LOOP GROUPS, CONJUGATE QUADRATURE FILTERS AND WAVELET APPROXIMATION Wayne Lawton Department of Mathematics National University of Singapore S ,

CQF’s AND ORTHONORMAL WAVELETS Scaling Function Filter Wavelet Function Dual Condition 1 Condition 2 and (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), S. Darlington, Synthesis of reactance four-poles, J. Mathematical Physics, 18(1939), 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), 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 Exact reconstruction techniques for tree structured subbandcoders, IEEE Transactions on Acoustics, ASSP-34(1986),

HISTORY: ORTHONORMAL WAVELETS Scaling Function Filters Wavelet A. Haar, Zur Theorie der orthogonalen Funktionenesysteme, Mathematische Annallen, 69(1910), Application: Used in early days ( s ?) 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, , Wadsworth, Belmont, Ca., 1983 Multiresolution Analysis  Scaling  CQF Spline 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),

FOURIER TRANSFORMS Scaling Equation Fourier Transform CQF Conditions Fourier Transform of Dual

REGULARITY AND SMOOTHNESS where Definition: A CQF c has regularity n > 0 if c has regularity n

OBJECTIVE there exists a finitely supported CQF whose Fourier transform has the form Theorem 1 then for every If a CQF c with regularity n has Fourier transform where 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 onand

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, 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 APPROXIMATION W. Lawton and D. Pollen, Group structures and invariant metrics for quadrature mirror filters and their Aware angular parameterization, Aware Inc. Tech. Report, Cambridge, Massachusetts, W. Lawton, Approximating wavelet conjugate quadrature filters using spectral factorization and lattice decomposition, Aware Inc. Tech. Report, Cambridge, Massachusetts, 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) W.Lawton,Necessary and sufficient conditions for constructing orthonormal wavelet bases". Journal of Mathematical Physics, 32#1(1991) W.Lawton,Multilevel properties of the wavelet- Galerkin operator, Journal of Mathematical Physics, 32# 6(1991)

HISTORY: CQF APPROXIMATION W. Lawton, “Conjugate quadrature filters",pages in Advances in Wavelets, Ka-Sing Lau (ed.),Springer-Verlag,Singapore,1999. A. Pressley and G. Segal, Loop Groups, Oxford University Press, New York, 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 DESIGN W. Lawton and C. A. Micchelli, "Construction of conjugate quadrature filters with specified zeros", Numerical Algorithms, 14,#4(1997), 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 then for every If a CQF c with regularity n has Fourier transform where Corollary If H has minimal phase (outer function) and P is chosen to have minimal phase then P approximates H P has NO zeros onand

MODULUS Construct Laurent polynomials 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 Construct functions in C(T)

MODULUS Construct (Weierstrass) Laurent Polynomial W Assertion 1. Assertion 2. Assertion 3. Construct (Fejer-Riesz) Laurent Polynomial P

PROOF OF THREE ASSERTIONS Herrmann, Daubechies showed that is an interpolatory filter (satisfies Bezout Identity) then so does Triangle inequality yields since Triangle inequality yields

LIE GROUPS Special Unitary Lie Group Exponential Map Lie Algebra

LOOP GROUPS pointwise multiplication Measurable loops parameterize equivalence classes of representations of Cuntz algebras We consider only the group of continuous loops Lie algebra is not onto, and sub-group, algebra, Laurent polynomials since

TROTTERS FORMULA Proposition 2. If andIf Proof. Trotter’s formula is standard, the extension follows after some computation using Leibnitz’s derivative formula for products Have continuous q-th derivatives andthen

SPECIAL LOOPS Define satisfy the following special loops Proposition 3. in

DENSITY Proposition 4. proposition 2 (Trotter) implies that Proof Clearly special matrices spanhence The result follows since Weierstrass approximation theorem implies that neighborhood of the identity contains an open in

LOOP REPRESENTATION OF A CQF We identify a CQF c with its Fourier transform and letdenote the Fourier transform of of the even, odd subsequences of c and define by Remark where is the polyphase representation of C Pollen (twisted) product

STABILIZER SUBGROUPS Proposition 5. Define Proof. Direct computation and both these subsets are subgroups of and

PHASE MAPS Proposition 6.Define the phase map then if Proof. Direct computation loops with and then if and only if are diagonal

APPROXIMATE PHASE Proposition 7. Proof. Let with Clearly and result follows by approximating by real linear combinations of special loops that exp maps into 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 construct whose modulus approximates

REGULAR SUBGROUPS DefinitionFor an integer n > 0 Proposition 8. Eachis a subgroup of and Proof. Direct computation

RESTORE REGULARITY Proposition 9. Regularity preserving CQF approximation is possible First modify the in proposition 7 by adding a linear combination of special loops chosen so 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