WAVELET ANALYSIS and its APPLICATIONS

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WAVELET ANALYSIS and its APPLICATIONS

• Wayne M. Lawton
• Department of Mathematics
• National University of Singapore
• 2 Science Drive 2
• Singapore 117543

Email wlawton_at_math.nus.sg Tel [phone] Fax
(65) 779-5452
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SEQUENCES
group of integers
field of real, complex numbers
vector space (over the real or complex numbers)
vector space of sequences
translation operator
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LINEAR OPERATORS
linear operator
is an algebra (related vector space and ring
structures)
extension defined by
embedding
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CONVOLUTION
sub-algebra generated by
Theorem 1.
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CONVOLUTION
Laurent polynomials
Convolution Operators Matrices over Laurent
Polynomials
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CONVOLUTION
Theorem 2.
is a unit (invertible) if and only if
is a unit, that is
Proof.
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CONVOLUTION
Theorem 3.
A row vector
can be extended to form a unit matrix if and only
if its entries
are relatively prime.
This is equivalent to either of the following two
conditions
The entries have no common nonzero complex roots
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MULTIRATE FILTERING
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FREQUENCY SEPARATION WITH PERFECT RECONSTRUCTION
Theorem 4.
is a low pass filter and
If
are relatively prime, then there exists
a high pass filter
such that
is a unit.
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PARAUNITARY MATRICES and CONJUGATE QUADRATURE
FILTERS
is a paraunitary matrix if
is a conjugate quadrature filter if
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PARAUNITARY MATRICES and CONJUGATE QUADRATURE
FILTERS
Theorem 5.
is a CQF
if and only if
is paraunitary.
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FILTER BANKS AND WAVELETS
Theorem 4 describes general biorthogonal wavelets
Theorem 5 describes general orthogonal wavelets,
such as those that were invented by Ingrid
Daubechies in 1988
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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.
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APPLICATIONS
The small support and vanishing moment properties
are useful for image compression.
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.
The adaptive properties of wavelets are useful
for detecting and classifying transient signals..
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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.
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BOOKS WAVELETS
Y. Meyer and R. Coifman, Wavelets
Calderon-Zygmund and multilinear operators,
Cambridge UP, 1990.
Y. Meyer, Wavelets and Operators, Cambridge UP,
1992.
I. Daubechies, Ten Lectures on Wavelets, SIAM,
1992.
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.
P. Wojtaszczyk, A Mathematical Introduction to
Wavelets, Cambridge Univ. Press, 1997.
S. Mallat, A Wavelet Tour of Signal Processing,
Academic Press, Boston, 1998.
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PAPERS WAVELETS
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 Rn as
unconditional bases for Hardy spaces, in Conf. In
Harmonic Analysis in Honor of A. Zygmund, vol.
II, 1983.
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.
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PAPERS WAVELETS
D. Pollen, SU(2,Fz,1/z) for F a subfield of C,
Journal of the American Mathematical Society,
3(1990) 611.
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.
G. Beylkin, R. Coifman, and V. Rokhlin, Fast
wavelet transforms and numerical algorithms,
Comm. Pure Appl. Math., 44(1991), 141-183.
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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.
J. Kautsky and Radka Turcajova, Discrete
biorthogonal wavelet transforms as block
circulant matrices, Linear Algebra and its
Applications, 224(1995) 393-413.
S. Suvorova, Applications of the Wavelet
Transform to Two Dimensional X-Ray Tomography,
PhD Dissertation, The Flinders University of
South Australia, October 1996
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PAPERS WAVELETS
W. Lawton, S. L. Lee, and Z. Shen, Convergence of
multidimensional cascade algorithm, Numerische
Mathematik 78(1998), 427-438.
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.
W. Lawton and C. A. Micchelli, Bezout identities
with inequality constraints, Vietnam Journal of
Mathematics 28(2000), 1-29.
W. Lawton, Infinite convolution products and
refinable distributions on Lie groups,
Transactions of the American Mathematical Society
352(2000), 2913-2936.