loop groups, conjugate quadrature filters and wavelet approximation wayne lawton department of...
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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
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