 + �02)! (�3 + �03)!

(2d + 2)!

which implies,

P(Bd(�1;�2;�3)

) = hBd(�1;�2;�3)

; Bd(�3;�1;�2)

iT =2jT j (�1 + �2)! (�2 + �3)! (�3 + �1)!

(2d+ 2)!> 0:

Therefore, according to the working rule provided at the end of the proof of Lemma 4.5,

we can always choose r3 = 1 in the S-orthogonalization of any orbit(Bd�). Figure 4 walks

us through the construction of barysymmetric scaling functions in case of d = 2.

We will denote by fBl; 0 � l < Mg the S-orthogonalized Bernstein basis of degree = d,

and �lTj;k the scaling functions de�ned in this section. Also, we will use the shorter term

\O-Bernstein polynomials" to stand for the S-orthogonalized Bernstein polynomials.

4.3.2 Mother

We now adapt the construction in Section 3.2.2 to construct barysymmetric multiwavelets,

we will reuse all the notations there.

In this section we will generally denote by

fgliji = 1; 2; 3; 0 � l < Mg

any barysymmetric set of 3M functions that can be combined with a basis of P (d) to span

Pd(T0)� Pd(T1)� Pd(T2)� Pd(T3)

2Traditionally, the Bernstein basis of degree d is de�ned as Bd�(�1; �2; �3) =

d!�1! �2! �3!

��11

��22

��33, which

make the basis a partition of unity, but this property is of no relevance to our construction and we simply

neglect the scalar coe�cients and de�ne the Bernstein basis as in Equation 4.29.

17

Split into barysymmetric orbits

( ) ( ) , ( ) ( ) ,( ) ( ) ,( )

,

( ),( )

,(

a b a b a b a b a b a b a b+ + − + + + + − − + + + =− −

=− −

− − − − − − − − −

τ τ τ τ τ τ τ τ τ

τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ

12

22

32

12

22

32

12

22

32 2 3 10

4

6 25 2 10

4

80 1 2 17 12 17 2

232

80

80 2 3 12 17 2

2 17 32

80

80 3 1 17 12

22 17 3

2 )

80

S - o r th o g o n a l iz e( )

,( )

,( )8 0 1 2 1 7 1

2 1 7 22

32

8 0

8 0 2 3 12 1 7 2

2 1 7 32

8 0

8 0 3 1 1 7 12

22 1 7 3

2

8 0

τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ− − − − − − − − −

( ) ( ) , ( ) ( ) ,( ) ( ) ,( )

,

, ,a b a b a b a b a b a b a b+ + − + + + + − − + + + =

− −=

− −

τ τ τ τ τ τ τ τ τ

τ τ τ τ τ τ12

22

32

12

22

32

12

22

32 2 3 10

4

6 25 2 10

4

1 2 2 3 3 1

S or th o go n a lize− τ τ τ12

22

32, ,

τ τ ττ τ τ τ τ τ

12

22

32

1 2 2 3 3 1

, ,

, ,

− − + − − + − − ++ − − + + + − − + + + −

5 52 234 5 52 234 5 52 23412

22

32

22

32

12

32

12

22

4 61 12 4 36 2

2 0 405 32 22 8 1 2 2 25 2 3 3 1 4 61 2

2 4 36 32 0 405 1

2 22 8 2 3 2 25 3 1 1 2 4 61 32 4 36 1

2 0

. . , . . , . .. . . . . , . . . . . , . .

τ τ τ τ τ τ τ τ ττ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ{ }. . .405 2

2 22 8 3 1 2 25 1 2 2 3τ τ τ τ τ τ τ− + +

{ | }τ τ τ λ λ λλ λ λ1 2 3 1 2 3

1 2 3 2+ + =

S - or th og on a lize

w . r . t.

1 2 2 3 3 1τ τ τ τ τ ττ τ τ

, ,

, ,12

22

32

Bernstein Polynomial (d=2)

Father Multiwavelets (d=2):S-Orthogonalized Bernstein Polynomial

Figure 4: Scaling Functions in Construction II (d=2)

18

Let fBlT g be the O-Bernstein polynomials mutilated to the triangle T (c.f. Equation

3.11.)

Let

gli(�1; �2; �3) =

8><>:BlTi((�1; �2; �3)MT!Ti) if �i � 1=2 and (�1; �2; �3) � 0;

�BlTi((�1; �2; �3)MT!Ti) if �i < 1=2 and (�1; �2; �3) � 0

0 if �j < 0 for any j

(4.30)

Notice that

1. fglig is just like fH lig in Equation 3.15, except O-Bernstein bases are used instead of

Legendre bases. Hence, fglig [ fBlT g spans Pd(T0)� Pd(T1)� Pd(T2)� Pd(T3).

2. orbit(gl1) = fgl1; gl2; gl3g := Gl+1 and hence

fglig = [Ml=1Gl:

In particular, fglig is a barysymmetric set.

Using equation 4.24 in Lemma 4.6, we S-orthogonalize each Gl against fBlT g and overwrite

Gl by the result. Then [M�1l=0 Gl is now a basis for W which is also barysymmetric. As in

construction I, we next arm [M�1l=0 Gl with more vanishing moments. In order to preserve

barysymmetric, we have a do a vector version of the corresponding procedure in construction

I. Assume we order the orbits of Bernstein polynomials on T of degree > d as

O = [O1;O2; : : :]: (4.31)

For example, if d = 2, then

O = [f�31 ; �32 ; �33 g; f�21 �2; �22 �3; �23 �1g; f�1�22 ; �2�23 ; �3�21 g; f�1�2�3g; f�41 ; �42 ; �43 g; : : :].

We use the following procedure:

1. Initiatization: L = [G1; G2; : : : ; GM ]. t = 1, Otarget = O1.

2. Find the �rst orbit, G, in the list Lwhich is non-orthogonal (in the sense of De�nition

4.8.) We assume such a G is always available. (In fact, that is at the case from the

actually calculation, at least for d = 0; 1; 2.)

3. reorder L as

L � [G; [LnG]]

4. \Orthogonalize [LnG1] against Otarget via G1" (c.f. Lemma 4.9)

5. L � [G2; G3; : : :], L0 � [L0; G1]. If G is now an empty list, t � t+ 1 goto 5, else

goto 1.

6. L � L0

Note that span(L) is una�ected and barysymmetry is preserved by the above operations,

but that each Gj+1 now has at least M +Pj

k=1 jOkj vanishing moments.

Finally, we apply step 2-6 of S-ORTHO to reverse(L) = [GM ; GM�1; : : : ; G1].

Of course, we can mutilate the so constructed basis to arbitrary triangle Tj;k using

Equation 3.11.

Figure 5 and 6 displays the multiwavelets from construction II for d = 1 and 2 respec-

tively.

19

0

0.5

1

β T

−2.5

−2

−1.5

−1

−0.5

0

g 10

−2.5

−2

−1.5

−1

−0.5

0

g 20

−2.5

−2

−1.5

−1

−0.5

0

g 30

Figure 5: Barysymmetric Multiwavelets, d = 0

4.3.3 Tradeo� between symmetry and vanishing moments

While the two constructions in Section 3.2.2 and Section 4.3.2 are similar in the overall

spirit, the latter yields bases which are barysymmetric while the former yields bases with

virtually no symmetry of any sort. On the other hand, the �rst construction yields bases

of the form fh1; h2; : : : ; h3Mg such that hi has M + i � 1 vanishing moment. Therefore

a so constructed basis has in average (5M � 1)=2 vanishing moments per element. The

second construction yields bases fg1; g2; : : : ; g3Mg such that g3i+1; g3i+2; g3i+3 possessM+3i

vanishing moments, for i = 0; 1; : : : ; (M � 1). (We are assuming all the orbits in Equation

4.31 has cardinality 3, which means we are slightly overestimating the number of vanishing

moments of the gi's.) Thus such a basis has in average (approximately) (5M�2)=6 vanishingmoments per element, which is about 3 times less than that in construction I.

In other words, there is a tradeo� in symmetry and vanishing moments. It occurs that

such a tradeo� is necessary from the experience in wavelet design in 1-D. See, for example,

Chapter 8 of [8].

It is also not hard to modify construction II in order to design multiwavelets that possess

strong-barysymmetry, but that would involve sacri�cing even more vanishing moments.

The relative importance of symmetry and vanishing moments occurs to be application-

dependent. In applications where visual quality is involved symmetry would be important,

whereas in numerical applications like those in [5] vanishing moment is all that matters.

20

−4

−2

0

χ1

−4

−2

0

χ2

−4

−2

0

χ3

−10

−5

0

5

h1

1

−5

0

5

h2

1

−5

0

5h

3

1

−10

−5

0

5

h1

2

−5

0

5

h2

2

−5

0

5h

3

2

−10

−5

0

5

h1

3

−5

0

5

h2

3

−5

0

5h

3

3

Figure 6: Barysymmetric Multiwavelets, d = 1

21

5 Two-scale Relations, Filter Banks and Pyramid Algorithms

Given an arbitrary non-degenerate triangle T , and its midpoint subdivision T = T0 [ T1 [T2 [ T3. Construction I or II gives us an orthonormal basis f�lT j0 � l < M � 1g for Pd(T )and an orthonormal basis fhliji = 1; 2; 3; 0 � l < M �1g for �3

i=0Pd(Ti)Pd(T ). Therefore,there exists sixteen M by M matrices Hi, G1;i, G2;i, G3;i, i=0,1,2,3, such that

�T =3X

i=0

Hi �Ti

hj =3X

i=0

Gj;i �Ti for j = 1; 2; 3:

By orthogonality, the �lter banks fHi; Gk;ig can easily be calculated by

(Hi)p;q = h�pT ; �qTii

(Gk;i)p;q = hhpk ; �qTii: (5.32)

Analogous to Equation 2, the 4M by 4M matrix

U =

26664

H0 H1 H2 H3

G1;0 G1;1 G1;2 G1;3

G2;0 G2;1 G2;2 G2;3

G3;0 G3;1 G3;2 G3;3

37775 (5.33)

is orthogonal. This is equivalent to saying we have two di�erent orthonormal bases, f�lT ; hligand f�lTig for �3

i=0Pd(Ti).

An important observation is that fHi; Gk;ig is completely independent of T . It can be

seen easily from Equation 5.32 that the values of the integrals on the right hand sides are

all independent of T .

6 Generalizations

6.1 Smooth Biorthogonal Duals

The multiwavelet bases constructed in this paper are discontinuous and orthonormal. It is

possible to construct their smooth biorthogonal counterparts via moment-interpolation as

described in [10].

6.2 Spherical Multiwavelets

Although we have been focusing on the planar case. The construction we have presented can

potentially be extended to other surfaces. Notice that the basic components of constructions

I and II are nested triangulation, barycentric coordinates and an inner-product.

We can, for instance, apply the above construction to construct multiwavelets on a

sphere. A nested triangulation can be obtained by starting from an icosahedron and subdi-

vide exactly as in the planar case, except triangles are now understood as geodesic triangles

on the surface of the sphere. This type of nested triangulations of the sphere have been

used, for example, by Swelden and Schr�oder in their construction of spherical wavelets [14].

Barycentric coordinates and Bernstein polynomials can also be de�ned on the sphere, see

[1]. Of course, L2(S2) is equipped with a well-de�ned inner-product.

22

7 Acknowledgement

The �rst author would like to thank Interval Research Inc. for the support of this work

during Summer 1996. He would also like to thank David Donoho for bringing up his

attention to multiwavelets and re�nement subdivision schemes.

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