Wavelets: A Primer

If
= 1

resp.

~ \ =1\
(1)

is necessary and sufficient for 5.1.(2) and 5.1.(3). The third and last condition that we have to take into account is maybe less obvious than the first two, but it is the most crucial of them all: We have to make sure that the inclusions 5.1.(1) are guaranteed. The verification of the following lemma is left to the reader:

(5.2) Assume that a

5.2 The scaling function

127

(5.3) For the inclusion Va C V-I it is necessary and sufficient that an identity of the form

V2

¢(t)

=

L

hk¢(2t-k)

(almost all t

E

JR.)

(2)

k=-=

is valid with a coefficient vector h.

r

E

l2 (Z).

The relations 5.1.(5) and 5.1.(6) imply

so in order for ¢ E Vo C V-I to hold, condition (2) is necessary. Conversely, the identity (2) implies for arbitrary l ¢(t -l) =

L=

V2

hk ¢(2t - (k

+ 2l))

E

Z the identity

(almost all t E JR.) ,

k=-<>o

and as a consequence one has 00

¢O,l =

L

hk ¢-I,k+21 E V-I

Vl EZ.

k=-=

Under such circumstances it is clear that arbitrary linear combinations of the 4>0,1 are lying in V-I as well, thus Va C V_I is proven. ~ The identity (2) goes by the name of the scaling equation; as we have said, it controls the entire multiresolution analysis. As a matter of fact, we shall see in Theorem (6.1) resp. 6.1.(2) that the coefficient vector h. determines the scaling function ¢ uniquely. The coefficients hk also appear in the corresponding algorithms; in fact, they determine more or less everything. When doing numerical computations, one does not need the scaling function ¢ nor the corresponding mother wavelet (that we shall construct in due course) at one's constant disposaL This is in marked contrast to Fourier analysis, where one has to compute function values eit; time and again. The scaling equation describes a kind of "self-similarity". It can be compared with the equation r

K = Uh(K) i=I

5 Multiresolution analysis

128

appearing in the theory of fractal sets, resp. of iterated function systems, to be exact. The Ii in this latter equation are contracting similarities of the euclidean plane; along the same vein the maps T 1-+ t := ~(T + k), playing a key role in wavelet theory, are contracting similarities of the real axis. - That the scaling function 4> should have the reproducing property (2) is obviously a very strong restriction on the possible choices for such a function. The hk cannot be chosen arbitrarily, either. Indeed, we have to make sure that the 4>o,k form an orthonormal basis of Vo. Since the scalar product in L2 is translation invariant, the equations

VnEZ are necessary and sufficient for that. In conjunction with (2) this leads to

80n

=

=

J

4>(t - n) 4>(t) dt = 2 ~ hk hl k,l

~ hk hl ~

J

4>(2t - 2n - k) 4>(2t -l) dt

J

4>(t' - 2n - k) 4>(t' -l) dt' =

~ hk hl82n+k,l ~

We see that in order for the 4>o,k to be orthonormal it is necessary that the hk satisfy the so-called consistency relations 00

~ hk hk+2n

(5.4)

=

VnEZ;

80n

k=-oo

in particular, one must have

L::k Ih k [2 =

l.

While we are at it, we are going to prove a certain linear relation among the hk ; the condition q -I 0 appearing therein is of no importance because of (1).

(5.5) Suppose that h

I

E

l1(Z) and that

J 4>(t) dt =: q -I O.

Then

Integrating the scaling equation (2) from -N to N with respect to t gives

j

N -N

4>(t) dt

=

..J2 ~ k

hk

jN -N

4>(2t-k) dt

=

j2N-k ~ hk 4>(t') dt' . (3) v2 k -2N-k 1

/0

5.2 The scaling function

129

Since

I[::~: ¢(t') dt'! ~ 1I¢1I1

VkEZ,

we can apply the theorem of Lebesgue to the sum on the right hand side of (3). Letting N --* 00 in (3) we obtain

from which the theorem follows. But we should be careful: Even if we have a coefficient vector h. E l2(Z) that satisfies the relations (5.4) and (5.5), we can by no means be sure that there exists a usable function ¢ fulfilling the scaling equation (2). Let us assume for the moment that a multiresolution analysis according to (a)-(c) above is given to Ufl. If we write (2) in the form

then we see that according to general principles about orthonormal bases one has the formula (k E Z) . (4) The scalar product (¢, ¢-l,k) can only be =I 0, if the supports of ¢ and of ¢-l,k overlap. Thus formula (4) allows us to conclude the following:

(5.6) If the scaling function ¢ has compact support, then only finitely many hk are different from O. But one can say even more. To this end, for arbitrary functions f: lR define the quantities

a(f)

:= inf{x

I f(x) =I O} 2: -00,

b(f)

:= sup{x

~

C we

I f(x) =I O} ~ 00 .

Thus a(J) and b(J) are respectively the "left end" and the "right end" of the support of f. In the following theorem we assume for simplicity that ¢ is a bona fide function, not a mere L2-object.

130

5 Multiresolution analysis

(5.1) lithe scaling function has compact support, then the quantities a:= a(
b(-I,k) =

1

"2 (b(
On account of (5.6), the integers kmin :=

I i= O} ,

I i= O}

min{k hk

k max := ma:x{k hk

are well defined. Considering the right hand side of the identity (2) as a superposition of congruent graphs, translated with respect to each other by steps of and taking the a(·) and the b(·) on both sides we see that the following is true:

l,

The last two equations give at once has hahb i= 0 as a bonus.

kmin

= a,

k max

=

bj

in particular, one

-1

Taking into account that only the hk are going to playa role in the numerical algorithms, the last two propositions make it obvious that constructing scaling functions with compact support is not a mere academic exercise. But we still have a long way to go until we are there. := ¢Haar := 1[0,1[

considered in Example 5.1.
(t) == (2t) + (2t -1)

resp.

1



1

= .J2 -1,0 + .J2 -1,1

(see Figure 5.1). Thus in the case at hand we have 1

ho = hI =

.J2'

hk = 0 Vk E Z \ {O, 1} .

(5)

It is easily verified that the statements (5.4), (5.5) and (5.1) are confirmed by this example. 0

:

5.2 The scaling function

131

1 1 - -.....,...------,


I I I I

o

1

Figure 5.1

To conclude this section, we take up a problem that we have postponed so far: We have to formulate precise assumptions on the scaling function that guarantee separation 5.1.(2) and completeness 5.1.(3) of the resulting family (V; Ij E Z). The following theorem shows that under very mild technical assumptions on , condition (I), listed at the beginning of this section, is indeed the only condition for these axioms to hold. (5.8) Assume that the scaling function E L2 satisfies an estimate of the form C (t E JR) (6) I{t)) $ 1 + t2 and that the fa,mily (O,k IkE Z) is an orthonormal basis of Va. Then, first, one has separation: (7)

nv;={O}; j

and second, if and only if the integral also has

r

Uj V;

J

¢(t) dt =: q has absolute value 1, one

= L2, i.e., completeness.

Any I E Va has a representation of the form I = L:k Ik ¢O,k with Because of (6) we have the further estimate

2:k )/kj2 = 11/112 < 00.

Vt E JR (with another C), and this implies, by Schwarz' inequality, that

I/(t) I

s I: Ilkl !(t k

k)1

s C 11/11

(almost all t E JR) .

132

Since

5 Multiresolution analysis

I

E Vo

was arbitrary, we therefore can say that

1111100

:= esssup

II(t)1

c 11111

~

tEIR

For a given 9 E Vj the function following:

I

:= g(2 j .) is in Vo , whence we can say the

Now, if such a 9 belongs to all Vj (j > 0) simultaneously, then this is possible only if IIglioo = 0, whence 9 = O. This proves (7). The space V:= Uj Vj is invariant with respect to the translations Tk (k E Z) and the dilations D 2 j (j E Z); on the other hand, the step functions with jumps at the binary rationals k . 2j are dense in £2. To prove the second statement it is therefore enough to prove the following: The function

I

:= 1[-1,1[ belongs to

V, if and only if Iql =

1.

The relation I E V can be expressed as follows: The function I is arbitrarily well approximated in the £2-sense by its projections P_jl when j - t 00, i.e.,

By general principles this is equivalent with

(8) Keep j > 0 fixed for the moment. By 5.1.(7) we have

P-jl =

L Ck 4>-j,k , k

and consequently

II P_j11l2 =

L ICkl k

2 •

5.2 The scaling function

133

The Ck can be computed as follows:

Ck

=

j1 4>-i,k(t) dt -1

= Ti/ 2

j

= 2i/2j1

4>(2it - k) dt

-1

(9)

N-k

4>(t') dt' ,

-N-k

where we have written 2i =: N as an abbreviation. In the following, the letter C denotes various positive constants that may depend on the chosen scaling function 4>, but not on j (resp. N) and k, and the letter e denotes various complex numbers of absolute value ~ l. From (6) we deduce for arbitrary a

1

> 0 the estimate

1

00

Itl~a

14>(t)ldt<2

a

C

C

2"dt=-. t a

(10)

In order to obtain additiop.al manrevering space in the subsequent convergence discussion we now choose'an c E ]0,1]. It then follows that there is an MEN with

r

Jltl~M

14>(t)ldt

~

c .

(11)

We are now going to estimate the integral on the right hand side of (9). We may assume from the outset that N := 2i ~ M and distinguish the following three cases: (a) If Ikl ~ N - M, then one has -N - k ~ -N + (N - M) = -M and analogously N - k ~ N - (N - M) = M. Because of (11) we therefore may conclude that

Ck

=

Ti/2

(if + 8c) ,

and from this we easily obtain

(b) If N -M <

Ikl ~ N

+M, then

Il:~: 4>(t) dtl ~ J/4>(t)1 dt = C implies the estimate

ICkl ~ 2-i / 2 C.

134

5 Multiresolution analysis

(c) If JkJ > N + M and, e.g., k > 0, then for the upper limit of the integral in question, one has N - k -M < O. This implies in view of (10) that the corresponding Ck can be estimated as follows:

:s

Summing over all such k One obtains

Taking into account the respective numbers of k's in the two cases (a) and (b) we arrive at the following representation of JJPj fJJ2:

II P _jf1l 2 =

2: ICkJ2 k

=

(2. (2j -M) + I) Tj(Jq12 +cee) + T j e(4MC + ~)

= (2JqJ2 +cee) + T Letting j

- t 00

~)

we can draw the conclusion that .lim

)->00

As e

j e(2M(lqI2 + C) +4MC +

JJP-jf1l 2 = 2lqJ2 + cee .

> 0 was arbitrary we see that (8) is valid if and only if Jql = 1.

5.3 Constructions in the Fourier domain Multiresolution analysis is "invariant" with respect to (a) integer translations of the time axis and (b) dilations by powers of 2. In order to make the best use of this inner symmetry we shall transfer the actual construction of admissible scaling functions 4> and corresponding mother wavelets 'IjJ into the "Fourier domain". As a consequence, e.g., the orthonormality of the 4>o,k = 4>(' - k) has to be expressed in terms of properties of ¢; of course we also need a Fourier version of the scaling equation, and so On.

5.3 Constructions in the Fourier domain

135

For an arbitrary function ¢ E L2 one may write

The integral on the right hand side can be thought of as an integral over Z x [0, 27r J. If one interchanges the order of integration, the function

{p(e)

:=

2:)¢'(e + 27rl)/2 I

appears as the new inner integral. By Fubini's theorem {P is defined almost everywhere, first on [0, 27r], then on all of JR, is 27r-periodic, and one has

We first prove the following lemma:

(5.9) The integer translates ¢k := ¢(. - k) of an arbitrarily given function ¢ E L 2 constitute an orthonormal system if and only if the following identity holds: (almost all

eE JR) .

(1)

r

For symmetry reasons it is enough to consider the scalar products of the form {¢o, ¢k}. They are computed as follows:

This implies that the orthonormality condition {¢o, ¢k} ~ 1 {P(k) = 27r

and the latter obviously means {p(e)

80k

==

= 80k is equivalent to

VkEZ, 2~ almost everywhere.

136

5 Multiresolution analysis

The next point on our agenda is the scaling equation

¢(t) == V2'Lhk¢(2t-k)

(almost all t E 1R.) .

(2)

k

Taking the Fourier transform on both sides of (2) we obtain, using the rules (R1) and (R2), the identity

Looking at this formula we are led to introduce (at first only formally) the function

H(~):= ~L hk e-ikf; ; v2

(3)

k

we call it the generating function of the multiresolution analysis under consideration. Because of Ilh.H = 1, the series (3) is almost everywhere convergent, by Theorem (2.4), and defines H as an actual 21l"-periodic function. If only finetely manyhnniknd ( Itdt t: • lnll

5.3 Constructions in the Fourier domain

137

(5.10) The generating function H of a multiresolution analysis satisfies the identity (almost all wE lR) . This of course implies that H is uniformly bounded on lR:

JH(w)J ~ 1

(w E lR) .

(5)

Furthermore, since ¢(O) =J. 0 by 5.2.(1), it follows from (4) that H(O) (5.10) in turn implies H(Jr) = O.

= 1, and

Our next goal is to describe the space W o, i.e., the orthogonal complement of Vo in the larger space V-I> as explicitly as possible. Having such a description in hand we shall be able to give an explicit formula for a possible mother wavelet 'if; belonging to the given scaling function
f =

f

E V-I has a representation of the form

L fk
fk

= (j,
(k E Z) ,

k

and taking the Fourier transform on both sides we obtain (cf. the same calculation for the scaling function
i(~) = ~ L

fk e- ik f,/2

¢(~)

.

(6)

k

Therefore we introduce (analogously to H above) the function

(7) In this way formula (6) becomes

(8) The series appearing in (7) is convergent for almost every ~ E Rj21l'; therefore we can say that the representation (8) is valid for almost all ~ E lR. The above chain of arguments can be reversed: If (8) is true for some function mf E L; , then f E V-I' A function

f

E

Wo C V-I is orthogonal to

170,

and as a consequence one has

(j,
5 Multiresolution analysis

138

for such

I,

and the latter is possible only if the periodic function

2: j(~ + 211"l) 1>(~ + 211"l) l

vanishes for almost all ~ E R/211". In the last sum we again separate the partial sums corresponding to even resp. odd values of l, then we express j by means of (8) and analogously 1> by means of (4), noting that and H are 211"-periodic. Altogether we obtain the following chain of equations, where in the end we again make use of (5.9):

mf

o ==

2: j(~ + 47l"l) ¢(~ + 411"l) + 2: fc~ + 211" + 411"l) ¢(e + 211" + 411"l) l

=

l

2: mf (~) H(~) l¢(~ + 211"l) 12 l

+

2:mf(~ +11") H(~ +11") I¢(~ +11" + 211"l) 12 l

=

(mf(~)H(~)+mf(~+1I")H(~+1I")). 2~·

It turns out that we have proven the following identity: (almost all w E R) .

(9)

Formulas (5.10) and (9) together can be paraphrased as follows: For (almost) every fixed w the vector

H :=

(H(w),H(w+1I"))

is a unit vector in the unitary space ([:2, and the vector

is orthogonal on H. It is easy to see that H and the further vector

together form an orthonormal basis of ([:2. This implies by general principles that illf =

>.(w)H',

(10)

5.3 Constructions in the Fourier domain

139

where the coefficient >.(w) is given by the formula

The function w f--+ >. (w) satisfies the identity >. (w + 1r) there is a 21r-periodic function v(·) such that

== - >. (w), consequently (11)

Inserting this into (10) and extracting the first coordinate we obtain the following representation of m f :

Introducing this into (8) we finally get for

1the expression (almost all e E JR.) _

(12)

This line of reasoning leids us to the following theorem:

(5.11) A function f E L2 belongs to the space Wo, if and only if there exists a function v(-) E L~, such that [ can be written in the form (12)_

r

We have already shown that f E Wo implies the existence of a 21r-periodic function v: JR. -+ C such that [has a representation ofthe form (12)_ Solving (11) for v(·) we get the expression vee) = e- i f.!2>.(e/2), and we infer from (10) that

This implies

Conversely, if (12) is true for some v(-) E L~, then we have (8) with

140

Multiresolution analysis

5

Because of (5) we may conclude that mf E L~, and this in turn implies

f E V-I- Furthermore, we have

proving that the vector m f is orthogonal on H for almost all w _ This means that (9) is true for almost all w; on the other hand, for an f E V-I this is equivalent to f l- Va-.J Inspired by the identity (12) we now define the mother wavelet 'ljJ corresponding to the given 4> by the following formula:

(13) It appears that in doing so we are successful:

(5.12) IE the mother wavelet 'I/J isdeiined by (13), then the system of functions ('l/Jo,k

IkE Z)

constitutes an orthonormal basis ofWo_

I

According to (5.9) the orthonormality of the 'l/Jo,k is proven by the following calculation:

:Llij;(e + 27rl) 12 = :L Iij;(e + 47rlW + :L \ij;ce + 27r + 47rl)\2 I

I

=

I

!H(~ + 7r) \2:L \¢(~ + 27rl) \2 + \H(~) r:L \¢(~ + 7r + 21rl) \2 I

=

(lH(~ + 1r)

r+ \H(~) \2) :1r

I

==

2~ -

As 1 E L~, it follows from (5.11) that 'I/J is indeed in W o , whence all integer translates 'l/Jo,k belong to Wo as well_ On the other hand, consider an arbitrary f E Wo _ By Theorem (5.11) resp_ (12) and (13) we know that there is a v(-) E L~ such that (almost all

eE JR) _

(14)

The function v(-) can be developed into a Fourier series Ek Vk e- ike , and by Carleson's theorem (2.4) this series converges almost everywhere to v(e)- It follows that we can replace (14) by

ice)

=

I: Vk e-ike ij;(e) k

(ahnost aU

eE JR) -

5.3 Constructions in the Fourier domain

141

Now, this is nothing more than the Fourier transform of the representation

J(t) =

"L

1/k

'if;(t - k)

resp.

k

the series appearing on the right converging in L2. Altogether this proves that the 'if;o,k do indeed form an orthonormal basis of Wo . ~ The scaling function does not determine the corresponding mother wavelet 'if; uniquely, thus formula (13) can be modified to a certain degree. For instance, amending it by factors eiO! e-iNf. with a E JR, NEZ, is allowed. An additional factor e-iNf. in :;j produces a translation of the graph of 'if; by N units to the right. In this way, depending on circumstances, one can achieve that 'if; has the same support as . Formula (13) gives only the Fourier transform of the wavelet 'if;. In order to obtain the function 'if; itself we have to translate (13) back into the time domain. Using (3) we get. '.

where at the very end we performed the substitution k := -k' - 1 (k' E Z). Therefore (13) can be replaced by

(15) According to the rules (Rl) and (R3) the last formula is nothing other than the Fourier transform of the representation

'if;(t) =

v'2"L(

_1)k-l h-k-l

(2t - k) .

(16)

k

In order to get a well-structured set of formulas we set

(-1)

k-l-h-k-l =: 9k.

(17)

::>

14:;!

LVIUltlresolutlOn analySis

In this way (16) becomes '1/J(t) =

v2L9k ¢(2t -

(18)

k) ,

k

an identity that has the same structure as the scaling equation 5.2.(2). Another admissible definition of the 9k would have been (19) If, e.g., only the hk for D ~ k ~ 2N -1 are different from zero, then (19) implies the same state of affairs for the 9k, and all summations in the corresponding algorithms (see Section 5.4) range over the index set {D, 1, ... ,2N - I}. Let us summarize the results obtained so far in the following theorem:

(5.13) Assume that (Vj Ij E Z) is a multiresolution analysis with scaling function ¢ and generating function H, and let the mother wavelet '1/J be defined by (13) resp. by (16). Then the function system j

-"/2 (t-k.2 ) '1/Jj,k(t) .= . 2 J '1/J 2j'

is an orthonormal wavelet basis of L2(~).

r

Consider a fixed j E Z. Since according to (5.12) the '1/Jo,k constitute an orthonormal basis of Wo, it is an easy consequence of the principle 5.1.(9) and a small calculation that ('1/Jj,k IkE Z) is an orthonormal basis of W j . The theorem now follows from Proposition (5.1). .-J

CD

As our first example we take up the Haar multiresolution analysis again, cf. Example 5.2.Q). This time we are in a position to construct the mother wavelet '1/J following the prescriptions of the general theory. It is easy to verify that ¢ := ¢Haa:r has as its Fourier transform the function

¢(~) =

1 sin(~/2) e-i~/2

.J2i

~/2

(20)

On the other hand we now insert the values of the hk' as computed in 5.2.(5), into (3) and obtain the following generating function: H(~) = -

1

1 ~ "c/2 -(1 + e-''') = cos - e-'" . 2

v2v2

"C

(21)

5.3 Constructions in the Fourier domain

143

It is easily seen that the functional equation (4) is fulfilled in this case. The recipe (13) now gives

which is the same as 1.6.(1), up to a factor _ei ( This means that the 'IjJ we have constructed here is translated one unit to the left and is multiplied by -1 with respect to the "official" Haar wavelet. This fact is corroborated, if we now compute the 9k by means of (17): -

9-1

1

= ho = .../2'

-

9-2

= -hI

1

= -

.../2 '

all remaining 9k being zero. This gives ~

1

'IjJ = j2 ¢>-1,-1 -

1

j2 ¢>-1,-2

resp. 'l/J(t) = ¢>(2t + 1) - ¢>(2t + 2), as announced above. The reader may convince himself on his own that the alternative definition (19) of the 9k (in the case at hand we have N = 1) would have led to the "official" 'l/JHaar , whose support coincides with that of ¢>Haar' 0

@ As our second example we present the so-called Meyer wavelet. For its construction we again make use of the auxiliary function

vex)

,~ Haxz

-15x' +

6'"

(x S 0) (0 x S 1) (x 2:: 1)

s

shown in Figure 4.6 (this v(.) has nothing to do with the vO's appearing in Theorem (5.11»). We set 1

¢(~).-

v'21r _1_

~

o

(I~I S

cos(~v( ~1~1-1)) 2 21r

2;)

e: s I~I S \1r) (I~I 2::

\?r)

144

5 Multiresolution analysis A

'I7=4\t;,-27f)

,..----

1/V21r

/

/

o Figure 5.2

(see Figure 5.2). This defines a functionj> E L2 about which we can say the following right away: From the fact that ¢ has compact support it follows that ¢ E Coo, and because of ¢ E C2 the assumption 5.2.(6) of Theorem (5.8) is satisfied by ¢; furthermore, one has

J

¢(t) dt = V2rr¢(O) = 1,

as is required for Uj 10 = L2, see (5.8). In view of Proposition (5.9) we now have to examine the function

~(t;,)

:=

L I¢(t;, + 21rlW . l

A short glance at Figure 5.2 shows that it is sufficient to verify condition (1) in the f.-interval [2;, 4;]. In this interval only the two terms corresponding to l = 0 and l =-1 contribute anything to ~(f.) at all. Because of

2~ If. - 27f! - 1 = 1 - (2~f. - 1)

< C < 41l") ( 21l" 3 -<" - 3

and v(l - x) == 1 - v(x)

(x E JR)

it follows that 1 cos ~(t;,) = 21r

2(7f"2v (321l"f. -

1))

is valid for 2; ~ f. ~ 4;, as required.

1sm . 2(7f + 21r "2v (321l"f. - 1))

1

145

5.3 Constructions in the Fourier domain

Figure 5.3

The scaling function for the Meyer wavelet

We now define (out ofthe blue) the 27r-periodic function

lI(f.)

:=

..j2;

I: if;(2f. + 41fl)

(22)

I

(this is, for any given f. E JR, a finite sum!) and assert that Hand ¢ are in fact related to each other by the functional equation

as called for by the general theory.

I

The function f.

f-+

if;(~) has as its support the interval [- 8; , 8;]. On the

other hand, all functions if;( . + 47rl) belonging to an l on this interval. Therefore we already know that

i= a are identically zero

k

is true. This

implies that the right hand side of (23) has for all f. the value

if;({), as stated.

But on the support [-

t, 4;]

of if; the identity if;(~)

;:::;

-.J

146

5 Multiresolution analysis

1

0.5

-4

Figure 5.4

1/2

The Meyer wavelet

According to what we just have proven, the function 1> satisfies a scaling equation as well, so that now all circumstances required for a multiresolution analysis are established. Formula (13) gives the following expression for an admissible mother wavelet in this case:

e~/2 H(~ + 7f) ¢(~)

ie /2 L ¢(f. + 27f + 47fl) ¢(~)

= .j2;e

I

=

.j2;e~/2 (¢(f. + 27f) + ¢(f. -

27f))

2

¢(~)

The corresponding 'f/; is called the Meyer wavelet. One easily verifies that it is, up to the ''phase factor" eie /2 , nothing other than the DaubechiesGrossmann-Meyer wavelet 4.3.(11) corresponding to the step sizes (j := 2, f3 := 1. We refer the reader to Example 4.3.(!) for details. There the 'f/;j,k constituted only a frame. Thanks to the additional factor eie /2 provided by the general theory we now even have an orthonormal wavelet basis. In Figures 5.3 and 5.4 the scaling function 1> as well as the Meyer wavelet ¢ are shown in the time domain. 0 At the beginning of Section 5.2 we put on record that a scaling function 4> has to meet three (sets of) requirements: First, the 1>o,k have to constitute an orthonormal system, second, there is the normalization condition 5.2.(1) securing separation and completeness, and third, there is of course the scaling equation. We conclude the current section by showing how a given 1> that

5.3 Constructions in the Fourier domain

147

satisfies only the second and the third of these conditions can be improved in such a way that the resulting ¢# is a scaling function belonging to the same exhaustion (Yj Ij E Z) of L2 and such that its integer translates ¢# (. - k) are in fact orthonormal.

(5.14) Assume that the function ¢ E Ll n L2 satisfies a scaling equation as well as the condition J ¢(t) dt =1= 0, and let the spaces Yj C L2 be defined by 5.1.(5)-(6). If there are constants B 2:: A > 0 such that (almost all

EE JR.) ,

then the following are true:

(a) The family (¢( . - k) IkE Z) is a Riesz basis of Va ; in particular, it is a frame for Va with frame constants 27rA and 27rB. (b) If one defines the function ¢# via its Fourier transform by

then ¢# determines a multiresolution analysis with the same spaces Yj. This means, in particular, that the functions (¢# ( . - k) IkE Z) constitute an orthonormal basis of Va.

r

(a) We have to show that for arbitrary

1

:=

L::>k¢(' -

k) E

Va

27rB

L I kl

k

the following inequalities are true: 27rA

L I kl 2 ::; C

111112 ::;

k

The Fourier transform of 1 is given by

i= (Lcke-ike)J;(E), k

therefore we have

C

k

2 •

148

5 Multiresolution analysis

In an analogous manner one argues with respect to A, and (4.6) shows that the ¢( . - k) are a fortiori forming a frame. (b) Because of ¢ E L\ the function ¢ is continuous, and this implies in turn the continuity of one after another of 1 v'21T~

,

The two functions v'21T~ and 1/v'21T~ belong to L~. Denoting the Fourier coefficients of 1/v'21T~ by ak, we have (almost all

eE R)

(almost all

eE R) .

and consequently

¢#(E,) = Lake-ik~ ¢(e) k

Translating the last equation into the time domain we come to the conclusion that ¢# =

L ak ¢( . - k) Eva, k

vt

and this in turn implies c Va. In an analogous manner, using the Fourier expansion of v'21T~, one proves the inclusion Va C It follows that each one ofthe spaces ~# coincides with the corresponding Vj.

vt.

That the ¢#(. - k) are orthonormal is an immediate consequence of (5.9). But our proof is not completely finished. It still remains to show that ¢# satisfies the normalization condition 5.1.(1), which means the same as saying that the Vj fulfilled the separation and the completeness axioms to begin with. Because of Va c V-I the modified scaling function ¢# satisfies a certain scaling equation as well, whence also an identity of the form (4): (24) By assumption on ¢ we have ¢(O) -10 and consequently ¢#(O) -10 as well. Therefore we may conclude from (24) that H#(O) = 1 and, what's more, that

5.4 Algorithms

149

H# is continuous in a neighbourhood of O. Since H# satisfies the identity (5.10) we must have H#(1f) = O. We now assert that the following are true: ¢#-(21fl)

=

0

Vl E Z \ {O} .

r

For any given l =1= 0 there is an r E N and an n E Z such that l = 2r(2n+1). Ifwe apply (24) recursively r times, we get r-l

''¢#(21fl)

=

IIH#(2 r - j (2n+1)1f) .H#((2n+1)1f),¢#((2n+1)1f)

=

0,

j=l

since H# vanishes at odd multiples of 1f. In view of what we have just shown, we now have

as required by 5.1.(1).

5.4 Algorithms At this point we pause for a moment in our pursuit of the general theory, in order to present at long last the "fast algorithms" that we have repeatedly announced in earlier sections. In the framework of multiresolution analysis such algorithms lend themselves almost automatically, contrary to Fourier , analysis, where it took centuries from its invention (by Euler) until the advent 'of the FFT . . Maybe the reader has found the numerous factors v'2 and ~ appearing in the foregoing sections to be kind of a nuisance, and he very likely might have , thought that such factors could have been avoided by arranging definitions and notations more carefully. The truth of the matter is that the agreements we made are very sound: Everything is set up in such a way that these annoying factors do not occur anymore where it really matters, to wit, in repetitive numerical calculations.

150

5 Multiresolution analysis

The motor propelling the fast wavelet algorithms is the scaling equation

1>(t)

V2 L

=

hk 1>(2t - k),

(1)

k

paired with the analogous equation for written in the form

'I/J(t)

=

'I/J.

The latter can by 5.3.(18) be

V2 Lgk 1>(2t -

k) ,

(2)

k

the gk appearing in (2) being related to the hk according to 5.3.(17) or 5.3.(19). From (1) we deduce, for arbitrary j E Z, nEZ, the identity

This may be written in the form Vj, Vn,

1>j,n = L hk 1>j-I,2n+k

(3)

k

that is to say, as a recursion formula for 1>j-I,. one obtains from (2) the formula

'V't

1>j, •. In an analogous way

Vj, Vn,

'l/Jj,n = Lgk1>j-I,2n+k

(4)

k

which leads from the array 1>j-I,. to the array 'l/Jj, •. We are now going to analyze a time signal f E L2, and having done that we are going to synthesize it back to its original appearance. In the whole process there will be a finest scale to be considered; we may assume that it belongs to the value j = o. Therefore the analysis begins with the data

aO,k

:=

(j,1>o,k)

:=

J

f(t) 1>(t - k) dt .

These values could be determined, e.g., by numerical integration. It may also be the case that f is only given in the form of a discrete array (J(k) IkE Z) to begin with. In such circumstances one simply puts

aO,k

:=

f(k)

(k

E

Z) .

5.4 Algorithms

151

This is not so farfetched in view of the fact that J ¢(t) dt = 1, particularly in the case when ¢ has a narrow support and subsequent values of ! do not differ much from each other. Be that as it may, for the remaining discussion our basic assumption on ! can be summarized as follows:

I: aO,k ¢O,k .

Po! =

k

The wavelet analysis now proceeds in the direction of increasing j, and this means in the direction of ever longer waves resp. toward more drawn-out features of the signal!. We describe right away the step j - 1 ..,... j. Let j ;::: 1 and assume Pj-I! =

I: aj-l,k ¢j-l,k ,

(5)

k

where the values aj-l,k are known and stored in an array. Intuitively speaking, the image Pj-I! encompasses all features of! having a spread of size;::: 2j - 1 on the time axis; see our detailed explanations in this regard in Section 5.l. Our first task is the computing of the quantities aj,n (n E Z). Using (3) we obtain aj,n := (j, ¢j,n) =

I: hk (j, ¢j-I,2n+k) , k

so

that we can write down the following recursion formula for the step from to aj,_ :

aj-I,_

aj,n =

I: hk aj-I,2n+k k

The array aj,_ encodes the next coarser approximation of !, to wit

p.! J

= ~ '"' aJ,. k

A. . k • 'i'J,

k

The approximations Pj-I! and Pj ! are related to each other by the formula

Qj denoting the orthogonal projection onto Wj. The image Qj! contains all features (details) of ! that have a time spread of size rv 2j /..;2. Since ('if;j,k IkE Z) is an orthonormal basis of Wj, we can write Qj! =

L dj,k 'if;j,k , k

152

5

Multiresolution analysis

and on account of (4) the coefficients appearing here are given by

L

dj,n = (f, 'l/Jj,n) =

9k (f,4>j-I,2n+k) .

k

Expressing the scalar products on the right by means of (5) we therefore obtain the following formula for the "diagonal" step from aj-I,. to dj ,.: dj,n =

L

9k aj-I,2n+k

k

The information about the time signal ! that was extracted in the transition from Pj ! to Pj-I! is now stored in the array dj ,.. Contrary to the "temporary" quantities aj,k , the dj,k are actual wavelet coefficients. Altogether we obtain the following cascade, in the course of which at each step the signal ! is made coarser by a factor of two and at the same time details having a time spread of size 2j / V2 are extracted: IV

ao,.

Ii ---t

al,.

~g

Ii ---t

az,.

~g

Ii

a3,.

~g

dz,.

dl,.

Ii ---t

Ii

---t

---t

~g

~g

d3,.

aJ,.

dJ,.

(6) The wavelet analysis (6) of the given time signal! is terminated after J steps, where the number J comes out in a natural way, see below. We now address the following question: How many arithmetical operations were necessary for this analysis? In order to fix ideas we assume from the outset that the scaling function 4> has compact support. We know from (5.7) that in this case the numbers a( 4» and b( 4» are integers. In keeping with the notation used in certain famous examples later on we assume that

a(4)) = 0,

b(4)) = 2N -1,

N~l.

It follows from (5.7) that only the hk with 0 ~ k ~ 2N - 1 are different from 0, and the same is true for the 9k, if we agree on 5.3.(19). We introduce the following piece of notation: If x. is an arbitrary array over the index set Il, then the formulas supp(x.)

c [p, q[ ,

length(x.)

~

q- p

5.4 Algorithms

153

express the fact that at most the Xk with p ~ k < q are nonzero and that at most q - p individual entries are considered resp. stored at all. (The numbers p and q need not be integers.) The array ao,. encodes all the information that we are going to use about the time signal f. For simplicity, we assume, e.g., supp(ao,.) C

[0, 2J[ ,

length(ao,.) = 2J

.

We assert that under the described circumstances the supports of the arrays fJj,. can be bounded as follows: supp(aj,.) C [-2N + 2, 2J - j [

(j '2 0) .

(7)

r

For j = 0 the assertion is true by assumption. For the step j - 1 -v+ j we may suppose that j '2 1 and that supp(aj-l,.) C [-2N + 2, q[ ,

Because of

q := 2J -(j-l) .

2N-l

aj,n =

L

hk aj-l,2n+k ,

k=O

a component aj,n can be

1= 0 only if the two sets

{2n,2n + 1, ... ,2n + 2N -I}

and

[-2N + 2, q[

have a nonempty intersection, and for the latter it is necessary and sufficient that the inequalities

2n< q

/\

2n + 2N - 1 '2 -2N + 2

hold. The first of these says n < q/2 = 2J - j , the second n '2 -2N + ~. Thus we may conclude that supp(aj,.) is bounded as stated in (7). ...J Formula (7) suggests that we terminate the process after J steps, since from then on supp(aj,.) stays put at [-2N + 2, OJ. How many multiplications have been carried out up to this point? (For the sake of simplicity we are disregarding the additions here.) The computation of an individual value aj,n requires at most length(h.) multiplications. On the other hand we conclude from (7) that length(aj,.) ~ 2 J -

j

+ 2N -

2

(j '2 0) ,

= 2N

5 Multiresolution analysis

154

and for length(dj,.) we obviously have the same bound. Altogether we obtain the following upper bound for the total number f.L of multiplications required for the complete analysis of the given signal f: J

f.L::; 2·2N· 2:)2 J - j +2N-2) =4N(2J -1+J(2N-2)). j=l

This implies f.L ::; 21ength( h.) length( ao,.) (1

+ 0(1)) ;

that is to say, the number of required operations is linear in the input length. Starting from ao,. and proceeding in the described way we have computed in J ~ 1 steps the coefficient arrays

(the intermediate or "temporary" arrays ao,., ... , aJ-l,. are no longer needed). The total length of these arrays is about equal to length(ao,.), so that at first glance we have gained nothing in terms of storage requirements. But we have to bear in mind that the individual coefficient arrays dj,. will contain long sequences of negligible entries dj,k, depending on the fine structure of the time signal f in different regions of the t-axis. By disregarding all dj,k whose absolute value is below a certain threshold and releasing the corresponding storage cells one is able to achieve spectacular compression ratios without Significant loss of information. For instructive examples in this regard, we refer the reader to [19]. Now for the synthesis: Here we obtain an algorithm of a similar simplicity. Since the step j -1 ~ j amounts to replacing the orthonormal basis CPj-l,. of YJ-l by the likewise orthonormal basis CPj,. U 'l/Jj,. , the reverse step j ~ j -1 does not necessitate the inversion of a certain matrix. The details are as follows: One has

L aj,k hk + L dj,k 'l/Jj,k k

k

and consequently aj-l,n = (Pj-d, CPj-l,n) =

L aj,k (CPj,k, CPj-l,n) + L dj,k ('l/Jj,k, CPj-l,n) . k

k

The scalar products appearing on the right can be read off from (3) and (4):

5.4 Algorithms

155

so that altogether the following synthesis formula emerges: aj-l,n =

L hn- 2k aj,k + L gn-2k dj,k k

k

In this way we obtain as a counterpart to (6) an "upward" cascade that takes the coefficient arrays aJ,. , dJ,., dJ-I,., ... , d2,., dl ,. as its input and finally returns ao,., i.e., Pof, as its output: aJ,.

h ~

aJ-I,.

/g dJ,.

h ~.

aJ-2,.

h

h

~

~

/g

/g d2,.

dJ-I,.

al,.

h ~

ao,.

/g d 1,.

We leave it to the reader as an exercise to compute the total number 11 of multiplications required ~for such a synthesis. The resulting figure will be about twice as large as the 11 from the "downward" cascade (6). The boxed formulas show that we need only a table of the hk and the gk in order to be able to begin with concrete numerical work. Neither the scaling function 1> nor the mother wavelet Whave to be stored, be it numerically or otherwise, nor do they have to be recomputed on end at runtime. (By the way, one does not need to understand anything of the underlying theory either. .. ) In [DJ one finds a great number of such tables; they relate to various wavelets 'if; that for the one reason or another have proved their worth. The following example of such a table belongs to the so-called Daubechies wavelet 3W having support [0,5] : k

hk

0 1 2 3 4 5

.3326705529500825 .8068915093110924 .4598775021184914 -.1350110200102546 -.0854412738820267 .0352262918857095

gk

= (-I)kh 5 _k

.0352262918857095 .0854412738820267 -.1350110200102546 -.4598775021184914 .8068915093110924 -.3326705529500825

(8)

We shall construct this wavelet in 6.2.@ ab ovo, only there we shall see how the values of the hk tabulated above come about.

5 Multiresolution analysis

156

CD (Continuation of 5.3.@) We have not yet computed the hk corresponding to the Meyer wavelet. That's what we are going to do now. The generating function H(·) is given by 5.3.(22) and is an even function, as is 4>. Thus on account of 5.3.(3) we obtain successively hk

= .../2 2 2j'lr H(f.) eikf; df. = .../2j'lr H(f.) cos(kf.) df.

-'lr 2~ -'lr -.../21'lr V21fL: ¢(2f. + 4~l) cos(kf.) df. . ~

=

~

0

l

In the last sum, only the term corresponding to l = 0 is contributing anything to the integral, whence we obtain hk = h-k =

r ¢(2f.) cos(kf.) dE; .

2 -..fo Jo

These integrals now have to be computed numerically. In view of the function v(·) used in the construction, the resulting ¢ has 4-clicks at the two points ± and 3-clicks at the two points ± ~; apart from that it is infinitely differentiable. This implies (cf. Example 1.2.@) that for k ~ 00 the hk decay only like 1/k4. The numerical computation results in the following values:

2;

k

hk = h-k

k

hk = h-k

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

.748791 .442347 -.039431 -.127928 .033278 .057120 -.024807 -.025310 .016000 .009538 -.008556 -.002451 .003416 .000058 -.000647 .000225

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

-.000329 .000061 .000333 -.000231 -.000059 .000174 -.000115 -.000027 .000115 -.000067 -.000028 .000066 -.000040 -.000015 .000046 -.000027

o

6 Orthonormal wavelets with compact support

6.1 The basic idea We are confronted with the task of producing scaling functions ¢: lR -..,. C having the following properties: (a)

¢ E L2 ,

(b)

¢(t)==h~hk¢(2t-k)

(c)

!

supp(¢) compact,

¢(t) dt = 1 resp.

¢(O) =

(d) !¢(t)¢(t-k)dt=OOk

¢(~)=H(~)¢(~),

resp.

k

vk,

resp.

~1¢(~+21rlW==~. 211"

k

If all these conditions are met, then Theorem (5.13) will provide us with an orthonormal basis of wavelets 'l/Jj,k having compact support.

Condition (a) immediately implies ¢ E L1 and ¢ E Coo; furthermore, we know from (5.6) that only finitely many hk are nonzero. It follows that the generating function

H(~)

:=

..2.- ~ hk e-ik~ V2k

is a trigonometric polynomial satisfying the identity

(1) and having the special values H(O)

= 1, H(1I") = 0;

see (5.10).

The systematic construction of polynomials with these properties is an algebraic problem that we shall take up in the next section. For the moment we assume that we have such an H at our disposal, and we begin our undertaking by showing that the corresponding scaling function ¢, if there is one at all, is uniquely determined by H. Applying (b) recursively r times we obtain

6 Orthononnal wavelets with compact support

158

and, therefore, because of (c), 1

~

¢(~) = V2-ff r~~ }1 H(2j) r

,

(2)

if the infinite product converges. In this regard, we show: (6.1) Assume that the generating function H E 0 1 satisfies the identity (1) as well as H(O) = 1. Then the product (2) converges locally uniformly on IR to a function ¢ E L2.

I

Setting max IH'(~)I =: M

e

and using the mean value theorem of differential calculus we obtain IH(~)

-1\

= IH(~) - H(O)I ~ M I~I

(~ E

R),

therefore we may conclude

- 1\ I H(~) 23

< -

M23I.~I

~

(j

0) .

Because 2: j2 1 2- j

= 1, this implies by general principles that the product (2) is converging locally uniformly to a continuous function ¢: R - t Co In order to prove ¢ E £2 we have to modify the limiting process leading from H to

¢ slightly by means off a

"cut-off function". To this end we set

1o(~)

vk 1[_1r,1r[(~)

:=

and define recursively, as in (b),

(3) This implies ~

fr(~)

=

1 !2;

~

II H(2j) .1[_2 1r,2 1r[ • ... 2?r r

r

(4)

r

j=l

For any given

~ E

R there is an - 2r ?r

TO

~ ~

such that

< 2r ?r

VT

> TO ,

showing that the "cut-off factor" in (4) has no effect as soon as Therefore the comparison with (2) proves (~E

T

>

TO.

R) ,

moreover, we have locally uniform convergence of the fr as well. The next point on the agenda is the following lemma:

6.1 The basic idea

159

(6.2) For each r ~ 0 the family (jr(' - k) IkE

r

Z)

is an orthonormal system.

Because of Proposition (5.9) the assertion of the lemma is equivalent to

(r Now the recursion formula (3) for the mula for the functions
<pr(e)

=

~

0) .

(5)

f.. implies the following recursion for-

L If..(e + 411"l) 12 + L If..(e + 211" + 411"l) 12 l

l

Since statement (5) is obviously true in the case r (1) together imply that .it is true for all r ~ o.

= 0, the last equation and -.l

,

In particular we have IIfrll 2 = 1 for all r ~ O. Using Fatou's lemma we therefore may draw the following conclusion about the limit function J;:

This proves ¢ E L2. The existence of a scaling function ¢ corresponding to the given H being established, we now have to take care of supp(¢). How can we be certain that the scaling function (2) indeed has compact support, given that only finitely many hk are nonzero? The functions fr that were used in the proof of Theorem (6.1) and are converging to ¢ in L2 certainly do not have compact support; in fact, theX are holomorphic functions of the complex variable since the sets supp( fr ) are compact. In order to get control over supp( ¢) we have to argue directly in the time domain. So let us assume that

e,

I

a(h.):= min{k hk =F O}

= 0,

I

b(h.):= max{k hk =F O}

= 2N -1.

(6)

If the resulting ¢ has indeed compact support, then we know from (5.7) that the latter is bounded below by a(¢) = 0 and above by b( ¢) = 2N -1. We now construct a second sequence (gr Ir ~ 0) that converges in some sense to ¢;

160

6 Orthonormal wavelets with compact support

but this time we make sure that the supports of all gr are lying in the interval [0,2N - 1] we are aspiring to. For the definition of such a sequence we recall the reproducing property of if> encoded in the scaling equation 5.2.(2). It can be expressed as follows: The scaling function is a fixed point of the transformation 2N-l

Sg(t) :=

..J2

L

hk g(2t - k) .

k=O

In functional analysis the common procedure to determine a fixed point of some mapping S is the following: One chooses a suitable starting point go and defines recursively a sequence (gr Ir ~ 0) by the formula

(r

~

(7)

0) .

If one is lucky, this sequence converges to ''the'' fixed point if> of S. In view of 5.2.(1), in the case at hand we choose go := 1[0,1[ and define the sequence (gr Ir ~ 0) by (7). The first thing we prove is

SUpp(gr) C [0,2N -1]

Vr~

O.

(8)

Because N ~ 1, the assertion is true for r = o. If (8) is valid for a certain r, then the value gr+l(t) = Sgr(t) has to be 0, unless the two sets

I

{2t - (2N -1), ... ,2t -1,2t}

and

[0,2N-1]

have a nonempty intersection; for this to be the case, the inequalities

2t

~

0

2t - (2N -1) 5: 2N -1

must hold, which is the same thing as saying that 0 5: t 5: 2N - 1. The effect of S in the Fourier domain is obviously given by

and iterating this r times produces for our gr the formula

.J

6.1 The basic idea

161

Now by 5.3.(20) we have

90(';)

_1_ e-i (.!2 Sinc(i)

=

and therefore lim

r-+oo

.j21r

90(~) r 2

2

(9)

= _1 .

.j21r

This implies that at least in the Fourier domain we have what we hoped for, that is to say

the convergence being locally uniform on lR. How well the 9r themselves converge to depends strongly on the regularity properties of , and these we don't know. At the moment the ''function'' is but an L2 object. Nevertheless, it makes sense to talk about the support of . The statement ;:supp(
[

1(tWdt = 0

JIR\[0,2N -I]

is guaranteed, and for the latter it is sufficient for to be orthogonal on all test functions u E C 2 having co~pact support disjoint from [0, 2N -1]. This is precisely what we are going td prove the following lemma:

in

(6.3) Let u be a C 2 -function having a support that is compact and disjoint from the interval [0, 2N - 1]. Then

(,u) =

J

(t)u(t)dt = O.

I

Let an c be given. By assumption on u we know that U E Ll, thus there is an M > 0 such that

1

leI2:M

lu(.;) Id'; ~

c .

Such an M being fixed, one can find an r :;::: 0 such that

162

6 Orthonormal wavelets with compact support

furthermore, we deduce from 5.3.(5) and (9) that 'It; E lR, 'IT 2: 0 .

In view of (8) the supports of 9r and u are disjoint, therefore we can write

(¢,u)

= (9r,U) + (¢ -

9r,U)

= 0 + (J; - 9n u) ,

so that we obtain the estimate

\(¢, u)1 <

1M I¢'(t;) -

9r(t;)lIu(t;) Idt; + {

-M

J~~M

(I¢'(t;) I + 19r(t;)\) lu(t;) \ dt;,

< (lIulll + ~)c. Since c > 0 was arbitrary, we must have (¢, u)

= 0, as stated in the lemma..J

Altogether, we have arrived at the following theorem:

(6.4) Assume that the coefficient vector h. is bounded by (6) and that the corresponding function H satisfies the identity (1), as well as H(O) = 1. Then the scaling equation admits a unique solution ¢ E L2, and ¢ has compact support in the interval [0, 2N - 11. By the way, the iteration procedure that we have used in the proof of (6.4) can easily be implemented for the actual numerical construction of ¢ as well. Figures 6.1 and 6.3 show the approximating step functions 9r together with the limiting scaling function ¢.

3

Figure 6.1

Iterative construction of Daubechies' scaling function 2

6.1 The basic idea

163

In view of (6.1) resp. (6.4) the scaling function 4> is uniquely determined by H and explicitly given by (2). Therefore the following procedure suggests itself: One chooses a trigonometric polynomial H that satisfies the identity (1), as well as H(O) = 1, and defines 4> by (2). Then (a), (b) and (c) at the beginning of this section are fulfilled automatically; it remains to prove (d). The following example shows that the consistency conditions encoded by (1) are necessary, but unfortunately not sufficient for (d).

CD

Taking off from Example 5.3.CD we define

H(f.) := ~ (1 + e-3i~) = e-3i~/2 cos 3f. . 2 2 The identity (1) is fulfilled in this case:

The uniquely determined solution of the functional equation (b) that also satisfies (c) can be written down explicitly; it is ,

Taking the inverse Fourier transform one gets

(0::; t < 3) (otherwise) It is easy to see that the functions 4>o,k = 4>(' - k) (k E Z) are not orthonormaL On the other hand it can be shown that the 'l/Jj,k derived from this particular 4> constitute a tight frame for L2, see [D], Proposition 6.3.2.

0

Various additional assumptions on H have been proposed to make property (d) come true, as a matter of fact the gap is not wide. We shall treat two such attempts in what follows. The following variant is due to Mallat [12]: (6.5) Assume that the generating function H E 0 and H(O) = 1, as well as the additional condition

1

satisfies the identity (1)

(10) and let if; be defined by (2). Then the functions 4>o,k (k orthonormal basis of Vo .

E

Z) constitute an

164

6 Orthonormal wavelets with compact support

I

We have to show that the orthonormality (6.2) of the functions];. ( . - k) is preserved in the limit. That's where the extra hypothesis (10) comes in.

If 1.;1 ~ 7r, then one has H(';/2 j ) :I 0 for all j ~ 1, and this implies by definition of the convergence of an infinite product that ¢;(.;) :I o. Because of the locally uniform convergence of (2) we know that ¢; is continuous, therefore we can find a 5 > 0 such that

(1.;1

~ 7r) .

(11)

A moment's reflection will show that the function];. can also be written in the following alternative way:

(otherwise) In view of (11) this implies that the universal estimate

is valid, so that in the concluding formula

J

¢(t) ¢(t - k) dt =

J

1¢;(.;Weikf; d';

= r->oo lim

J1];.(';)1

2

eikf; d';

= OOk

Vk

E

7l

we are allowed to apply Lebesgue's theorem (on limits under the integral sign) .

.-J

CD

(Continued) In order to see what went wrong in this example we compute

~ I2 = 27r 1 Ifr(';)

r

r

j=l

j=l

ITI H (.;) 12 = 27r 1 IT 2 3.. 2j cos 2j+1

Now consider the points ';r := one has

i 2r 7r

C

(r ~ 1). According to the last formula

6.1 The basic idea

165

er

for all r ~ 1. Since the tend to infinity when r -T that the 1];.1 2 have a common integrable majorant.

00,

it seems inconceivable

The deeper reason for the phenomenon observed here is the following: The action D: 'R./21r -T 'R./21r , has a closed orbit

(eo,··· ,en-I),

ek

:=

Dek-I Vk,

en

=

eo

(12)

F:,

~}. It is a sidesuch that IH(ek)1 = 1 for all k, namely the two-cycle effect of condition (10) to make orbits of this kind impossible. This can be seen as follows: Condition (10) implies

< c"(: < 311") 2

IH(e) 1< 1 the variable

( 1!: 2 -

e being understood modulo 21r.

'

Let (12) be an arbitrary closed

orbit of D. In the (necessarily periodic) binary representation of

g~

modulo 1

each of the two sequences 01 or 10 must occur somewhere. But this implies that after finitely many steps a point Dj eo falls into the interval [~, 3;]; therefore the orbit under discussion necessarily contains points ej for which 0 one has IH(ej) I < 1. Lawton [11] has found a condition of a more algebraic nature that likewise guarantees the orthonormality of the functions ¢>O,k. We again assume (6); then by Theorem (6.4) we have a(¢» = 0 and be¢»~ = 2N -1. At stake are the numbers

am

:=

(¢>, ¢>O,m)

=

J

¢>(t) ¢>(t - m) dt

Because of supp(¢» c [0, 2N -IJ all am with Iml zero. Due to the scaling equation 5.2.(2) one has

J J

= L ~

hk hi

2N -1 are automatically

¢>(2t - k) ¢>(2t - 2m -l) dt

am = 2 Lhk hi k,1

~

(m E Z) .

¢>(t') ¢>(t' + k - 2m - l) dt' = L hk hi a2m+l-k . ~

If we substitute the summation variable 1 according to 1 := n + k - 2m, where n is the new running variable, we obtain

am

=

L(Lhkhn+k-2m) an· n

k

(13)

166

6 Orthonormal wavelets with compact support

In this way the square matrix A are defined by

:=

[Amn] of order 4N - 3, whose elements

L hk hn+k-2m

Am,n :=

(lml, Inl < 2N -1)

(14)

k

comes into play. Formula (13) can now be read as O:m = I:n Amn O:n, meaning that the vector 0:. is an eigenvector of A corresponding to the eigenvalue 1. The special vector

/3.

:=

(0, ... ,0,1,0, ... ,0) ,

i.e.

f3m = 80m

(lml < 2N -1)

is an eigenvector of A corresponding to the eigenvalue 1 as well; for, because of (1) resp. (5.4), one has

L Amn f3n = Am,o = L hk hk -

2m

=

80,m

= f3m

( Iml < 2N -

1) .

k

n

After all this work, we are in a position to state the following theorem:

(6.6) Assume that the coefficient vector h. is bounded by (6), that the corresponding function H satisfies the identity (1) and H(O) = 1, and that is the scaling function determined by (2). H 1 is a simple eigenvalue of the matrix A, then the functions O,k (k E Z) are orthonormal.

I

By assumption on A there is a number c E C* such that 0:. = cf3.; that is to say, all O:m = (, O,m) corresponding to m =F 0 have the value 0 as stated, and 0:0 = c =F o. The computation carried out in the proof of (5.9) shows that under these circumstances the identity

holds. Now, if l = 2T(2n + 1) =F 0, then the calculation T-1

¢(27rl)

=

II H(2T-

j=l

j

(2n + 1)7r) . H(2n + 1)7r) ¢(2n + 1)7r)

=

0,

(15)

6.1 The basic idea

167

repeated from the proof of (5.14), shows that in fact c

=

27r 1¢(0)1 2

=1 .

CD (Continued) In this example we have N = 2, and the hk take the following values: 1 ho = h3 =

J2'

Inserting these into (14) one arrives at the matrix

(the rows and columns

a~e

numbered from -2 to 2), having the eigenvalues

-1,

1

1

-"2' "2'

1, 1 .

The eigenspace corresponding to the eigenvalue 1 is two-dimensional; it is 0 spanned by the vectors (1,2,0,2,1) and, of course, (0,0,1,0,0). So far we have not touched the question of how regular the scaling functions are that one obtains in this way. Figures 6.1 (resp. 6.5) and 6.3 show that 1 may indeed look quite jagged. Since such a 1 comes into being only as the limit of a certain "fractal" process, and is not at our disposal in the form of a simple expression, the investigation of its regularity, be it via the decay of ¢(~) for I~I - 00 or via a careful analysis of the operator S, is very delicate and requires subtle estimates of various sorts. In this way one is able to prove, e.g., that the Daubechies scaling function 31 and its corresponding mother wavelet 37/J are already continuously differentiable, and furthermore that the order of differentiability increases essentially linearly (with a proportionality factor rv 0.2) with N. For details we refer the reader to [DJ, Chapter 7, or to the paper [7J.

168

6 Orthonormal wavelets with compact support

6.2 Algebraic constructions In view of the results presented in the last section, only the following algebraic problem remains: We have to find trigonometric polynomials that satisfy the identity H(f.) := ~ hk e- ike

I:

V2k

and, of course, the condition H(O) = 1. We shall insist here on real coefficients hk; the corresponding scaling functions if> as well as the mother wavelets 'if; will then be real-valued as well. According to 5.3.(13) the Fourier transform of 'if; is given by

Now, on account of what we said in Section 3.5 (see, e.g., Theorem (3.13)), we are interested in our wavelet 'if; having an order N as high as possible, and according to 3.5.(3) this is equivalent to the requirement that ~ should vanish of an order N as large as possible at f. = O. As a consequence the generating function H should have a zero of order N » 1 at f. = 7r, a fact that we express most elegantly by writing N?l. Instead of looking for H we switch for a moment to the function

(1) that would have to satisfy the linear identity

(2) For symmetry reasons the function M is a polynomial in cos f. , and M contains the factor

Therefore we may write

(3)

6.2 Algebraic constructions

169

where P is a certain polynomial as welL Now we introduce a new variable y by letting y := sin2 ~. This leads to A(~) = p(cos~) = P(l - 2y) =: P(y) ,

(4)

where again P is a certain polynomiaL In this way (3) becomes M(~)

Because of

= (1 _ y)N P(y) .

+ 1r) =sm . 2 "2~ =y cos 2 (~ -2-

and

A(~ +1r) = p(-cos~) = P(2y -1) = P(l- 2(1- y») = P(l- y), the identity (2) takes the following form when expressed in terms ofthe variable y: (5) This formula is valid for 0 :::; y :::; 1 at first, but by general principles on holomorphic functions we may conclude that it is true for arbitrary y E C. By the theorem on decomposition into partial fractions there are uniquely determined coefficients Ck , C k such that

and for symmetry reasons one has Ck = C k for all k. Clearing denominators, we can infer that there is a polynomial PN of degree :::; N - 1 such that

holds, and PN is the only polynomial solution of (5) having a degree:::; N - l. Now it easy to see that any solution P of (5) satisfies the identity

P(y) == (1 - y)-N (1 _ yN P(l _ y») as welL In particular, this is the case for PN, and this allows us to draw the following conclusion:

PN(y)

=

if:- 1 PN(y)

=

I: (-N) k (_y)k = I:

N-l

N-l

k=O

k=O

(N + k-1) yk . k

(6)

170

6 Orthonormal wavelets with compact support

Here we have made use of the fact that the part of PN carrying the factor 1 PN . The solution of (5) having the smallest yN gives no contribution to possible degree now has been determined explicitly: It is the right hand side of (6). Now let P be an arbitrary solution of (5). Then

i:-

and consequently P(y) - PN(y)

= yN P*(y)

for some polynomial P*. If we insert this into (7) again, we obtain P*(y)

+ P*(l- y) == 0,

which is equivalent to p*(y)

=

R(l- 2y)

= R(cos~) ,

R odd.

Since we can perform the same computations backward as well, all in all the following theorem has been proven:

(6.7) A trigonometric polynomial M(·) satisfies the identity (2) if and only if it has the following form:

M(~) = (coS2~) N p(sin2~)

.

Here P(y)

= PN(y) + yN R(l - 2y) ,

where PN is given by (6) and R is an arbitrary odd polynomial. In view of (1) such a function M(·) is of use only if P satisfies the additional condition P(y) ;::: 0 Letting P := PN, this condition is obviously satisfied. So much for the admissible functions M, these being related to H by (1). In order to get the generating functions H themselves, we must, so to speak, "take the square root of M ". In doing this we only have to bother about the factor

P(sin2~) = p(cos~) = A(~)

introduced in (3). For carrying out this task a surprising lemma of Riesz will come to our help. It reads as follows:

6.2 Algebraic constructions

171

(6.8) If n

A(~) = Lak cosk~, k=O

and if A(~) ~ 0 for real~, in particular A(O) = 1, then there is a trigonometric polynomial n

L b e-ik~

B(~) =

k

k=O

with real coefficients bk and B(O) = 1, such that A(~)

identically in

I

B(~) B( -~)

==

(8)

,

~.

The function A(·) possesses a product representation of the form n

~ A(~) = an

II (cos~ -

(9)

Cj) ,

j=1

the Cj being real or else appearing in complex conjugate pairs. We introduce the complex variable z by writing e-i~ =: Z; then (9) goes over into n

A(~)=anII

(Z +Z-1

)

2

(10)

-Cj.

j=1

In investigating the individual factors appearing in (10), we need the well known properties of the mapping z I--t (z + Z-1 ) /2 as well as the identity

z + Z-1

----=2=--- -

s + s-1 2

1

== - 2s (z - s) (z

-1

(a) If Cj E R and ICjl ~ 1, then there is an s Therefore we obtain, using (11):

z + Z-1

- - - - Cj =

2

(b) If Cj

E

Rand

ICjl

1

-_.

2s

E

R* such that

1

eia:

S+S-1 2

# 0) .

Cj =

=COSct.

(11)

(s+s-I)/2.

)

(z - s) . (z- - s .

< 1, then there is an s = Cj=

(zs

- s)

# ±1 such that

172

6 Orthononnal wavelets with compact support

This implies that A(';) contains a factor cos'; - COSQ, and the latter is not compatible with A(';) ~ 0 (.; E JR), unless this factor occurs an even number of times. Therefore there is a j' such that Cj' = Cj, and using (11) we obtain the identity 1 (Z+2Z-

-Cj)

(z+2z -

1

-Cjl)

=

4e;ia (z - eia ) (Z-1 - eia)(z - eia )(z-1 - eia )

=

~(z - eia)(z _ e- ia ) (Z-l _ eia ) (Z-1 _ e- ia ) 4

= ~.(z2-2ZCOSQ+1).(Z-2-2z-1COSQ+1). (c) If Cj tj JR, then there is, first, a j' such that s E C* such that 2

Cj'

=

Cj'

=

Cj

and, second, an

8+8- 1

2

Using (11) again we get

All things considered, it follows that it is possible to combine and to regroup the factors appearing in (10) in such a way that the resulting representation of A(';) assumes the following form:

Here Q(z) = L:~=o qkzk is a polynomial with real coefficients qk, and the constant C E C" is obtained by collecting an and the various numerical factors that have appeared in (a)-(c). The extra condition A(O) = 1 gives C = 1/(Q(1))2. It follows that, if we set B(';) := Q(e-i~)IQ(l), then (8) is valid; therefore the lemma is proven. -.J The decomposition (8) is not uniquely determined, since in the cases (a) and (c) interchanging sand S-1 leads to another decomposition ofthe corresponding partial product of A(·). This, albeit modest, flexibility can be used to make

173

6.2 Algebraic constructions

the resulting scaling function and in consequence the related mother wavelet more symmetrical. We shall not pursue this matter any further. Assume that N is given. If we choose for simplicity P := PN , then A(.) becomes a polynomial of degree N - 1 in cos~ and B(·) a polynomial of degree N - 1 in e-i~. In this way the generating function

is of degree 2N - 1 in e-i~, and the support of the corresponding scaling function (=: N¢» turns out to be the interval [0, 2N - 1]. The mother wavelets N'I/J derived from the N¢> are called Daubechies wavelets.

CD

In the case N = 1 we of course obtain the Haar wavelet. Formula (6) gives P1 (y) == 1, and this in turn implies p(cos~) == 1, B(~) == 1, so that we finally get 1

·c

H(!;.) = 2(1 + e- t ... )

,

o

which is in agreement with 5.3.(21).

The case N = 2 shall be dealt with in detail in the next section; the case N = 3 appears as Example @ below. In [D], Table 6.1, the coefficient vectors (hk 10 :'S k :'S 2N - 1) corresponding to the Daubechies wavelets N'I/J are given to 16 decimal places for 2 :'S N :'S 10. In [L], Table 2.3, one finds these coefficients to six decimal places for N from 2 to 5.

@ We now describe in detail the case N P(y) :=P3 (y) =

= 3, choosing

G) + G)y+ (:)y 2 =1+3 +6y2 . Y

Inserting 2 ~ 1 ·c ·c y = sin - = -( -e-·. . + 2 - e·. . ),

2

4

into (4) we get

A(~)

=

~e-2~ _ -49 e-i~ + 19 8

4

_ ...

Figure 6.2 confirms that A(~) is 2:: 0 throughout so that it makes sense to proceed with our computation. In the case at hand, the function B(·) has the

174

6 Orthononnal wavelets with compact support y

1

o

211"

Figure 6.2

form B(~) = bo + b1e- ie + b2e- 2ie , so that we have to compare coefficients in the identity

°e + b2 e- 2°e °e 2°e , )(bo + b1e' + b2e • )

(b o + b1e-'

3 2°e - '49 e-'"·c + 4: 19 - ... = Se-'

For symmetry reasons it is enough to check the coefficients corresponding to e- 2ie , e- i { and 1. In this way we obtain the three equations

(12) Because A(O) = P(O) = 1, Lemma (6.8) guarantees that we can find real solutions (b o, b1 , b2) that satisfy the additional condition bo + b1 + b2 = 1. If we use this condition to eliminate bo + b2 from the second equation in (12), we get for bl the quadratic equation by - bl - ~ = 0, and this in turn leads to b _ l±v'lO 12 '

We leave it to the reader to pursue the upper choice of the sign here; it will result in complex solutions bo and b2 • This means that we definitively have b1 = (1 - v'lO) /2, and because of the first equation in (12) we can say that bo and b2 are the two solutions of the quadratic equation

Choosing arbitrarily (well, not quite ... ) one ofthe two possible assignments, we get

V5 + 2v'lO + 1 - 2v'lO e-ie + 1 + v'lO - 4V5 + 2v'lO e -2ie ,

B(..C) -_ 1 + v'lO + 4

6.2 Algebraic constructions

175

so that we finally obtain

C+

H(f,)

;-i€

r

B(e)

.!.(1 3 -i€ 8 + e +...) =

1 + J10 + V5

32

(1 + J10 +4V5 + 2J1O + 1-2J10 e-i€ + ...)

+ 2J1O

+

5 + J10 + 3V5 + 2J1O -i€ 32 e

+ ....

From the part of H that is actually printed out here one can immediately read off ho and hI: 1 + J10 + V5 + 2J1O ho = v1n2 L, 32

hI =

= 0.33267 . .. ,

J2 5 + J10 + 3V5 + 2J1O = 32

0.80689 ... ,

both in agreement with Table 5.4.(8). We leave it to the reader as an exercise to compute the remaining hk as well and so convince herself that we have indeed determined the coefficient vector h. corresponding to the Daubechies wavelet 31/J. Figures 6.3 and 6.4 show the functions

0

34> and 31/J in the time domain.

1-+------::#----'>-

3

Figure 6.3

The Daubechies scaling function

34>

4

5

6 Orthonormal wavelets with compact support

176

1

1

Figure 6.4

4

5

The Daubechies wavelet 3'1/1

6.3 Binary interpolation In the two foregoing sections we obtained scaling functions and corresponding wavelets by means of constructions in the Fourier domain, and also as limiting functions of an iteration procedure. In neither approach, however, did we discuss the convergence behaviour in the time domain. Now there is a third, called the direct method for constructing scaling functions . This method yields without a limiting process the exact values (x) at all "binary rational" points x E JR, and it is with the help of this method that one obtains the best regularity results, e.g. for the Daubechies wavelets N'lj;. In order to fix ideas, we assume that an N > 1 has been chosen once and for all and, furthermore, that

a(h.)

=

0,

b(h.) = 2N -1,

as agreed upon in connection with the Daubechies wavelets. The following abbreviations will prove useful:

{O, 1, ... ,2N - I}

=: J ,

6.3 Binary interpolation

177

For the description of the binary rational numbers we use the handy notation

therefore we have the inclusions Z

= lJ)o

C lJ)1 C ... C lJ)r C lJ)r+1 C . . . C lJ) ,

and lJ) is dense in R. The scaling equation now has the form 2N-l

¢(t) =

v'2

L

ho h2N-

hk ¢(2t - k),

1 =1=

0.

(1)

k=O

The "direct method" is founded on the following three simple facts: • • •

If t E lJ)r for some r ~ 1, then the numbers 2t - k (k E J) belong to lJ)r-l· If t < 0, then the numbers 2t - k (k E J) are < 0 as well. If t > 2N -1, then,the numbers 2t - k (k E J) are> 2N -1 as well.

On account of these facts the scaling equation (2) allows us to compute the values of ¢ successively on w

••

,

and therefore on all oflJ), if only these values have been determined on lJ)o = Z beforehand. Moreover, if ¢(k) = 0 for k E Z2N-l to begin with, then automatically ¢(t) = 0 for all t E lJ)
¢(k) := 0

(k

Z\J)

E

is in agreement with (1). Therefore, we are left with the system of homogeneous equations ¢(j) = v'2 2:k hk ¢(2j - k), or, equivalently, 2N-l

¢(j)

= v'2

L

h 2j -k ¢(k)

(0 :::; j :::; 2N - 1) ,

(2)

k=O

for the vector (¢(j) Ij E J) =: a. This means that the (J x J)-Matrix Bjk

:=

v'2 h2j- k

((j, k) E J x J)

should have an eigenvector a corresponding to the eigenvalue 1. In this regard we shall prove the following:

178

6 Orthonormal wavelets with compact support

(6.9) The matrix B has 1 as an eigenvalue in any case. If this eigenvalue is simple, then there is exactly one corresponding eigenvector a such that

L kEJ ak=l. r-

(3)

As an illustration of this theorem we show here the matrix B in the case

N=3:

B

v'2

ho 0 0 0 0 0 h2 hI ho 0 0 0 h4 h3 h2 hi ho 0 0 h5 h4 h3 h2 hI 0 0 0 h5 h4 h3 0 0 0 0 h5 0

(4)

For the proof we argue about the column sums of B. To this end we consider again the generating function H, as given in 5.3.(3). Because of

we have, in addition to (5.5), the equation

so that the following is true:

A glance at (4) shows that the matrix B has (at least in the case N = 3) constant column sums 1. Of course, this is true in general:

(k even) (k odd) and it is easy to verify that for each k E J the sum extends over all h2l =f. 0 resp. all h2l+1 =f. O. What we have found can be expressed in other words as follows: The vector e := (Iii E J) is an eigenvector of the matrix B', corresponding to the eigenvalue 1. Such being the case, the matrix B has 1 as an eigenvalue as well, and there is a corresponding eigenvector a =f. o. For the proof of the second part of the theorem, we note the following: By general prinCiples (see [6], §58, Theorem 1), our space X is the direct sum

179

6.3 Binary interpolation

of two B-invariant subspaces U and V such that B - Ix is nilpotent on U and invertible on V. Therefore the characteristic polynomial q(>..) of B can be decomposed as q(>..) = (>.. - l)m q1(>"), where m := dim(U). Now by assumption on q(.) we have m = 1; therefore U = < a> and dim(V) = dim(X) -1. To any y E V there is an x E V such that y = Bx - x, and from this we conclude that

(e,y) = (e,Bx) - (e,x) = (B'e,x) - (e,x)

=0.

This proves V c < e >1., by counting dimensions we therefore have V <e>1.. Because a tj. V, this implies

~

~kEJ

=

= (e, a) # 0 ,

ak

which is enough to show that the sum on the left can be normalized to 1.

-1

Condition (3), resp. I:kEJ 4>(k) = 1, does not come out of the blue. As a matter of fact, one ha~the following theorem (cf. (6.1)):

(6.10) Suppose that the generating function H is as in Theorem (6.1) and that if; E L2 is defined by the infinite product 6.1.(2). If 4> is in reality a continuous function, satisfying an estimate of the form

c

14>(t)1 ~ 1 + t2

(t E JR.) ,

then the following identity holds: 1

L4>(x-k)

(x E JR.) •

(5)

k

I

By assumption on 4> the auxiliary function

g(x) := L 4>(x - k) k

is a continuous periodic function of period 1 and has Fourier coefficients Cj

=

=

t

Jo

g(x) e-2j1rix dx

J

4>(x) e-2j1rix dx

=L k

[1 4>(x - k) e- 2j1ri (x-k) dx

Jo

= v2-irif;(2j1f) = 80j

(j

E

Z) ,

6 Orthonormal wavelets with compact support

180

where in the end we have used 6.1.(15). From this it follows that 9 has the constant value 1, as stated. .-J For N ~ 2 the Daubechies scaling functions N