The Discrete Wavelet Transform for a Symmetric ...python.rice.edu/~johnson/papers/SymDMWT.pdf · The Discrete Wavelet Transform for a Symmetric - Antisymmetric Multiwavelet Family

The Discrete Wavelet Transform for a Symmetric-

Antisymmetric Multiwavelet Family on the Interval

Haixiang Wang and Bruce R. Johnson*

Rice University

ABSTRACT

An interval basis is constructed from the Chui-Lian multiwavelet family of

approximation order three. To the symmetric and antisymmetric pairs of scaling functions

and wavelets are added edge functions which preserve the approximation order and

provide the basis for an orthogonal Discrete Multiwavelet Transform on the interval. This

transform applies to data sequences with no special constraints on behavior at the interval

endpoints.

This work was supported by grants from the National Science Foundation (#CHE-

O111008) and the Robert A. Welch Foundation.

The authors are with the Department of Chemistry and the Rice Quantum Institute, MS

#600, Rice University, Houston, TX 77005-1892. Telephone: 713-348-5103; Fax: 713-348-

5401; email: [email protected].

Index terms: Multiwavelet, Interval multiwavelet, Dilation Eigenvalues, Approximation Order.

EDICS Classification: SP 2.4.4, 2-MWAV

I. INTRODUCTION

Multiwavelet families, characterized by multiple wavelet parent functions, can possess

advantages over single wavelet families such as shorter filter lengths and definite symmetry or

antisymmetry. As in the case of compact support single wavelet families, the implementation of

a Discrete Multiwavelet Transform (DMWT) for finite data streams can proceed via adaptations

such as periodization or symmetric reflection of the signal [1]. An alternative implementation

which leaves the data in place (except for prefiltering) adds a few special functions at the edges,

allowing a multiresolution analysis (MRA) strictly limited to the interval [2], [3], [4], [5], [6],

[7]. The number and nature of the additional functions is usually chosen to preserve the

approximation order (number of vanishing wavelet moments) of the regular wavelet family at all

points in the interval. Multiwavelet families on the interval have also been derived recently for

both biorthogonal [8], [9] and orthogonal [9], [10] cases. The particular example in paper I [10]

was based on the symmetric-antisymmetric Chui-Lian family of multiplicity two and

approximation order three [11]. Three new scaling functions and three new wavelets were

constructed at each edge, just sufficient to preserve the approximation order throughout the

interval.

This turns out, however, to be insufficient for implementation of a DMWT on the interval

[12]. The essential problem is that each transform step (filtering to coarser scales) fails to

preserve the total number of basis functions. This problem may be eliminated by constructing

four instead of three edge functions of each type, i.e., one more than the approximation order.

This number is, in fact, the approximation order that the Chui-Lian filter length would have

allowed were it not for the symmetry-antisymmetry condition imposed. Other cases in which the

approximation order is chosen non-maximally in order to satisfy particular design goals are

certainly familiar, e.g., Coiflets [13] and balanced multiwavelets [14], [15], [16], [17], [18]. It is

thus expected that the present specific example of edge function construction for a basis with

non-maximal approximation order will be of more general use. It is shown that a satisfactory

extension of the Chui-Lian multiwavelets to the interval can be defined based on dilation

eigenvalues, that is, eigenvalues of the two-scale dilation matrix for the regular scaling functions

truncated to lie within the interval.

An issue which is important but separate from the above is the need for prefiltering of the

data. This is critical for multiwavelets applied to single input channels [19], [20], [21], [22],

[23], [24], [25], and can be useful even in single wavelet cases [26], [27]. A projection-based

prefilter family which produces exact output for low-order polynomial signals was previously

derived for Chui-Lian and other multiwavelets [27]. Adaptation of the prefilter to an interval

basis is straightforward [10], involving little more than evaluation of a few moments of the edge

functions. Multiwavelet decomposition and reconstruction of a variable-frequency signal on an

interval is examined in a case where no special conditions on the signal are required at the

endpoints of the interval.

II. CHUI-LIAN MULTIWAVELETS

The multiscaling functions {φ1 x( ),φ2 x( )} and multiwavelets {ψ1 x( ),ψ 2 x( )} defined by

Chui and Lian on [0, 3] are shown in Fig. 1. These satisfy the two-scale matrix dilation relations

φ1 x( )φ2 x( )

=

k=0

3∑ ckφ1 2x − k( )φ2 2x − k( )

,

ψ 1 x( )ψ 2 x( )

=

k=0

3∑ dkφ1 2x − k( )φ2 2x − k( )

, (1)

where the 2x2 matrix filters

c0 =1

40

10 − 3 10 5 6 − 2 15

5 6 − 3 15 5 − 3 10

, c1 =

1

40

30 + 10 5 6 − 2 15

−5 6 − 7 15 15 − 3 10

,

c2 =1

40

30 + 3 10 −5 6 + 2 15

5 6 + 7 15 15 − 3 10

, c3 =

1

40

10 − 3 10 −5 6 + 2 15

−5 6 + 3 15 5 − 3 10

, (2)

d0 =1

40

5 6 − 2 15 −10 + 3 10

−5 + 3 10 5 6 − 3 15

, d1 =

1

40

−5 6 + 2 15 30 + 3 10

15 − 3 10 5 6 + 7 15

,

d2 =1

40

−5 6 + 2 15 −30 − 3 10

−15 + 3 10 5 6 + 7 15

, d3 =

1

40

5 6 − 2 15 10 − 3 10

5 − 3 10 5 6 − 3 15

, (3)

satisfy constraints maintaining the orthogonality of the basis on different scales

k=0

3∑ ck ⋅ ckT =

k=0

3∑ dk ⋅dkT = 2,

k=0

3∑ ck ⋅dkT = 0. (4)

Normalized functions on level j with support k / 2 j , k + 3( ) / 2 j[ ] are given by

φ jkα t( ) = 2 j / 2 φα 2 j t − k( ), ψ jkα t( ) = 2 j / 2ψα 2 j t − k( ). (5)

A multiresolution expansion of the function f t( ) with coarsest level j = 0 then takes the form,

f t( ) =k=−∞

+∞∑α =1

2∑ φ0kα f φ0kα t( )+j =0

+∞∑k =−∞

+∞∑α =1

2∑ ψ jkα f ψ jkα t( ). (6)

In calculations, the summations over values of j and k at each scale are truncated according to the

localized scale variations. For monomials of order p ≤ 2, the wavelet projections are exactly zero

by construction

t p =k =−∞

+∞∑α=1

2∑mp kα φkα t( ), (7)

where the projection coefficients are given by the integrals

m pkα = ∫ t p φkα t( )dt , (8)

available from [10] and [11].

The problem that arises in the presence of an interval edge is that some functions at each

scale will have tails on both sides of the boundary. Figure 2 shows this for the Chui-Lian case

restricted to t ≥ 0, where the relevant functions are those with k = –2 and –1. The positive-t tails

are important to the expansion of tp, p ≤ 2, just to the right of the origin, while the negative-t tails

are unwanted. The restriction of Eq. (7) to positive t is simply

Θ t( )t p =k=−2

+∞∑α =1

2∑ mp kα Θ t( )φkα t( ), (9)

where Θ(t) is the Heaviside step function and the sum over k starts at the first functions which

have tails to the right of the origin. These tails, Θ t( )φkα t( ), are not mutually orthogonal nor

even strongly linearly independent for k = –2 and –1 as a consequence of the truncation.

Nevertheless, they provide what is missing from the set of k ≥ 0 functions for the latter to form

an exact expansion basis for the monomials t0, t1, t2 everywhere to the right of the origin.

Following the procedure used for single wavelet families [4], paper I grouped the six tail

functions Θ t( )φ−2,α t( ), Θ t( )φ−1,α t( ) , and φ0,α t( ) together to form the three linear combinations (p

= 0, 1, 2)

φpL t( ) =

k =−2

0∑α =1

2∑ mp kα Θ t( )φk,α t( ) . (10)

Combined with the regular functions φk ,α t( ) for k ≥ 1, this produces an exact expansion basis for

the monomials t0, t1, t2 everywhere to the right of the origin. The three new edge functions φpL t( )

take the exact form t p for 0 ≤ t ≤ 1, i.e., before the beginning of the contributions of the inner

functions. These new functions are still not orthogonal, but may be made so by Gram-Schmidt

orthogonalization. Three orthogonal edge functions ΦnL t( ) , n = 0, 1, 2, were accordingly

constructed with leading behaviors tn at the origin. (Other choices are possible, but will differ

only by a 3x3 orthogonal transformation.) A right-hand edge can also be accommodated, in

which case the symmetry of the regular Chui-Lian basis allows right-edge functions ΦnR t( ) to be

the mirror images of the ΦnL t( ) . Orthogonal edge wavelets are then readily determined [4], [7],

[10] by consideration of the differences in detail between the edge functions with arguments t

and 2t. A multiresolution interval basis is thus constructed where, at each level j, the edge

scaling functions and wavelets are scaled and translated copies of those for j = 0, and the two-

scale relations are modified only near the edges. It is assumed as usual that there is a sufficient

number of regular basis functions between left and right edges at level j = 0 that the

corresponding edge functions do not overlap.

In using this basis for multiresolution expansions on a finite interval [12], it became

clear that each lowpass/highpass stage of the wavelet transform failed to preserve the total

number of basis functions. This may be seen as follows. Suppose that one has at level j = 0 the

α = 1 and 2 scaling functions with k = 1, 2, …, kmax , i.e., 2kmax inner plus three left- and three

right-hand edge functions. With equal numbers of inner and edge wavelets, there are 4kmax +12

basis functions at j = 0. The corresponding j = 1 scaling functions with spacing 1/2 on the same

overall interval can be determined to require k = 1, 2, …, 2kmax + 4 , for a total of

2(2kmax + 4) + 6 = 4kmax +14 inner and edge functions. This is two more than the total number

of functions produced at j = 0 as output from the lowpass and highpass operations.

In contrast, such a problem does not arise for scalar Daubechies families of (even) filter

length L. These possess L/2 vanishing wavelet moments and require L/2 scaling functions and

L/2 wavelets at each edge to maintain the approximation order throughout the interval. For k =

1, 2, …, kmax inner scaling functions and wavelets at j = 0, the corresponding values at j = 1 must

be k = 1, 2, …, 2kmax + L . There are thus 2(kmax + L) total scaling functions at j = 1 and

2(kmax + L) scaling functions and wavelets at j = 0. This ensures that the wavelet transform can

be expressed as an orthogonal transform and can be straightforwardly inverted. The same

situation unfortunately does not hold for the earlier-derived Chui-Lian interval multiwavelet

family.

From the standpoint of counting, it is easy to see that four edge functions of each type

(scaling function and wavelet) at each edge would correct the situtation for the Chui-Lian

multiwavelet family. The total number of basis functions both before and after the wavelet

transform step would then be 4kmax +16. In this regard, Chui and Lian point out in their

construction of the inner functions that approximation order four (up to cubic polynomials)

would have been obtained had the requirements of symmetry and antisymmetry not been

imposed [11]. The minimal requirements for an interval basis would then have been four scaling

functions and four wavelets at each edge. Furthermore, suppose one desires to use the function

tails touching t = 0 from both sides to construct separate but adjoining edge bases for postiive

and negative t. Inspection of Fig. 2 shows that the eight regular functions corresponding to k = –

3, –2, –1, 0 would need to be utilized. The choice of four edge functions on each side of t = 0

would then maintain the density of orthogonal basis functions in the vicinity of the origin.

To summarize, counting arguments appear to favor one more scaling function (and one

more wavelet) at each edge than required to maintain the approximation order. They do not,

however, indicate precisely how this is to be accomplished. This issue appears not to have been

considered before, though it is not a problem restricted solely to multiwavelets. In the next

section, a particular solution is given for the Chui-Lian multiwavelet family.

III. DILATION EIGENVALUES

Following Cohen, et al., [4] and Monasse and Perrier, [7] the three nonorthogonal

functions φpL t( ) in Eq. (10) may be written in an alternative form using Eq. (9),

φpL t( ) = Θ t( ) t p −

k=1

∞∑α =1

2∑m p kα φk ,α t( ), (11)

Beyond t = 3 the two terms on the right-hand side completely cancel each other, whereas

truncation effects prevent complete cancellation of the k = 1 and 2 components of the last term in

the region t ≤ 3. The right-hand side for 0 ≤ t ≤ 3 may be expressed in terms of functions of 2t

using the ordinary two-scale relations,

φpL t( ) = 2− p 2t( ) p −

k=1

2∑α =1

2∑ m p kα′k =0

3∑′α =1

2∑ c ′k ;α ′α φ2k+ ′k , ′α 2t( ). (12)

Using Eq. (11) with t replaced by 2t and performing algebraic manipulations and cancellations

simliar to those of Ref. [7], Eq. (12) can be brought to the form,

φpL t( ) = ap pφ p

L 2 t( )+k =1

3∑α =1

2∑ bp,kα φkα 2t( ), (13)

where b is a 3x6 constant matrix and a is a diagonal matrix with nonzero elements

ap p = 2− p . (14)

With respect to the edge functions entering the two-scale relations, there is no mixing and the a

matrix components scale as do the corresponding monomials.

One natural candidate to consider for a fourth nonorthogonal vector is the extension of

the definition of φpL t( ) to p = 3. Since the function t3 has components in the wavelet subspace,

Eq. (7) does not hold for the Chui-Lian wavelet family. (Actually, close examination shows that

it does hold at integer and half-integer values of t, but not at points in between.) Nevertheless,

the projection coefficients m3 kα are easily calculated for k = –2, –1, 0, [10] and so one may

define a function φ3L t( ) through Eq. (10). When the two-scale relations are used on the six

positive-t tail-functions Θ t( )φk ,α t( ), however, the resulting linear combination of tail functions

Θ t( )φk ,α 2t( ) cannot be reduced to a linear combination of the smaller set of four functions φpL t( ) ,

p ≤ 3. This would be required in order to obtain a two-scale relation like Eq. (13) with a 4x4 a

matrix. We may say that the edge dilations in the nonorthogonal basis are not closed, and so this

avenue dead ends.

What values are then allowed for the scaling such that the edge-edge portion of the two-

scale relations results in a 4x4 a matrix? We ignore for now the form of the b matrix which

multiplies only regular scaling functions for k ≥ 1 and will not contribute to the two-scale

relations in the immediate vicinity of t = 0. The previous construction has already found the

three scaling factors 1, 1/2, 1/4. To find all possible values, we assume a nonorthogonal basis

vector of the form

φL t( ) =k =−2

0∑α =1

2∑ µ kα Θ t( )φk,α t( ), (15)

where the µkα are to be determined. Using the two-scale relations on the φk ,α , this may be re-

expressed as

φL t( ) =k =−2

0∑α =1

2∑ µ kα Θ t( )l = 2k

2k +3∑β=1

2∑ cl− 2k;α β φ l, β 2t( )

=l =−2

0∑β =1

2∑Θ t( )φ l, β 2x( )k=−2

floorl

2

∑ cl− 2k;α β

α =1

2∑ µkα

+l =1

3∑β=1

2∑Θ t( )φl ,β 2 t( )k= floor

l− 2

2

0∑α =1

2∑ cl−2k ;α β µ kα , (16)

where the floor function is the nearest integer (independent of sign) less than or equal to its

argument. The requirement is now imposed that the linear combination of tail functions for l ≤ 0

be a simple multiple of the original function evaluated at 2t,

l =−2

0∑β =1

2∑Θ t( )φ l, β 2t( )k=−2

floorl

2

∑ cl− 2k;α β

α =1

2∑ µkα = λl =−2

0∑β=1

2∑Θ t( )φl ,β 2t( )µ lβ . (17)

Equating the coefficients of the different scaling functions leads to a system of eigenvalue

equations,

C T − λI( )⋅ µµµµ = 0 (18)

where

C =c2 c3 0

c0 c1 c2

0 0 c0

(19)

is the portion of the composite dilation matrix [cf., Eq. (1)] with the φk ,α t( ) (rows) and φl ,β 2t( )

(columns) restricted to –2 ≤ k, l ≤ 0, that is, connecting the tail functions between two adjacent

scales.

Three of the "dilation eigenvalues" of C (as well as C T ) are found to be the known

values, 1, 1/2, 1/4, and the corresponding non-orthonormal right eigenvectors of C T are the φ0L ,

φ1L , φ2

L previously given in Eq. (10). The additional eigenvalues are (a)1/2, (b)

1 / 2 − 3 / 2 10 ≅ 0.0257 , (c) –1/8. The eigenvectors for the latter two cases are simple and have

only k = 0 components,

µµµµ b = 0, 0, 0, 0, 15 − 2 6( )/ 9,1{ }T, (20)

µµµµ c = 0, 0, 0, 0, 3 / 2,1{ }T. (21)

Case a, however, is a repeat of one of the earlier eigenvalues, and a straightforward attempt to

solve for the eigenvector leads back to the earlier eigenvector rather than a new one. Instead, a

new vector may be generated by allowing for mixing of functions with the same eigenvalue λ =

1/2. For example, assume that, in the region near t = 0 where the regular scaling functions do not

contribute,

φaL t( ) =

12φa

L 2t( )− 12φ1

L 2t( ). (22)

The general solution to this equation is given by

µµµµ a =13

52

, 0, − 13

52

, 0,45− 1

10,

4

15− 4

25

T

+ ξ − 2 + 5

3,1,

2 + 5

3,1, 6 + 15,1

T

, (23)

where ξ is a free parameter. The three nonorthogonal functions φaL , φb

L , φcL (with ξ = 0 in case

a) are illustrated in Fig. 3. The a matrix for the entire set of six functions takes a diagonal form

except for the degenerate members,

a =

1 0 0 0 0 0

0 1/ 2 0 0 0 0

0 0 1 / 4 0 0 0

0 −1/ 2 0 1/ 2 0 0

0 0 0 0 1 / 2 − 3 / 2 10 0

0 0 0 0 0 −1/ 8

. (24)

Any one of φaL , φb

L , φcL may be chosen as the fourth vector and will lead to closed two-

scale relations between φpL t( ) and φp

L 2t( ) , p = 0, 1, 2, 3. All three vanish at t = 0. Of the

choices, the function φbL is concentrated farthest away from t = 0 and interferes the least with the

association of the monomials t p with the first three functions φpL close to t = 0. In analogy, one

expects φbL to behave in some manner as tη where the power

η =ln 1 / 2− 3 / 2 10( )/ ln2 ≅5.2844 . The actual behavior is that φbL is increasingly oscillatory

but with extremely small amplitude as t → 0. At the dyadic rationals, however, the first three

values are always consistent with the power law behavior. If h = 1 / 2n , then one always finds

that φbL 0( ) = 0 and φb

L 2h( ) /φbL h( )=2η = 1/ 1 / 2− 3 / 2 10( ), no matter how large n is taken.

This means that φbL 4h( ) , φb

L 8h( ) , etc., are also in conformity with the power law behavior,

though the points in between are not. While investigations have been made into cases (a) and

(c), case (b) is adopted from here on.

IV. ORTHOGONALIZED EDGE FUNCTIONS

Choosing φ3L =φb

L , a set of four orthogonal functions ˆ ΦL = Φ0L ,Φ1

L ,Φ2L ,Φ3

L{ }T can be

obtained from the intermediate basis ˆ φL = φ0L ,φ1

L ,φ2L ,φ3

L{ }T by Gram-Schmidt orthogonalization.

This process starts with the last vector Φ3L ∝φ3

L first so that only the last-created function Φ0L

possesses a component of the function φ0L that has nonvanishing behavior at the origin. The

transformation takes the form

ˆ ΦL = T ⋅ ˆ φL , (25)

where T is may be obtained from the inner product matrix in the nonorthogonal edge basis. The

corresponding inner products for the tail function basis have already been evaluated [10] and, in

conjunction with Eqs. (10), (15) and (20), leads to the φpL inner product matrix

SL =

227 − 10

12

11− 10

3

3

227 − 10

12

11− 10

3

1276 −185 10

180

7 6 + 2 15

1211− 10

3

1276 −185 10

180

136 − 26 10

9

11 6 + 4 15

1232

7 6 + 2 1512

11 6 + 4 1512

52

. (26)

T is then obtainable from the Cholesky factorization of SL written in the form

SL = T−1 ⋅T−1T . (27)

from which one obtains the upper triangular matrix

T =

2.603339382928659 –7.801793778192170 4.896079164872692 –1.727579066875647

0 4.435038322916785 –4.121459009851160 2.150027680114256

0 0 1.013620631065421 –1.433811000716809

0 0 0 0.6324555320336759

(28)

The resulting orthogonal functions are shown in Fig. 5.

The two-scale relations for the ΦnL are derived from the corresponding ones for theφp

L in

Eq. (13). The 4x4 matrix a is diagonal with elements 1, 1/2, 1/4, 1 / 2 − 3 / 2 10 . The b matrix

can be calculated from the last term of Eq. (16). One obtains

ˆ ΦL t( ) = A ⋅ ˆ ΦL 2t( ) + B ⋅ ˆ φ 2t( ), (29)

where ˆ φ = φ1,1,φ1,2,φ2,1,φ2,2,φ3,1,φ3,2{ }T and

A = T ⋅a ⋅T−1

=

1 0.8795632878614199 –0.04634410027189778 –0.4336776980658760

0 0.5 1.016519120550821 0.6919832305713737

0 0 0.25 0.5085946884788026

0 0 0 0.02565835097474310

, (30)

B = T ⋅ b

=

0.1812840211039820 0.05028172775872669 –0.01668648325962257

–0.4700624510385532 –0.1034875534523519 0.04782947856311466

1.275331944409571 –0.09464145843777787 –0.1267567242955789

0.1423493603424239 0.1743416490252569 1.386969104937278

0.02154215725709569 0.002455796686582593 0.001902251933761148

0.06174759131090924 –0.007039198922189607 –0.005452540039256310

–0.1636422274055273 0.01865514372824096 0.01445021219611660

0 0 –0.1581138830084190

. (31)

The orthonormality of the edge functions is reflected in the condition

A ⋅AT + B ⋅BT = 2I. (32)

The complete scaling function dilation matrix including four edge functions on each side is

C =

A B

c0 c1 c2 c3

c0 c1 c2 c3

Oc0 c1 c2 c3

c0 c1 c2 c3

BR A

, (33)

where only nonzero matrix blocks are shown. By virtue of the behavior of the regular scaling

functions under reflection, the right-hand BR matrix is related to the left-hand matrix by

Bn, kmax +1−k ,αR = −1( )α −1

Bn,k ,α . (34)

The matrix C, after division by 2 , is an orthogonal matrix.

The procedure of Cohen, et al., [4], [10] is followed to obtain the corresponding edge

wavelets. Primitive nonorthogonal wavelets are first defined which span the differences in

scaling function detail between scales t and 2t, and the resulting functions are then

orthogonalized. This yields ˆ ΨL = Ψ0L ,Ψ1

L ,Ψ2L ,Ψ3

L{ }T in terms of two-scale relations similar to

those above,

ˆ Ψ L t( ) = F ⋅ ˆ ΦL 2t( ) + G ⋅ ˆ φ 2t( ) , (35)

F =

1 –0.8795632878614199 0.04634410027189778 0.4336776980658760

0 0.4502630845326074 –0.9477446331229220 0.9259083478119553

0 0 0.04090766042270790 0.1653725680436606

0 0 0 0.03201958893798465

, (36)

G =

–0.1812840211039820 –0.05028172775872669 0.01668648325962257

–0.1862695758096610 –0.08152599693036275 0.01207937550875358

–0.01805573443469635 1.375100352035823 –0.1718229577467140

0.1776407224391046 0.2175645637575901 –0.04557968003749467

0.02154215725709569 –0.002455796686582593 –0.001902251933761148

0.01559440672599612 –0.001777755677385011 –0.001377043626427567

–0.2218224846163747 0.02528766809326757 0.01958774347792920

1.375999119229812 –0.1568633093434542 0.005196063972409472

. (37)

The new matrices satisfy the additional equations

F ⋅FT + G ⋅GT = 2I, (38)

F ⋅AT + G ⋅BT = 0 . (39)

The complete wavelet two-scale matrix D then takes the same form as C but with the

substitutions ck → dk, A → F, B → G. The resulting wavelet functions are shown in Fig. 6.

For implementation of the proper initialization (prefiltering) of data, it is necessary to

calculate the monomial moments of the edge scaling functions. As it turns out [10], these

moments may be evaluated in terms of the known integrals mp kα of the inner functions [Eq. (8)]

and the above A and B matrices,

M p nL = dt∫ t p Φn

L t( )

=′n =0

3∑ 2 p+1 I − A( )n ′n

−1

k=1

3∑α=1

2∑ B ′n ;kα m p kα (40)

Numerical values of these new moments are given in Table I. Armed with these results,

quadrature weights can be constructed for evaluation of projection integrals over all the edge and

regular scaling functions. For the inner functions, assuming uniform sampling, this takes the

form

φ j kα t( )∫ f t( )dt ≈q=1

r∑ω jα q fk + δ + q −1

2 j

, (41)

whereδ is a small shift determined by the location from which monomial moments are

calculated for k = 0 and the k–independent quadrature weights ω jα q can be expressed in terms of

Lagrange interpolating polynomials and the k = 0 moments [see Eq. (42) of I]. The errors of the

quadrature approximations for non-polynomial functions converge with decreasing step h = 2− j

as hr . (In the scalar wavelet case, the Mallat algorithm [27, 28] corresponds to the least accurate

one-point quadrature.) Correspondingly, the edge function projection integrals can be

approximated as

Φ j nL t( )∫ f t( )dt ≈

q =1

r∑ω j n qL f

q −1 + ′δ2 j

, (42)

where the new weights are expressed in terms of Lagrange interpolating polynomials and the

edge moments of Eq. (38) [see Eq. (44) of I]. The optional shift ′δ (upon which the weights

implicitly depend) can be used for overall alignment of the basis functions and the data samples.

Coarser-scale scaling function and wavelet projections can be calculated by use of the two-scale

matrices. A C++ program, MultiWavePack, is under development which can already provide

numerical values for the interval Chui-Lian quadrature weights and those of other wavelet

families [29]. Together, Eqs. (41) and (42) provide a class of prefilters for the data which

provide exact projections on the interval multiwavelet basis for low-order polynomial signals. As

described, the basis is not maximally-decimated; this can be trivially remedied by starting from a

coarser basis and adjusting the quadrature appropriately. While there are a variety of other

multiwavelet prefilters available, for instance [19], [20], [21], [22], [23], [24], [25], the present

choice is unusual in its easy adaptation to an interval basis.

Reconstruction is accomplished by evaluation of the wavelet series in Eq. (6) using

samples of the basis functions at the dyadic rationals i / 2 j . This requires only the values of the

inner and edge functions at the integers, followed by iterative use of the appropriate two-scale

relations to obtain the values at successively finer scales 1 / 2 j . In its simplest implementation,

this is equivalent to the inverse DMWT using the matrices CT and DT as usual, followed by a

postfilter based on the samples of the scaling functions. It should be noted, however, that the

terms in the wavelet series can easily be calculated on finer grids than the basis function spacing

2− j (wavelet interpolation). The non-zero integer samples of the regular Chui-Lian multiscaling

functions are given by φ1 2( )= φ1 1( )= 1 / 2 , φ2 2( ) = −φ2 1( )= 5 / 3 / 2 , while the values of the

edge functions at the integers are given numerically in Table 2.

V. EXAMPLE

The interval MRA is implemented for the variable-frequency function Doppler used in

investigations of wavelet denoising [30],

f t( ) = t 1− t( ) sin 2π 1+ εt + ε

, (43)

with ε = 0.05. The wide range of frequencies involved provides a good test for the interval

multiwavelet basis. The range of t is restricted to the interval [0.03, 0.80] so that the endpoint

values are nonzero (see Fig. 7a). The truncated signal is projected using five-point quadratures

onto an interval scaling function basis with 512 subintervals, corresponding to 1 ≤ k ≤ 508.

Scale iteration using low-pass and high-pass filters produces an MRA spanning six octaves, from

j = 0 at the coarsest level ( kmax = 4) to j = 5 at the finest level ( kmax = 252). The decomposition

of the original signal into contributions from j = 0 interval multiscaling functions and j = 0 to 5

interval multiwavelets is shown in Fig. 8. A logarithmic t scale is used to visually accommodate

the large dynamic range of frequencies. It is clearly seen how significant the leftt-most wavelets

become as finer scales are included.

The sum of all the different components in the MRA (precisely the same as the original

single-resolution projection as a consequence of the orthogonality of the DMWT constructed

above) is shown together with the original signal in Fig. 7b. In all regions except for the very

high frequency left-hand edge, the results are graphically indistinguishable. The absolute errors

are graphed in Fig. 7c. The errors incurred at the left-hand edge are comparable with those in the

adjacent inner region, which increase more or less systematically as the signal features become

sharper. The errors in the right-hand edge region behave as 8−J when all wavelets up to j = J are

included, the base error of 1/8 corresponding to the expected behavior for a discrete wavelet

series which exactly interpolates up to quadratic polynomials. The decreased error in the nearby

inner region corresponds to the fact mentioned earlier that the regular Chui-Lian multiscaling

functions on scale 0 provide an expansion for cubic polynomials which is exact at the integers

and half-integers but not at points between. At the discrete samples used in the inner region for

Fig. 7, the error is therefore lower, i.e., 16−J . Nevertheless, the fractional error of the

reconstruction at the right-hand edge is down to ~10–7. This is far more than sufficient to

demonstrate the ability of the interval basis to yield an accurate MRA for fixed-length signals

without special extensions of the data. The limitations shown at the left-hand edge, due to

increased feature sharpness relative to a fixed sampling rate, are of course universal to all

methods.

VI. SUMMARY

By construction of a new set of edge scaling functions and wavelets properly accounting

for dilation eigenvalues at the boundaries of a finite interval, an extension of the Chui-Lian

multiwavelet family has been designed which preserves the approximation order three on the

interval without the need for periodic extension. The distinguishing feature of the new set is that

the number of edge functions of each type is required to be greater than the approximation order.

A variable-frenquency signal with significant dyanmic range and without special conditions on

endpoint values has been chosen to examine quantitatively the behavior of the associated

DMWT. With the usual provision that the sampling rate is not too low, the signal reconstructed

from the contributions of different scales of the MRA has been shown to be capable of high

accuracy for either inner or edge regions.

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TABLE I

LOWEST ORDER EDGE SCALING FUNCTION MOMENTS

TABLE II

EDGE SCALING FUNCTION VALUES AT INTEGERS ΦnL i( ).

n i 0 1 20 2.603339382928659 –0.245149538886205 –0.0222486443461631 0 0.242360100522656 0.0637726380841532 0 1.061115317459909 –0.1690089657274383 0 –0.020949955843121 0.795546625084605

p Mp0L Mp1

L Mp2L

M p 3L

0 0.384122026715947 0.675719265516541 0.892106351732471 0.7745966692414831 0.000000000000000 0.225477194826657 0.916807519157840 1.3119444045789272 0.000000000000000 0.000000000000000 0.986562397559821 2.2365904745371123 0.0420232694126764 -0.152944454965251 1.077771189640070 3.8228165407162904 0.0909575134268806 -0.276035052169062 1.172846974202943 6.5248618997349055 0.142677882469103 0.386981079864648 1.263182670548571 11.066985163929626 0.203644287601671 -0.510551852571727 1.383523503326036 18.52834930917891

1

0

3210

t

1 (t)

-1

0

1

3210

t

1 (t)

-1

0

1

3210

t

2 (t)

-1

0

1

3210

t

2 (t)

Figure 1. The scaling functions and wavelets of the Chui-Lian multiwavelet family orthogonal

on the interval [0, 3].

43210-1-2-3t

k = –3

–2

–1

0

1

Figure 2. Restriction to t ≥ 0 for the case of Chui-Lian multiscaling functions. All functions or

partial functions for t < 0 (dotted lines) are neglected. The included tails for k = –2 and –1 (solid

lines) are no longer mutually orthonormal. The k = 0 functions are used with the latter to

construct modified edge functions as explained in the text, while those for k = 1 (dashed lines)

and higher are unmodified "inner" functions.

3

2

1

0

3210t

0L (t)

1L (t)

2L (t)

Figure 3. The three nonorthogonal scaling functions φpL t( ) behaving near the origin as t p for p =

0, 1, 2.

2

1

0

-1

3210t

aL (t)

bL (t)

cL (t)

Figure 4. The three choices for a fourth nonorthogonal edge scaling function to be added to those

shown in Fig. 3.

2

1

0

3210t

0L (t)

1L (t)

2L (t)

3L (t)

Figure 5. Final orthogonal left-hand edge scaling functions for the Chui-Lian interval

multiwavelet basis. Right-hand analogs are mirror images of these functions.

2

1

0

-1

3210t

0L (t)

1L (t)

2L (t)

3L (t)

Figure 6. Final orthogonal left-hand edge wavelet functions for the Chui-Lian interval

multiwavelet basis. Right-hand analogs are mirror images of these functions.

-0.4-0.20.00.20.4

0.80.60.40.2t

-0.4-0.20.00.20.4

87654321t × 10

10-11

10-8

10-5

10-2

87654321t × 10

(a)

(b)

(c)

Figure 7. Example of reconstruction of sampled signal. The variable-frequency signal Doppler

[30], as shown in (a), is nonzero at endpoints of the chosen interval [0.03, 0.8]. The reconstructed

signal using the interval MRA based on 512 sub-intervals (Fig. 8) is shown by open circles in (b)

on a logarithmic t-scale. Solid circles, completely obscured except for the very left-hand edge,

represent the original samples. The absolute error relative to the original samples, graphed in (c)

on a log-log plot, exhibits steady decrease with decreasing sharpness of the signal features.

87654321t 10

j = 0

j = 0

j = 1

j = 2

j = 3

j = 4

j = 5

Figure 8. Decomposition of Doppler signal shown in Fig. 7a into contributions from different

scales j of interval multiwavelets. The uppermost trace corresponds to j = 0 scaling functions

while all others correspond to wavelets.