MA5242 Wavelets Lecture 3 Discrete Wavelet Transform

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MA5242 Wavelets Lecture 3 Discrete Wavelet
Transform

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

Email matwml_at_nus.edu.sg Tel [phone]
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Riesz Representation
Theorem. If
is a finite dimensional unitary space
there exists an antilinear isomorphism
such that
Proof. Let
be an ONB for
and define
Then
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Adjoint Transformations
Theorem Given unitary spaces
and a linear
there exists a unique
transformation
linear transformation
(adjoint of T)
with
Proof. Define
by composition
let
be the Riesz Rep. transformations and define
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Problem Set 1
1. Assume that
and that
are ONB for unitary spaces
is linear. Derive the relationship
between the matrices
that represent
with respect to these bases.
2. Prove that a transformation
is unitary iff
3. Derive the Riesz Representation, Adjoint and
matrix representations, and characterization for
orthogonal transformations for Euclidean spaces.
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General Discrete Wavelet Transform
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Convolution Representation
where a,b,c,dare infinite sequences that extend
the finite sequences
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Orthogonality Conditions
Theorem. The wavelet transform matrix is unitary
iff
for all
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Laurent Polynomials
Definition A Laurent polynomial is a function
that admits a representation
where c is a finitely supported sequence.
Definition For a sequence c let
and define the unit circle
Theorem For seq. a, b,
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Conjugate Quadrature Filters
Definition A sequence c that satisfies the
quadratic equations necessary for a wavelet
transform matrix to me unitary is called a
Conjugate Quadrature Filter
Theorem. A sequence c is a CQF iff it satisfies
Theorem Prove that if c is a CQF and if d is
related to c by the equation on the previous page
then d is also a CQF and the WT is unitary
Theorem If c is a CQF then the WT is unitary if
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Problem Set 2
1. Derive the conditions for a WT to be unitary.
2. Prove the theorems about Laurent polynomials
and the two theorems on the preceding page.
3. Prove that c, d form a unitary WT iff
(the modulation matrix) is unitary for all
4. Prove that d on the previous page is the same
as
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Moment Conditions
Definition. d has -1lt p vanishing moments if
Theorem. If c,d gives a unitary WT then d has -1lt
p vanishing moments
has a factor
iff
has a factor
iff
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Moment Consequences
Theorem. If d has -1lt p vanishing moments and
is supported on the set 0,1,,2N-1 then
if the finite sequence d(k),d(k-1),,d(k-2N1)
can be represented by a polynomial having degree
lt N
Proof.
by the binomial theorem and vanishing moments.
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Riesz-Fejer Spectral Factorization
Theorem. A Laurent polynomial N is
on
iff there exists a LP P such that
Proof. Let
be the set of roots of
where
is the multiplicity of
Since N is real-valued
furthermore, since N is non-negative the are
even hence paired, now choose P to contain one
root from each pair and the result easily follows.
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Daubechies Wavelets
Theorem. If c is a CQF supported on 0,1,,2N-1
and
then
has a factor
satisfies and is uniquely determined by the
equations
Furthermore,
and c can be chosen by the R.-F. Theorem.
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Problem Set 3
1. Prove all of the Theorems after Problem Set 2.