LOOP GROUPS, CONJUGATE QUADRATURE FILTERS AND WAVELET APPROXIMATION

Wayne Lawton

Department of Mathematics

National University of Singapore

S14-04-04, matwml@nus.edu.sg

http://math.nus.edu.sg/~matwml

CQF’s AND ORTHONORMAL WAVELETS

Rxkxkcxk

),2()(2)( Scaling Function

Znnknckck

),()2()( 21

Filter CZ: c

O.N.}:)({ Zkkx

k

kcdxx 1)()2 CQF(1)(

Rxkxkdxk

),2()(2)( Wavelet Function

Dual Zkkdkd k ,)1(1)(

Condition 1

Condition 2and

)R(for OB is ,:)2(2 22/ LZnmnxmm

(CQF 1)

HISTORY: LOSSLESS ANALOGUE CQF’s

O. Brune, Synthesis of finite two terminal network whose driving point impedance is prescribed function of frequency, J. Mathematics and Physics, 10(1931),191-235.

S. Darlington, Synthesis of reactance four-poles, J. Mathematical Physics, 18(1939), 257-353.

Application: Analogue filterbanks constructed from LC-circuits (inductors, capacitors) preserve power and were vital for early radio receivers – they correspond to digital IIR (recursive) CQF’s

HISTORY: DIGITAL CQF’s

O.Herrmann, On the approximation problem in nonrecursive digital filter design, IEEE Trans- actions in Circuit Theory, CT-18(1971), 411-413.

M.J. Smith and T.P.Barnwell, A procedure for designing exact reconstruction filter banks for tree structured subband coders, Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing, San Diego, March 1986.

Exact reconstruction techniques for tree structured subbandcoders, IEEE Transactions

on Acoustics, ASSP-34(1986),434-441.

HISTORY: ORTHONORMAL WAVELETS

Scaling Function

Filters 21

21

21 )1(,)0(,)1()0( ddcc

)1,0[

Wavelet

A. Haar, Zur Theorie der orthogonalen Funktionenesysteme, Mathematische Annallen, 69(1910), 331-371.

)1,[),0[ 21

21

Application: Used in early days (1940-1950s ?)

at the Jet Propulsion Lab to compress video data collected by unmanned aircraft (drones)

HISTORY: ORTHONORMAL WAVELETS

J. O. Stromberg, A modified Franklin system and higher-order spline systems on R^n as unconditional bases for Hardy spaces, Conf. in Harmonic Analysis in Honor of Antoni Zygmund, II, 475-493, Wadsworth, Belmont, Ca., 1983

Multiresolution Analysis Scaling CQFSpline MA: Stromberg, Battle-Lemarie

Fourier MA: Paley-Littlewood, Shannon

CQF Scaling Multiresolution Analysis I. Daubechies, Orthonormal bases of compactly supported wavelets, Communications on Pure and Applied Mathematics, 41(1988), 909-986

FOURIER TRANSFORMS

Scaling Equation

R,)()0(ˆ)()(ˆ1

2/ yeCdxexy

k

iyixy k

T,)()( zzkczCk

kFourier Transform

1|)(||)(| 22 zCzC

CQF Conditions

1|)0(| C

)()( zzDzD Fourier Transform of Dual

REGULARITY AND SMOOTHNESS

where

R,)(sinc)(ˆ1

2/2

n2/ yeHey

k

iyyiy k

Definition: A CQF c has regularity n > 0 if

)(RC n

2/)1()( zzU ,HUC n

c has regularity n

nkdxxx k 0,0)( c has regularity n

1

0

2/2log

1 |)(|suplim1j

k

iyyjj

k

eHn)R( C

OBJECTIVE

there exists a finitely supported CQF whose Fourier transform has the form

Theorem 1

PU n

,HUC n then for every 0If a CQF c with regularity n has

Fourier transform

where

T:|)()(|max|||| zzPzHPH

Corollary Infinite supported and non-orthonormal tight frame scaling functions and wavelets can be ‘nicely’ approximated by compactly supported orthonormal scaling functions and wavelets.

P has NO zeros on T and

HISTORY: CQF PARAMETERIZATION

D. Pollen, The unique factorization for the topological group of coefficient vectors for one-dimensional, multiplier-two scaling function, wavelet systems, Aware Inc. Technical Report, Cambridge, Massachusetts, 1988.

D. Pollen, SU(2,F[z,1/z]) for F a subfield of C, J. American Mathematical Society, 3(1990),611.

Represents a CQF by a loop in the unit quaternion group (=SU(2)), lattice factorization

Enabled efficient implementation of long chaotic DWT filters – netted $3 million

HISTORY: CQF APPROXIMATIONW. Lawton and D. Pollen, Group structures and invariant metrics for quadrature mirror filters and their Aware angular parameterization, Aware Inc. Tech. Report, Cambridge, Massachusetts, 1988.

W. Lawton, Approximating wavelet conjugate quadrature filters using spectral factorization and lattice decomposition, Aware Inc. Tech. Report, Cambridge, Massachusetts, 1988.

Greedy algorithm, peals off each lattice factor, fails to converge, loses regularity.

Spectral factorizes convolution of a Fejer Kernel with |C|^2, loses phase and regularity.

HISTORY: CQF-WAVELET RELATIONS

W.Lawton,Tight frames of compactly supported affine wavelets,Journal of Mathematical Physics, 31# 8(1990)1898-1901.

W.Lawton,Necessary and sufficient conditions for constructing orthonormal wavelet bases". Journal of Mathematical Physics, 32#1(1991)57-61.

W.Lawton,Multilevel properties of the wavelet-Galerkin operator, Journal of Mathematical Physics, 32# 6(1991)1440-1443.

HISTORY: CQF APPROXIMATIONW. Lawton, “Conjugate quadrature filters",pages 103-119 in Advances in Wavelets, Ka-Sing Lau (ed.),Springer-Verlag,Singapore,1999.

A. Pressley and G. Segal, Loop Groups, Oxford University Press, New York, 1986.

Proved that polynomial loops are dense in loop group in SU(2) by first approximating modulus, then phase. Loses regularity.

Density of loop groups already proved

– but by using Trotter’s Approximation.

HISTORY: CQF DESIGNW. Lawton and C. A. Micchelli, "Construction of conjugate quadrature filters with specified zeros", Numerical Algorithms, 14,#4(1997), 383-399

W. Lawton and C. A. Micchelli, "Bezout identities with inequality constraints",Vietnam Journal of Mathematics, 28#2(2000),1-29.

Uses Weierstrass and Bezout and Spectral Factorization to construct dim=1 CQF’s whose zeros include a specified set of zeros

.Use a matrix method that refines dim > 1 result and enables use of Quillen-Suslin theorem to design interpolatory filters for dim > 1.

STRATEGY

Modify the Bezout Identity / matrix method

used in the second Lawton-Micchelli paper

Approximate Modulus & Preserve Regularity

Use a loop group method that combines both

the Lawton & the Pressley-Segal methods

Approximate Phase & ‘Slightly Lose’ Regularity

Restore the Regularity

Uses nonlinear-perturbation – jets, implicit function theorem, topological degree

MODULUS

there exists a finitely supported CQF whose Fourier transform has the form

Proposition 1

PU n

,HUC n then for every 0If a CQF c with regularity n has

Fourier transform

where

|||||||| 22 PH

Corollary If H has minimal phase (outer function) and P is chosen to have minimal phase then P approximates H

P has NO zeros on T and

MODULUS

)()(,4

21)( 12

1

01 zVzVzz

k

knzV

kn

k

Construct Laurent polynomials)()(,||)( 12

21 zUzUUzU n

to construct $W \in

\label{EW_approx}

||E - W|| < \delta.

\end{equation}

%

Since $E$ is real-valued and satisfies $E(-z) = -E(z), \ \ z \in \T$ we can choose

$W$ to be real-valued and satisfy $W(-z) = -W(z), \ \ z \in \T.$ We construct

Laurent polynomials

%

\begin{equation}

\label{W1}

W_1 := V_1 - U_2 \, W

\end{equation}

%

$$W_2(z) := W_1(-z), \ \ z \in \T.$$

unique positive root of

1

|||| 2H

Construct functions in C(T)

)()(,||)1( 1212

1 zHzHVHH

R)()(1221 zEzEHVHVE

MODULUS

Construct (Weierstrass) Laurent Polynomial W

R)()(,|||| zWzWWE

Assertion 1.

)()(, 12211 zWzWWUVW

Assertion 2.

Assertion 3.

12211 WUWU

1W

|||||| 12 WH

Construct (Fejer-Riesz) Laurent Polynomial P

12|| WP

PROOF OF THREE ASSERTIONSHerrmann, Daubechies showed that

is an interpolatory filter (satisfies Bezout Identity) 12211 VUVU then so does

11VU

|||| 11 WH

Triangle inequality yields

EH

H

VV

UU 1

2

1

12

21

WUVWzWzWWUVW 12212211 )()(, 11WU since

12211 HUHU

EUVH 211

1|||| 2U 11 1 HV

01 W

Triangle inequality yields

|||||| 12 WH

LIE GROUPSSpecial Unitary Lie Group

C,,

BAAB

BAg

1det,:)2( * gIgggSU

Exponential Map

)2()2(:exp SUsu

Lie Algebra

C,R,

crirc

cirh

0trace,0:)2( * hgghsu

,θ sincIθcosexp hirc

cir

22 ||θ cr

LOOP GROUPS )2(T:L SUa pointwise multiplication

L:exp

Measurable loops parameterize equivalence classes of representations of Cuntz algebras

,exp0

0

z

z

We consider only the group of continuous loops

Lie algebra continuous:)2(T: su

is not onto, and

sub-group, algebra, Laurent polynomialspp ,L

pp Lexp

since

since

,L0

0exp p

zz

zz

TROTTERS FORMULA

Proposition 2. If

and itez

m ,...,1

NmNNN

m

k k 11

11

expexplimexp

If

NmNNq

q

N

m

k kq

q

dt

d

dt

d 11

11

expexplimexp

0q

Proof. Trotter’s formula is standard, the extension follows after some computation

using Leibnitz’s derivative formula for products

m ,...,1 Have continuous

q-th derivatives and then

SPECIAL LOOPS

Define

satisfy

02

20n

n

uzu

uzM

1

RRMzuuzzuuz

zuuzzuuzunnnn

nnnn

Rθ,θsinθcosθexp MIM

the following special loops

Proposition 3.

21

21

21

21

,10

0)( R

zzD

Z},,{ 221 nu i

11 )1()1(

DRRMDzuuzzuuz

zuuzzuuzunnnn

nnnn

inp

DENSITY

Proposition 4.

proposition 2 (Trotter) implies that

pp L closure)exp(

Proof Clearly special matrices span hence

pL closureL

p

The result follows since

pp L closureexp closure Weierstrass approximation theorem implies that

expneighborhood of the identity

contains an openI in L

LOOP REPRESENTATION OF A CQF

C

pL][ C

We identify a CQF c with its Fourier transform and let

T,)()(

)()()]([ 1

z

zCzCz

zzCzCzC

T),()(]][[)()]([ 21 zzDRzCzDzC

oe C,C denote the Fourier transform of

of the even, odd subsequences of c and define

by

Remark

eo

oe

CC

CCC 2]][[where

is the polyphaserepresentation of C

Pollen (twisted) product ]][[]][[]][[ 1 ERCEC

STABILIZER SUBGROUPS

][][:S FFaLa

Proposition 5.

Define

Proof. Direct computation

]F[]F[:LS aar

)()(:S zazaa

111121111 )()(,)()(:S DzazazazaaDr

and both these subsets are subgroups ofL

CQFFLF a of ansformFourier tr theis :F

and

PHASE MAPS

}0)(:{\,||

))(( wawTza

aza

Proposition 6. Define the phase map

then if

Proof. Direct computation

][][ FbCa F, FC

)()( DC Lb,a

loops with

and

then

if and only if

)()(

)()()(2

zCzC

zFzFza

)()(

)()()(2

zCzC

zFzFzb

are diagonal

APPROXIMATE PHASE

Proposition 7.

Proof. Let

, with exp,exp baClearly )()(),()( zzzz and result follows by approximating

,by real linear combinations of special loopsthat exp maps into rS,S

FL closureF p

and, by multiplying by a monomial, makes the winding number of the expressions on the right sides of proposition 6 equal to zero. Compute

Use proposition 1 to constructF.F

pLFC whose modulus approximates .|| F

REGULAR SUBGROUPS

)(),()()(:S 12 TCGzGzUzaLa nn

Definition For an integer n > 0

Proposition 8. Each is a subgroup ofnS

]F[]F[SSa nn an

and

L

Proof. Direct computation

regular is :FF nFFn

]F[]F[SSa nn anr

RESTORE REGULARITYProposition 9. Regularity preserving CQF approximation is possible

First modify the , in proposition 7 by addinga linear combination of special loops chosen so

rS,S

Proof. It suffices to construct n-regular a, b

that exp maps them into

coefficients as vectors in V and first n Taylor coefficients at z=1 of upper right entries of a,b

as a (nonlinear) mapping f of V into V. Jet theory implies f is a local homeomorphism of 0, degree theory implies Trotter approx of f also has a root.

Regard set of