WAVELET ANALYSIS and its APPLICATIONS

Wayne M. Lawton

Department of Mathematics

National University of Singapore

2 Science Drive 2

Singapore 117543Email wlawton@math.nus.sg

Tel (65) 874-2749Fax (65) 779-5452

SEQUENCES

vector space (over the real or complex numbers)

group of integers

C,RZ

Vfield of real, complex numbers

VZ:a )Z(V vector space of sequences

)Z(V)Z(V:T translation operator

Zj),Z(Va,a)Ta( 1jj

LINEAR OPERATORS

embedding

linear operator

dxddxddd C,RMC,RV

)V,V(Lin VV:M

)Z(V)Z(V:M~ extension defined by

Zj),Z(Va,MaaM~

jj

is an algebra (related vector space and ring structures)

))Z(V),Z(V(Lin)V,V(Lin

CONVOLUTION

))Z(V),Z(V(Lin))Z(V),Z(V(Conv

Theorem 1.

)V,V(LinP,TPP k

2Nk

1Nk

kk

sub-algebra generated by1T,T),V,V(Lin

))Z(V),Z(V(ConvP

TPPTand))Z(V),Z(V(LinP

CONVOLUTION

k kjkj aP)Pa(

Laurent polynomials

dd1dd ])T,T[C())Z(C),Z(C(Conv

]T,T[C 1

k kjkj QP)PQ(

Convolution Operators = Matrices over Laurent Polynomials

CONVOLUTION

Theorem 2.

is a unit (invertible) if and only if

dd1])T,T[C(P

]T,T[CPdet 1 is a unit, that is

Zk,}0{\C,TPdet k

Proof. I)P(det)Pcofactor(P

CONVOLUTION

Theorem 3.

can be extended to form a unit matrix if and only if its entries

d11,*1 ])T,T[C(P A row vector

]PPP[P d,12,11,1,*1 are relatively prime.

This is equivalent to either of the following two conditions:

]QQQ[Q d,12,11,1,*1

1QP)IdentityBezout(k k,1k,1

The entries have no common nonzero complex roots

MULTIRATE FILTERING

)Z(2Coaea

)Z(Ca

oaea

eGoGoTFeF

o)Ga(e)Fa(

FREQUENCY SEPARATION WITH PERFECT RECONSTRUCTION

)T(H)T1()T(F N is a low pass filter and

)T(H),T(H are relatively prime, then there exists

)T(K)T1()T(G N

If

a high pass filter

such that

eGoGoTFeF

is a unit.

Theorem 4.

PARAUNITARY MATRICES and CONJUGATE QUADRATURE FILTERS

is a paraunitary matrix ifdd1])T,T[C(P

R),d,C(SU)ie(P

]T,T[CQ 1 is a conjugate quadrature filter if

R,1|)ie(Q||)ie(Q| 22

PARAUNITARY MATRICES and CONJUGATE QUADRATURE FILTERS

]T,T[CQ 1Theorem 5.

)T(eQ)T(oQT

)T(oTQ)T(eQ

2

1)T(P 111

is paraunitary.

is a CQF

if and only if

FILTER BANKS AND WAVELETS

n )nx2(nQ)x(

Theorem 4 describes general biorthogonal wavelets

Theorem 5 describes general orthogonal wavelets, suchas those that were invented by Ingrid Daubechies in 1988

n )nx2(nQn)1()x(

}Zj,L:)jx2({ L )R(Lforbasislorthonormaanis 2

MULTILEVEL FILTERING

The original sequence is convolved with lowpass and highpass filters and subsequently downsaampled to form low and high frequency bands.

The same process is repeated to the low frequency bands in a recursive manner to obtain bands that are approximately one octave wide and whose centers are one octave apart.

For two-dimensional data, the pair of lowpass and highpass filters are applied in both a horizontal and vertical direction before recursively applying the same process to the low frequency band that is obtained.

APPLICATIONS

The small support and vanishing moment properties are useful for image compression.

The adaptive properties of wavelets are useful for detecting and classifying transient signals..

The small support and approximate octave frequency wavelet decomposition is useful for efficiently approximating Calderon-Zygmund operators. This is useful for tomographic reconstruction and for solving elliptic differential equations, either by sparsely representing boundary integral operators or by preconditioning differential operators.

BOOKS – FILTERBANKS

M. Vidyasagar, Control Theory Synthesis, A Factorization Approach, MIT Press, Cambridge, 1985.

P. P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice-Hall, New Jersey, 1993.

M. Vetterly and J. Kovacevic, Wavelets and Subband Coding, Prentice-Hall, NJ, 1995.

G. Strang and T. Nguyen, Wavelets and Filterbanks, Wellesley-Cambridge Press, MA, 1996.

BOOKS – WAVELETS

G. Kaiser, A Friendly Guide to Wavelets, Birkhauser, 1994.

C. S. Burrus, R. A. Gopinath and H. Guo, Introduction to Wavelets and the Wavelet Transform, Prentice-Hall, 1996.

I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992.

P. Wojtaszczyk, A Mathematical Introduction to Wavelets, Cambridge Univ. Press, 1997.

S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, Boston, 1998.

Y. Meyer, Wavelets and Operators, Cambridge UP, 1992.

Y. Meyer and R. Coifman, Wavelets Calderon-Zygmund and multilinear operators, Cambridge UP, 1990.

PAPERS – WAVELETS

I. Daubechies, Orthonormal bases of compatly supported wavelets, Comm. Pure Appl. Math., 41(1988), 909-996.

R. Glowinski, W. Lawton, M. Ravachol, and E. Tenenbaum, Wavelet solution of linear and nonlinear elliptic, parabolic, and hyperbolic problems in one space variable, Proc. 9th Intern. Conf. Computing Methods in Appl. Sci. & Engineering, Paris, France, January 1990.

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

J. O. Stromberg, A modified Franklin system and higher order spline systems on R^n as unconditional bases for Hardy spaces, in Conf. In Harmonic Analysis in Honor of A. Zygmund, vol. II, 1983.

PAPERS – WAVELETS

G. Beylkin, R. Coifman, and V. Rokhlin, Fast wavelet transforms and numerical algorithms, Comm. Pure Appl. Math., 44(1991), 141-183.

W. Lawton, Tight frames of compactly supported wavelets, J. Math. Physics, 31(1990) 1898-1901. Necessary and sufficient conditions for constructing orthonormal wavelets, J. Math. Physics, 32(1991) 57-61. W. Lawton and H. Resnikoff, Multidimensional wavelet bases, Aware Technical Report, 1991.A. S. Cavaretta, W. Dahmen and C. A. Micchelli, Stationary Subdivision, Memoirs of AMS, 93(453), 1991.

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

PAPERS – WAVELETS

A. Cohen, I. Daubechies, and J. C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math., 45(1992), 485-560.

W. Dahmen and A. Kunoth, Multilevel preconditioning, Numerische Mathematik 63(1992), 315-344.

S. Suvorova, Applications of the Wavelet Transform to Two Dimensional X-Ray Tomography, PhD Dissertation, The Flinders University of South Australia, October 1996

J. Kautsky and Radka Turcajova, Discrete biorthogonal wavelet transforms as block circulant matrices, Linear Algebra and its Applications, 224(1995) 393-413.

PAPERS – WAVELETSW. Lawton, S. L. Lee, and Z. Shen, Convergence of multidimensional cascade algorithm, Numerische Mathematik 78(1998), 427-438.

W. Lawton, Infinite convolution products and refinable distributions on Lie groups, Transactions of the American Mathematical Society 352(2000), 2913-2936.

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

B. Liu and S. F. Ling, On the selection of informative wavelets for machinery diagnostics, Journal of Mechanical Systems and Signal Processing, 13(1999) 145-162.