Algorithms of Wavelets and Framelets Bin Han Department of Mathematical and Statistical Sciences University of Alberta, Canada Present at Summer School on Applied and Computational Harmonic Analysis Edmonton, Canada July 29–31, 2011 Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 1 / 64
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Algorithms of Wavelets and Framelets
Bin Han
Department of Mathematical and Statistical SciencesUniversity of Alberta, Canada
Present at Summer School onApplied and Computational Harmonic Analysis
Edmonton, Canada
July 29–31, 2011
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 1 / 64
Outline of Tutorial: Part I
Algorithms for discrete wavelet/framelet transform
Perfect reconstruction, sparsity, stability
Multi-level fast wavelet transform
Oblique extension principle
Framelet transform for signals on bounded intervals
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 2 / 64
Notation
l(Z) for signals: all v = {v(k)}k∈Z : Z → C.l0(Z) for filters: all finitely supported sequencesu = {u(k)}k∈Z : Z → C on Z.For v = {v(k)}k∈Z ∈ l(Z), define
v ⋆(k) := v(−k), k ∈ Z,
v(ξ) :=∑
k∈Z
v(k)e−ikξ, ξ ∈ R.
Convolution u ∗ v and inner product:
[u ∗ v ](n) :=∑
k∈Z
u(k)v(n − k), n ∈ Z,
〈v , w〉 :=∑
k∈Z
v(k)w(k), v , w ∈ l2(Z)
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 3 / 64
Subdivision and Transition Operators
The subdivision operator Su : l(Z) → l(Z):
[Suv ](n) := 2∑
k∈Z
v(k)u(n − 2k), n ∈ Z
The transition operator Tu : l(Z) → l(Z) is
[Tuv ](n) := 2∑
k∈Z
v(k)u(k − 2n), n ∈ Z.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 4 / 64
Subdivision and Transition Operators
The upsampling operator ↑d : l(Z) → l(Z):
[v ↑d](n) :=
{v(n/d), if n/d is an integer;
0, otherwise,n ∈ Z
the downsampling (or decimation) operator↓d : l(Z) → l(Z):
[v ↓d](n) := v(dn), n ∈ Z.
Subdivision and transition operators:
Suv = 2u ∗ (v ↑2) and Tuv = 2(u⋆ ∗ v)↓2.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 5 / 64
Discrete Framelet Transform (DFrT): Decomposition
Let u0, . . . , us ∈ l0(Z) be filters for decomposition.
For data v ∈ l(Z), a 1-level framelet decomposition:
wℓ :=√
22 Tuℓ
v , ℓ = 0, . . . , s,
where wℓ are called framelet coefficients.
Grouping together, a framelet decompositionoperator W : l(Z) → (l(Z))1×(s+1):
Wv :=√
22 (Tu0
v , . . . , Tusv), v ∈ l(Z).
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 6 / 64
Reconstruction
Let u0, . . . , us ∈ l0(Z) be filters for reconstruction.
A one-level framelet reconstruction byV : (l(Z))1×(s+1) → l(Z):
V(w0, . . . , ws) :=
√2
2
s∑
ℓ=0
Suℓwℓ, w0, . . . , ws ∈ l(Z).
prefect reconstruction: VWv = v for any data v .
A filter bank ({u0, . . . , us}, {u0, . . . us}) has theperfect reconstruction (PR) if VW = Id l(Z).
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 7 / 64
Perfect Reconstruction (PR) Property
TheoremA filter bank ({u0, . . . , us}, {u0, . . . , us}) has the perfectreconstruction property, that is,
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 11 / 64
Role of√
22
in DFrT
TheoremLet u0, . . . , us ∈ l0(Z). Then TFAE:
(i) ‖Wv‖2(l2(Z))1×(s+1) = ‖v‖2
l2(Z) for all v ∈ l2(Z), that is,
‖Tu0v‖2
l2(Z)+· · ·+‖Tusv‖2
l2(Z) = 2‖v‖2l2(Z), ∀ v ∈ l2(Z);
(ii) 〈Wv ,W v〉 = 〈v , v〉 for all v , v ∈ l2(Z);
(iii) the filter bank ({u0, . . . , us}, {u0, . . . , us}) has PR:[
u0(ξ) · · · us(ξ)u0(ξ + π) · · · us(ξ + π)
] [u0(ξ) · · · us(ξ)
u0(ξ + π) · · · us(ξ + π)
]⋆
= I2,
{u0, . . . , us} with PR is called a tight framelet filter bank.Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 12 / 64
Orthogonal Wavelet Filter Bank
A tight framelet filter bank with s = 1 is called anorthogonal wavelet filter bank.
PropositionLet {u0, . . . , us} be a tight framelet filter bank. Then thefollowing are equivalent:
1 W is an onto orthogonal mapping satisfying〈Wv ,W v〉 = 〈v , v〉 for all v , v ∈ l2(Z);
2 for all w0, . . . , ws , w0, . . . , ws ∈ l2(Z),
{u0, u1} is the Haar orthogonal wavelet filter bank:
u0 = {12, 1
2}[0,1], u1 = {12,−1
2}[0,1]. (2)
({u0, u1}, {u0, u1}) is a biorthogonal wavelet filterbank, where
u0 = {−18, 1
4, 3
4, 1
4,−1
8}[−2,2], u1 = {1
4,−1
2, 1
4}[0,2],
u0 = {14, 1
2, 1
4}[−1,1], u1 = {1
8, 1
4,−3
4, 1
4, 1
8}[−1,3].
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 14 / 64
Illustration: I
Apply the Haar orthogonal filter bank to
v = {1, 0,−1,−1,−4, 60, 58, 56}[0,7]
Note that
[Tu0v ](n) = v(2n)+v(2n+1), [Tu1
v ](n) = v(2n)−v(2n+1), n ∈ Z.
We have the wavelet coefficients:
w0 =√
22{1,−2, 56, 114}[0,3], w1 =
√2
2{1, 0,−64,−2}[0,3].
Note that
[Su0w0](2n) = w0(n), [Su0
w0](2n + 1) = w0(n), n ∈ Z
[Su1w1](2n) = w1(n), [Su1
w1](2n + 1) = −w1(n), n ∈ Z.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 15 / 64
Illustration: II
Hence, we have√
22 Su0
w0 = 12{1, 1,−2,−2, 56, 56, 114, 114}[0,7],√
22 Su1
w1 = 12{1,−1, 0, 0,−64, 64,−2, 2}[0,7].
Clearly, we have the perfect reconstruction of v :
√2
2 Su0w0+
√2
2 Su1w1 = {1, 0,−1,−1,−4, 60, 58, 56}[0,7] = v
and the following energy-preserving identity
‖w0‖2l2(Z) + ‖w1‖2
l2(Z) = 161372
+ 41012
= 10119 = ‖v‖2l2(Z).
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 16 / 64
Diagram of 1-level DFrTs
input
√2u⋆
0
√2u⋆
1
√2u⋆
s
↓2
↓2
↓2
processing
processing
processing
↑2
↑2
↑2
√2u0
√2u0
√2u0
⊕ output
Figure: Diagram of a one-level discrete framelet transform using adual framelet filter bank ({u0, . . . , us}, (u0, . . . , us}).
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 17 / 64
Sparsity of DFrT
One key feature of DFrT is its sparse representationfor smooth or piecewise smooth signals.
It is desirable to have as many as possible negligibleframelet coefficients for smooth signals.
Smooth signals are modeled by polynomials. Letp : R → C be a polynomial: p(x) =
∑mn=0 pnx
n.
a polynomial sequence p|Z : Z → C by[p|Z](k) = p(k), k ∈ Z.
N0 := N ∪ {0}.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 18 / 64
Polynomial Differentiation Operator
Polynomial differentiation operator:
p(x − i d
dξ
)f(ξ) :=
∞∑
n=0
pn
(x − i d
dξ
)nf(ξ).
p(x − i d
dξ
)f(ξ) =
∞∑
n=0
(−i)n
n!p(n)(x)f(n)(ξ)
=
∞∑
n=0
xn
n!p(n)
(−i d
dξ
)f(ξ).
Generalized product rule for differentiation:
p(x − i d
dξ
)(g(ξ)f(ξ)
)=
∞∑
n=0
(−i)n
n!g(n)(ξ)p(n)
(x − i d
dξ
)f(ξ).
[p(−i d
dξ
)(e ixξf(ξ))
]∣∣∣ξ=0
=[p(x − i d
dξ
)f(ξ)
]∣∣∣ξ=0
.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 19 / 64
Convolution with Polynomials
LemmaLet u = {u(k)}k∈Z ∈ l0(Z). Then for any polynomialp ∈ Π, p ∗ u is a polynomial with deg(p ∗ u) 6 deg(p),
[p ∗ u](x) =∑
k∈Z
p(x − k)u(k) =[p(x − i d
dξ
)u(ξ)
]∣∣∣ξ=0
=∞∑
n=0
(−i)n
n!p(n)(x)u(n)(0) =
∞∑
n=0
xn
n!
[p(n)
(−i d
dξ
)u(ξ)
]∣∣∣ξ=0
.
Moreover, p ∗ (u ↑2) = [p(2·) ∗ u](2−1·),
p(n)∗u = [p∗u](n), p(·−y)∗u = [p∗u](·−y), ∀ y ∈ R.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 20 / 64
Big O Notation
For smooth functions f and g, it is often convenient touse the following big O notation:
f(ξ) = g(ξ) + O(|ξ − ξ0|m), ξ → ξ0
to mean that the derivatives of f and g at ξ = ξ0 agreeto the orders up to m − 1:
f(n)(ξ0) = g(n)(ξ0), ∀ n = 0, . . . , m − 1.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 21 / 64
Transition Operator Acting on Polynomials
TheoremLet u ∈ l0(Z). Then for any polynomial p ∈ Π,
Tup = 2[p ∗ u⋆](2·) = p(2·) ∗ u =∞∑
n=0
2(−i)n
n!p(n)(2·)u(n)(0),
where u is a finitely supported sequence on Z such thatˆu(ξ) = 2u(ξ/2) + O(|ξ|deg(p)+1), ξ → 0.
In particular, for any integer m ∈ N, TFAE:1 Tup = 0 for all polynomial sequences p ∈ Πm−1;2 Tuq = 0 for some q ∈ Π with deg(q) = m − 1;3 u(ξ) = O(|ξ|m) as ξ → 0;4 u(ξ) = (1 − e−iξ)mQ(ξ) for some 2π-periodic
trigonometric polynomial Q.Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 22 / 64
Vanishing Moments
We say that a filter u has m vanishing moments ifany of items (1)–(4) in Theorem holds.
Most framelet coefficients are zero for any inputsignal which is a polynomial to certain degree.
If u has m vanishing moments. For a signal v , if vagrees with some polynomial of degree less than mon the support of u(· − 2n), then [Tuv ](n) = 0.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 23 / 64
Coset Sequences
For u = {u(k)}k∈Z and γ ∈ Z, we define the associatedcoset sequence u[γ] of u at the coset γ + 2Z to be
u[γ](ξ) :=∑
k∈Z
u(γ + 2k)e−ikξ,
that is,
u[γ] = u(γ + ·)↓2 = {u(γ + 2k)}k∈Z.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 24 / 64
Subdivision Operator on Polynomials
Sup is not always a polynomial for p ∈ Π.
For example, for p = 1 and u = {1}[0,0], we have[Sup](2k) = 2 and [Sup](2k + 1) = 0 for all k ∈ Z.
LemmaLet u ∈ l0(Z) and q be a polynomial. TFAE:
(i)∑
k∈Zq(−1
2− k)u(1 + 2k) =
∑k∈Z
q(−k)u(2k),
that is, (q ∗ u[1])(−12) = (q ∗ u[0])(0);
(ii) [q(−i ddξ)(e
−iξ/2u[1](ξ))]|ξ=0 = [q(−i ddξ)u
[0](ξ)]|ξ=0;
(iii) [q(− i2
ddξ)u(ξ)]|ξ=π = 0.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 25 / 64
Subdivision Operator on Polynomials
TheoremLet u = {u(k)}k∈Z. For m ∈ N, TFAE:
1 SuΠm−1 ⊆ Π;2 Suq ∈ Π for some q ∈ Π with deg(q) = m − 1;3 SuΠm−1 ⊆ Πm−1;4 u has m sum rules:
u(ξ + π) = O(|ξ|m), ξ → 0;
5 u(ξ) = (1 + e−iξ)mQ(ξ) for some 2π-periodic Q;
6 e−iξ/2u[1](ξ) = u[0](ξ) + O(|ξ|m), ξ → 0.
Moreover, Sup = 2−1p(2−1·) ∗ u.Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 26 / 64
Linear-phase Moments (LPM)
LemmaLet u ∈ l0(Z). Let p be a polynomial and definem := deg(p). For c ∈ R, p ∗ u = p(· − c) ⇐⇒ if u hasm + 1 linear-phase moments with phase c:
u(ξ) = e−icξ + O(|ξ|m+1), ξ → 0.
PropositionLet u ∈ l0(Z) and c ∈ R. Then u has m + 1 linear-phasemoments with phase c ⇐⇒ Tup = 2p(2 · +c) for allp ∈ Πm. Similarly, u has m + 1 sum rules and m + 1linear-phase moments with phase c ⇐⇒Sup = p(2−1(· − c)) for all p ∈ Πm.Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 27 / 64
Symmetry
We say that u = {u(k)}k∈Z : Z → C has symmetry if
u(2c − k) = ǫu(k), ∀ k ∈ Z
with 2c ∈ Z and ǫ ∈ {−1, 1}.u is symmetric if ǫ = 1; antisymmetric if ǫ = −1.c is the symmetry center of the filter u.A symmetry operator S to record the symmetry type:
[Su](ξ) :=u(ξ)
u(−ξ), ξ ∈ R.
[Su](ξ) = ǫe−i2cξ.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 28 / 64
LPM and Symmetry
PropositionSuppose that u ∈ l0(Z) has m but not m + 1linear-phase moments with phase c ∈ R. If m > 1, thenthe phase c is uniquely determined by u through
c = i u′(0) =∑
k∈Z
u(k)k .
Moreover, if u has symmetry: u(2c0 − k) = u(k) for allk ∈ Z for some c0 ∈ 1
2Z, then c = c0 (that is, the phase
c agrees with the symmetry center c0 of u).
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 29 / 64
Example: B-spline filters
B-spline filter of order m: aBm(ξ) := 2−m(1 + e−iξ)m
aB4 (0) = 1, aB
4
′(0) = −2i , aB
4
′′(0) = −5, aB
4
′′′
(0) = 14i .
For p ∈ Π3,
[p ∗ aB4 ](x) = p(x) − 2p′(x) +
5
2p′′(x) − 7
3p′′′(x).
[TaB4p](x) = 2p(2x) + 4p′(2x) + 5p′′(2x) +
14
3p′′′(2x).
[SaB4p](x) = p(x/2) − p′(x/2) +
5
8p′′(x/2) − 7
24p′′′(x/2).
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 30 / 64
Multi-level Fast Framelet Transform: Decomposition
Let a be a primal low-pass filter and b1, . . . , bs beprimal high-pass filters for decomposition.
For a positive integer J , a J-level discrete frameletdecomposition is given by
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 31 / 64
Thresholding and Quantization
− ε
2ε
2− ε
2ε
2− q
2q2
Figure: The hard thresholding function, the soft thresholdingfunction, and the quantization function, respectively. Boththresholding and quantization operations are often used to processthe framelet coefficients in a discrete framelet transform.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 32 / 64
Multi-level Reconstruction
Let a be a dual low-pass filter and b1, . . . , bs be dualhigh-pass filters for reconstruction.
a J-level discrete framelet reconstruction is
vj :=
√2
2Savj−1 +
√2
2
s∑
ℓ=1
Sbℓwj−1;ℓ, j = 1, . . . , J .
a J-level discrete reconstruction operatorVJ : (l(Z))1×(sJ+1) → l(Z) is defined by
A fast framelet transform with s = 1 is called a fastwavelet transform.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 33 / 64
Diagram of Multi-level FFrT
input
√2a⋆
√2b⋆
1
√2b⋆
s
↓2
↓2
↓2
√2a⋆
√2b⋆
1
√2b⋆
s
↓2
↓2
↓2
processing
processing
processing
↑2
↑2
↑2
√2a
√2b1
√2bs
⊕processing
processing
↑2
↑2
↑2
√2a
√2b1
√2bs
⊕ output
Figure: Diagram of a two-level discrete framelet transform using adual framelet filter bank ({a; b1, . . . , bs}, (a; b1, . . . , bs}).
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 34 / 64
Stability
A multi-level discrete framelet transform employing adual framelet filter bank {a; b1, . . . , bs}, {a; b1, . . . , bs})has stability in the space l2(Z) if there exists C > 0 suchthat for J ∈ N0,
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 35 / 64
Stability
PropositionSuppose that a multi-level discrete framelet transformhas stability in the space l2(Z). Then all the waveletdecomposition operators must have uniform stability inspace l2(Z):
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 39 / 64
Lack of Vanishing Moments
LemmaLet ({a; b1, . . . , bs}, {a; b1, . . . , bs}) be a dual frameletfilter bank. If all b1, . . . , bs have m vanishing momentsand all b1, . . . , bs have m vanishing moments, then
(i) the primal low-pass filter a must have m sum rules,that is, a(ξ + π) = O(|ξ|m) as ξ → 0;
(ii) the dual low-pass filter a must have m sum rules,that is, ˆa(ξ + π) = O(|ξ|m) as ξ → 0;
(iii) aˆa has m + m linear-phase moments with phase 0:
1 − a(ξ)ˆa(ξ) = O(|ξ|m+m), ξ → 0.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 40 / 64
Example of B-spline Filters
Let a(ξ) = aBm(ξ) = 2−m(1 + e−iξ)m and
ˆa(ξ) = aBm(ξ) = 2−m(1 + e−iξ)m be two B-spline filters.
a(0)ˆa(0) = 1,
[aˆa]′(0) =i(m − m)
2,
[aˆa]′′(0) =(m − m)2 + m + m
4.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 41 / 64
Oblique Extension Principle
Theorem({a; b1, . . . , bs}, {a; b1, . . . , bs})Θ has the followinggeneralized perfect reconstruction property:
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 42 / 64
Vanishing Moments of OEP
LemmaLet ({a; b1, . . . , bs}, {a; b1, . . . , bs})Θ be an OEP-basedfilter bank. Suppose that all b1, . . . , bs have m vanishingmoments and all b1, . . . , bs have m vanishing moments,where m, m ∈ N. Then
Θ(ξ) − Θ(2ξ)a(ξ)ˆa(ξ) = O(|ξ|m+m), ξ → 0.
If in addition Θ(0) 6= 0, then the primal low-pass filter amust have m sum rules and the dual low-pass filter amust have m sum rules.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 43 / 64
{a; b1, . . . , bs}Θ having PR is called an OEP-basedtight framelet filter bank.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 45 / 64
Fejer-Riesz Lemma
LemmaLet Θ be a 2π-periodic trigonometric polynomial withreal coefficients (or with complex coefficients) such thatΘ(ξ) > 0 for all ξ ∈ R. Then there exists a 2π-periodictrigonometric polynomial θ with real coefficients (or withcomplex coefficients) such that |θ(ξ)|2 = Θ(ξ) for allξ ∈ R. Moreover, if Θ(0) 6= 0, then we can further
require θ(0) =√
Θ(0).
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 46 / 64
OEP-based Tight Framelet
LemmaLet {a; b1, . . . , bs}Θ be an OEP-based tight framelet
filter bank. If Θ(0) > 0, then Θ(ξ) > 0 for all ξ ∈ R andconsequently, there exists θ ∈ l0(Z) such that Θ = θ ∗ θ⋆
holds and θ(0) =√
Θ(0).
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 47 / 64
OEP with s = 1
TheoremLet ({a; b}, {a; b})Θ be an OEP-based dual framelet
filter bank with Θ(0) 6= 0. Then
Θ(2ξ)Θ(π) = Θ(ξ)Θ(ξ + π)
[ˆa(ξ)
ˆb(ξ)
ˆa(ξ + π)ˆb(ξ + π)
] [a(ξ) b(ξ)
a(ξ + π) b(ξ + π)
]⋆
=
[1 00 1
],
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 48 / 64
Continued...
Theoremwhere all filters are finitely supported and are given by
ˆa(ξ) := ˆa(ξ)Θ(2ξ)/Θ(ξ) andˆb(ξ) := ˆb(ξ)/Θ(ξ).
That is, ({a; b}, {a; b}) is a biorthogonal wavelet filterbank. Moreover,
(ii) If c − 12 ∈ Z, then Tuv is an N-periodic sequence:
[Tuv ](−12−c−k) = [Tuv ](N− 1
2−c−k) = ǫ[Tuv ](k),
and [⌈−14− c
2⌉, ⌊N
2− 1
4− c
2⌋] is its control interval.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 60 / 64
Table 1: Endpoint Nonrepeated (EN)
filter u u⋆ ∗ v with v extended by EN Tuv with v extended by EN
c = 0ǫ = 1
(2N − 2)-periodic,symmetric about 0 and N − 1,a control interval [0, N − 1].
(N − 1)-periodic,
symmetric about 0 and N−12
,
a control interval [0, N2
− 1].
c = 0ǫ = −1
(2N − 2)-periodic,antisymmetric about 0 and N − 1,a control interval [0, N − 1],[u⋆ ∗ v](0) = [u⋆ ∗ v](N − 1) = 0.
(N − 1)-periodic,
antisymmetric about 0 and N−12
,
a control interval [0, N2
− 1],
[Tuv](0) = 0.
c = 1ǫ = 1
(2N − 2)-periodic,symmetric about −1 and N − 2,a control interval [−1, N − 2].
(N − 1)-periodic,
symmetric about − 12
and N2
− 1,
a control interval [0, N2
− 1].
c = 1ǫ = −1
(2N − 2)-periodic,antisymmetric about −1 and N − 2,a control interval [−1, N − 2],[u⋆ ∗ v](−1) = [u⋆ ∗ v](N − 2) = 0.
(N − 1)-periodic,
antisymmetric about − 12
and N2
− 1,
a control interval [0, N2
− 1],
[Tuv]( N2
− 1) = 0.
The decomposition filter u has the symmetrySu(ξ) = ǫe−i2cξ, where ǫ ∈ {−1, 1} and c ∈ {0, 1}. v isa symmetric extension with both endpoints non-repeated(EN) of an input signal v b = {v b(k)}N−1
k=0 .
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 61 / 64
Table 2: Endpoints Repeated (ER)
filter u u⋆ ∗ v with v extended by ER Tuv with v extended by ER
c = 12
ǫ = 1
2N-periodic,symmetric about −1 and N − 1,a control interval [−1, N − 1].
N-periodic,
symmetric about − 12
and N−12
,
a control interval [0, N2
− 1].
c = 12
ǫ = −1
2N-periodic,antisymmetric about −1 and N − 1,a control interval [−1, N − 1],[u⋆ ∗ v](−1) = [u⋆ ∗ v](N − 1) = 0.
N-periodic,
antisymmetric about − 12
and N−12
,
a control interval [0, N2
− 1]
.
c =− 1
2ǫ = 1
2N-periodic,symmetric about 0 and N,a control interval [0, N].
N-periodic,symmetric about 0 and N
2,
a control interval [0, N2
].
c =− 1
2ǫ = −1
2N-periodic,antisymmetric about 0 and N,a control interval [0, N],[u⋆ ∗ v](0) = [u⋆ ∗ v](N) = 0.
N-periodic,antisymmetric about 0 and N
2,
a control interval [0, N2
],
[Tuv](0) = [Tuv]( N2
) = 0.
The decomposition filter u has the symmetrySu(ξ) = ǫe−i2cξ, where ǫ ∈ {−1, 1} and c ∈ {−1
2,12}. v
is a symmetric extension with both endpoints repeated(ER) of an input signal v b = {v b(k)}N−1
k=0 .
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 62 / 64
Example: Biorthogonal Wavelet Filter Bank
Since Su0 = 1 and Su1 = e−i2ξ, extend v b by bothendpoints non-repeated (EN):
b(ξ) = e−i(ξ+π)ˆa(ξ + π), ˆb(ξ) = e−i(ξ+π)a(ξ + π).Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 3 / 72
Orthogonal Filters and Refinable Functions
An orthogonal low-pass filter is defined by
|a(ξ)|2 + |a(ξ + π)|2 = 1
with standard high-pass filter:
b(ξ) = e−i(ξ+π)a(ξ + π).
Refinable function φa satisfies refinement equation:
φa = 2∑
k∈Z
a(k)φa(2 · −k).
For b = {b(k)}k∈Z, a wavelet function ψa,b by
ψa,b := 2∑
k∈Z
b(k)φa(2 · −k).
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 4 / 72
Basic Quantities
vm(a) := m, the highest order of vanishingmoments satisfied by u: u(ξ) = O(|ξ|m), ξ → 0.sr(a) := m, the highest order of sum rules satisfiedby u: u(ξ + π) = O(|ξ|m) as ξ → 0.lpm(a) := m, highest order of linear-phase momentssatisfied by u: u(ξ) = e−icξ + O(|ξ|m), ξ → 0.If u(m)u(n) 6= 0 and u(k) = 0 for all k ∈ Z\[m, n],we define filter support fsupp(u) := [m, n].The smoothness exponent of filter u is defined by
sm(u) := −1/2 − log2
√ρ(u),
where ρ(u) is spectral radius of (w(2j − k))−K6j ,k6K
with∑K
k=−K w(k)e−ikξ := |v(ξ)|2.Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 5 / 72
Other Quantities
Expectation of a filter u:
E(u) :=
∑k∈Z
|u(k)|2k‖u‖2
l2(Z)
, ‖u‖2l2(Z) :=
∑
k∈Z
|u(k)|2.
Variance Var(u):
Var(u) :=
∑k∈Z
|u(k)|2(k − E(u))2
‖u‖2l2(Z)
.
frequency separation indicator Fsi(u, v):
Fsi(u, v) :=
∫π
−π|u(ξ)|2|v(ξ)|2dξ
√∫π
−π|u(ξ)|4dξ
√∫π
−π|v(ξ)|4dξ
.
In particular,
Fsi(u) := Fsi(u, v) with v(ξ) := e−i(ξ+π)u(ξ + π).
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 6 / 72
Coset Sequences
coset sequence of u at γ is u[γ] := {u(γ + 2k)}k∈Z,
u[γ](ξ) =∑
k∈Z
u(γ + 2k)e−ikξ.
(a, a) is a pair of biorthogonal filters ⇐⇒
a[0](ξ)a[0](ξ) + a[1](ξ)a[1](ξ) = 1/2.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 7 / 72
Interpolatory filters
(a, a) is a pair of biorthogonal filters ⇐⇒ theircorrelation filter u := a⋆ ∗ a is interpolatory:
u(ξ) + u(ξ + π) = 1.
Equivalently, u is interpolatory if u[0](ξ) = 1/2 or
u(0) = 1/2 and u(2k) = 0 ∀ k ∈ Z\{0}.
u is an interpolatory filter ⇐⇒ (u, δ) is a pair ofbiorthogonal filters, where δ(0) = 1 and δ(k) = 0for all k ∈ Z\{0}.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 8 / 72
Interpolation
PropositionA filter u ∈ l0(Z) is an interpolatory filter if and only if
(Suv)↓2 = v , that is, [Suv ](2k) = v(k), ∀ k ∈ Z.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 9 / 72
Polynomials
For any real number m and any nonnegative integer n,
Pm,n(x) = (1 − x)−m + O(xn), x → 0.
More explicitly,
Pm,n(x) :=
n−1∑
j=0
(m + j − 1
j
)x j ,
where(c
0
):= 1,
(c
j
):=
c(c − 1) · · · (c − j + 1)
j!, j ∈ N, c ∈ R
and j ! = 1 · 2 · · · (j − 1)j .Basic identity:
(1− x)mPm,m(x)+ xmPm,m(1− x) = 1 ∀ x ∈ R,m ∈ N.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 10 / 72
Construction of Interpolatory Filters
TheoremFor every positive integer m, there exists a uniqueinterpolatory filter aI
2m such that aI2m has 2m sum rules
and vanishes outside [1 − 2m, 2m − 1]. Moreover, aI2m is
a real-valued filter, symmetric about the origin, and hasthe following explicit expression:
aI2m(ξ) = cos2m(ξ/2)Pm,m(sin2(ξ/2)).
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 11 / 72
Examples of Interpolatory Filters
aI2 = {1
4,12, 1
4}[−1,1],
aI4 = {− 1
32, 0,932,
12, 9
32, 0,− 132}[−3,3],
aI6 = { 3
512, 0,− 25512, 0,
75256,
12, 75
256, 0,− 25512, 0,
3512}[−5,5],
aI8 = {− 5
4096, 0,49
4096, 0,− 2454096, 0,
12254096,
12, 1225
4096, 0,− 2454096,
0, 494096, 0,− 5
4096}[−7,7],
aI10 = { 35
131072, 0,− 405
131072, 0, 567
32768, 0,− 2205
32768, 0, 19845
65536, 1
2,
1984565536, 0,− 2205
32768, 0,567
32768, 0,− 405131072, 0,
35131072}[−9,9].
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 12 / 72
The smoothness exponents sm(aI2m), variances Var(aI
2m),and frequency separation indicators Fsi(aI
2m).
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 13 / 72
Graphs of Refinable Functions
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.2
0
0.2
0.4
0.6
0.8
1
(a) φaI1
−3 −2 −1 0 1 2 3−0.2
0
0.2
0.4
0.6
0.8
1
1.2
(b) φaI2
−5 −4 −3 −2 −1 0 1 2 3 4 5−0.2
0
0.2
0.4
0.6
0.8
1
1.2
(c) φaI3
−6 −4 −2 0 2 4 6−0.2
0
0.2
0.4
0.6
0.8
1
(d) φaI4
−8 −6 −4 −2 0 2 4 6 8−0.2
0
0.2
0.4
0.6
0.8
1
(e) φaI5
−10 −8 −6 −4 −2 0 2 4 6 8 10−0.2
0
0.2
0.4
0.6
0.8
1
(f) φaI6
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 14 / 72
Convolution Method
TheoremLet a be an interpolatory filter having n sum rules. Letm be a positive integer and P be a polynomial satisfying
(1 − x)mP(x) + xmP(1 − x) = 1
(For example, P = Pm,m.) Define a filter u by
u(ξ) := (a(ξ))mP(a(ξ + π)).
Then u is an interpolatory filter and has mn sum rules.
If plug aI2(ξ) = cos2(ξ/2) as a, then we have aI
2m.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 15 / 72
Linear-phase of Filters
PropositionLet u ∈ l0(Z) and ξ0 ∈ (−π, π) such that u(ξ0) 6= 0.Write u(ξ0) = M0e
−iθ0 for some M0, θ0 ∈ R. Then thereare unique real-valued continuous functionsM , θ : (−π, π) → R such that
u(ξ) = M(ξ)e−iθ(ξ) ∀ ξ ∈ (−π, π)
withM(ξ0) = M0, θ(ξ0) = θ0.
Moreover, for a real-valued filter u, the filter u hassymmetry if and only if θ is a linear function.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 16 / 72
Linear-phase Moments
Propositionsr(u) = lpm(u) for any interpolatory filter u.
PropositionLet M ,N ∈ N and c ∈ R. For any subset Λ of Z suchthat #Λ = N, there exists a unique solution {ck}k∈Λ to
u(ξ) = e−icξ + O(|ξ|N), ξ → 0,
where u(ξ) := (1 + e−iξ)M∑
k∈Λ cke−ikξ.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 17 / 72
Filters Having Linear-phase Moments
TheoremFor any positive integers m and n, set c = 0, M = 2m,N = 2n − 1, and Λ = {1− n −m, . . . , n −m − 1}. Thenthe unique filter in Proposition, denoted by a2m,2n, musttake the form
a2m,2n(ξ) = cos2m(ξ/2)Pm,n(sin2(ξ/2)),
The filter a2m,2n has 2m sum rules, 2n linear-phasemoments with phase 0, is symmetric about the origin,and has filter support [1 − m − n,m + n − 1]. Moreover,when n = m, the above filter a2m,2m is obviously theinterpolatory filter aI
2m.Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 18 / 72
Examples of a2m,2n
a4,2 = { 116,
14,
38, 1
4,116}[−2,2],
a6,2 = { 164,
332,
1564,
516, 15
64,332,
164}[−3,3],
a6,4 = {− 3256,− 1
32,364,
932,
55128, 9
32,364,− 1
32,− 3256}[−4,4],
a8,4 = {− 1256,− 5
256,
− 5256,
564,
35128,
49128, 35
128,564,− 5
256,− 5256,− 1
256}[−5,5],
a8,6 = { 52048,
3512,− 15
1024,− 25512,
752048,
75256,
231512, 75
256,75
2048,
− 25512,− 15
1024,3
512,5
2048}[−6,6].
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 19 / 72
The smoothness exponents sm(a2m,2n), variancesVar(a2m,2n), and frequency separation indicatorsFsi(a2m,2n), where the filters a2m,2n. Note thatE(a2m,2n) = 0.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 20 / 72
Graph of Refinable Functions
−2 −1.5 −1 −0.5 0 0.5 1 1.5 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(a) φa4,2
−3 −2 −1 0 1 2 30
0.1
0.2
0.3
0.4
0.5
(b) φa6,2
−4 −3 −2 −1 0 1 2 3 4−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(c) φa6,4
−5 −4 −3 −2 −1 0 1 2 3 4 5−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
(d) φa8,4
−6 −4 −2 0 2 4 6−0.2
0
0.2
0.4
0.6
0.8
(e) φa8,6
−6 −4 −2 0 2 4 6−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(f) φa10,4
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 21 / 72
Filters a2m−1,2n
TheoremFor any positive integers m, n, set c = 1/2,M = 2m − 1, N = 2n − 1, andΛ = {2 − n − m, . . . , n − m}. Then the unique filter inTheorem, denoted by a2m−1,2n, must take the form
The filter a2m−1,2n has 2m− 1 sum rules, 2n linear-phasemoments with phase 1/2, is symmetric about the point1/2, and has filter support [2 − m − n,m + n − 1].
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 22 / 72
Examples of a2m−1,2n
a3,2 = {18,
38, 3
8 ,18}[−1,2],
a5,2 = { 132,
532,
516, 5
16,532,
132}[−2,3],
a5,4 = {− 5256,− 7
256,35256,
105256, 105
256,35256,− 7
256,− 5256}[−3,4],
a7,4 = {− 71024,− 27
1024, 0,21128,
189512, 189
512,21128, 0,
− 271024,− 7
1024}[−4,5],
a7,6 = { 6316384,
7716384,− 495
16384,− 69316384,
11558192,
34658192
, 34658192,
11558192,− 693
16384,− 49516384,
7716384,
6316384}[−5,6].
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 23 / 72
The smoothness exponents sm(a2m−1,2n), variancesVar(a2m−1,2n), and frequency separation indicatorsFsi(a2m−1,2n), where the filters a2m−1,2n.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 24 / 72
Graphs of a2m−1,2n
−1 −0.5 0 0.5 1 1.5 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
(a) φa3,2
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(b) φa5,2
−3 −2 −1 0 1 2 3 4−0.2
0
0.2
0.4
0.6
0.8
(c) φa5,4
−4 −3 −2 −1 0 1 2 3 4 5−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(d) φa7,4
−5 −4 −3 −2 −1 0 1 2 3 4 5 6−0.2
0
0.2
0.4
0.6
0.8
(e) φa7,6
−5 −4 −3 −2 −1 0 1 2 3 4 5 6−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
(f) φa9,4
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 25 / 72
Real-valued Orthogonal Filters
TheoremIf a ∈ l0(Z) is a real-valued orthogonal filter with m sumrules,
|a(ξ)|2 = cos2m(ξ/2)P(sin2(ξ/2))
for some polynomial P with real coefficients such thatbasic identity xmP(1 − x) + (1 − x)mP(x) = 1 and
P(x) > 0, ∀ x ∈ [0, 1].
Conversely, if both identities are satisfied for apolynomial P with real coefficients, then there exists areal-valued orthogonal filter a with a(0) = 1 such that ahas m sum rules.Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 26 / 72
Daubechies Orthogonal Filters
Take P = Pm,m. Then P(x) > 0, x ∈ [0, 1]. There is areal-valued sequence QD
m ∈ l0(Z) such that
|QDm(ξ)|2 = Pm,m(sin2(ξ/2)), QD
m(0) = 1, fsupp(QDm) = [0,m−1].
The Daubechies orthogonal filter of order m is
aDm(ξ) := 2−me i(m−1)ξ(1 + e−iξ)mQD
m (ξ), m ∈ N.
fsupp(aDm) = [1 − m,m], sr(aD
m) = m, and aDm satisfies
|aDm(ξ)|2 = aI
2m(ξ) = cos2m(ξ/2)Pm,m(sin2(ξ/2)).
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 27 / 72
Examples of Daubechies Filters
aD1 = {1
2, 1
2}[0,1],
aD2 = {1+
√3
8 , 3+√
38, 3−
√3
8 , 1−√
38 }[−1,2]
aD3 = {0.235233603893, 0.570558457918,
0.325182500266,−0.095467207782,
− 0.060416104155, 0.024908749869}[−2,3].
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 28 / 72
More Examples
QD4 = {−0.857191211347, 3.093477124385,
− 1.600848680204, 0.364562767168}[0,3],
QD5 = {0.618476735277,−2.424433845637,
5.051894897560,−2.688052234523,
0.442114447325}[0,4]
QD6 = {0.697110410451,−4.024690866806,
8.351866150884,−5.869382002997,
2.198116068769,−0.35301976030}[0,5]
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 29 / 72
where β ∈ Z is the left endpoint of filter support of a,1 θm,n(ξ) =
∑n−1j=0 λje
−ijξ with λ0, . . . , λn−1 uniquelydetermined by
θm,n(ξ) = e−i(c−β)ξ2m(1+e−iξ)−m+O(|ξ|n), ξ → 0,
2 θ(ξ) =∑ℓ−1
j=0 tje−ijξ for some ℓ ∈ N, where unknown
t0, . . . , tℓ−1 are to be determined by theorthogonality condition.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 36 / 72
Example
Let m = 2, n = 3 in Algorithm. Then
θ2,3(ξ) = (12c2− 5
2c + 11
4)+(−c2 +4c− 5
2)e−iξ +(1
2c2− 3
2c + 3
4)e−i2ξ.
For ℓ = 1, set c = 1. Then t0 = 3−√
158 . For c = 0 and
β = −1 satisfying c − β = c = 1, the real-valuedorthogonal filter aOL
2,3 is supported inside [−1, 4] with
sr(aOL2,3) = 2 and lpm(aOL
2,3) = 3 with phase c = 0.
By calculation, E (aOL2,3) ≈ 0.38393, Var(aOL
2,3) ≈ 0.465706,
Fsi(aOL2,3) ≈ 0.184371, and sm(aOL
2,3) ≈ 1.232138.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 37 / 72
Graphs
−2 −1 0 1 2 3 4 5
−0.1
0
0.1
0.2
0.3
0.4
0.5
(a) Filter aOL2,3, c =
0
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3
(b) aOL2,3, c = 0
−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
(c) φaOL2,3 , c = 0
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3
−1.5
−1
−0.5
0
0.5
1
(d) ψaOL2,3 , c = 0
−2 −1 0 1 2 3 4 5
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
(e) Filter aOL2,3, c =
− 15
−3 −2 −1 0 1 2 3
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
(f) aOL2,3, c = − 1
5
−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
(g) φaOL2,3 , c = − 1
5
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3
−1.5
−1
−0.5
0
0.5
1
(h) ψaOL2,3 , c = − 1
5
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 38 / 72
Symmetric Complex Orthogonal Filters
LemmaLet m be a positive odd integer and a ∈ l0(Z) havecomplex coefficients. Then a is a filter having symmetrya(1 − k) = a(k) and m sum rules ⇐⇒ there is apolynomial Q with complex coefficients such that
a(ξ) = 2−me iξ(m−1)/2(1 + e−iξ)mQ(sin2(ξ/2)).
Moreover, the filter a above is an orthogonal filter ⇐⇒for all x ∈ R,
(1 − x)m|Q(x)|2 + xm|Q(1 − x)|2 = 1, x ∈ R.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 39 / 72
Symmetric Complex Orthogonal Filters
LemmaLet P be a polynomial with real coefficients andP(0) 6= 0. Then P(x) = |Q(x)|2, x ∈ R for somepolynomial Q with complex coefficients andQ(0) =
√P(0) ⇐⇒ the polynomial P is nonnegative
on the real line:
P(x) > 0, ∀ x ∈ R.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 40 / 72
Symmetric Complex Orthogonal Filters
LemmaFor any positive odd integer m, the polynomial Pm,m
satisfies Pm,m(x) > 0 for all x ∈ R.
There is a subset Y Sm (S=Symmetry) of C such that
∣∣∣∏
y∈Y Sm
(1 + yx)∣∣∣2
= Pm,m(x).
aSm(ξ) := 2−me iξ(m−1)/2(1 + e−iξ)m
∏
y∈Y Sm
(1 + y sin2(ξ/2)).
1 aSm is orthogonal and aS
m(1 − k) = aSm(k);
2 |aSm(ξ)|2 = |aD
m(ξ)|2 = aI2m(ξ) and aS
m(0) = 1;3 sr(aS
m) = m, fsupp(aSm) = [−(m − 1)/2, (m + 1)/2].
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 41 / 72
Examples
Y S3 = {3−
√15i
2 }Y S
5 = {2.67984516848 + 1.60066496071i ,
− 0.179845168483− 2.67428235144i}Y S
7 = {3.22858920491 + 1.30036579467i ,
1.30198701500 + 2.95813693603i ,
− 1.03057621991− 2.49790321151i}
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 42 / 72
m ) 0.0136919 0.0308002 0.00796381 0.00003864 0.00583666 0.0207335 0.00466501 0.000048884
The smoothness exponents, variances, frequencyseparation indicators, and orthogonal filter indicators ofaIS
m and aISm . Note that E(aIS
m ) = E(aISm ) = odd(m)/2,
Fsi(aISm , b
ISm ) = Fsi(aIS
m , bISm ), and
Ofi(aISm , b
ISm ) = Ofi(aIS
m , bISm ).
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 59 / 72
Graphs
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−2
−1
0
1
2
3
4
5
(a) φaIS2
−1 −0.5 0 0.5 1 1.5 2
−6
−5
−4
−3
−2
−1
0
1
2
3
4
(b) ψaIS2 ,aIS
2
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
(c) φaIS2
−1 −0.5 0 0.5 1 1.5 2
−1.5
−1
−0.5
0
0.5
(d) ψaIS2 ,bIS
2
−3 −2 −1 0 1 2 3 4
−2
−1
0
1
2
3
(e) φaIS3
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3
−4
−3
−2
−1
0
1
2
3
4
(f) ψaIS3 ,bIS
3
−1 −0.5 0 0.5 1 1.5 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
(g) φaIS3
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
(h) ψaIS3 ,bIS
3
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 60 / 72
Graphs
−6 −4 −2 0 2 4 6−0.2
0
0.2
0.4
0.6
0.8
1
(a) φaIS6
−5 −4 −3 −2 −1 0 1 2 3 4 5 6
−1
−0.5
0
0.5
(b) ψaIS6 ,bIS
6
−5 −4 −3 −2 −1 0 1 2 3 4 5
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
(c) φaIS6
−5 −4 −3 −2 −1 0 1 2 3 4 5 6
−1.5
−1
−0.5
0
0.5
1
(d) ψaIS6 ,bIS
6
−6 −4 −2 0 2 4 6 8
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(e) φaIS7
−6 −4 −2 0 2 4 6−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
(f) ψaIS7 ,bIS
7
−5 −4 −3 −2 −1 0 1 2 3 4 5 6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
(g) φaIS7
−6 −4 −2 0 2 4 6
−1.5
−1
−0.5
0
0.5
1
1.5
(h) ψaIS7 ,bIS
7
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 61 / 72
CBC Algorithm
Let a ∈ l0(Z) with a(0) 6= 0 and a has a dual filter a.(S1) choose Λ ⊂ Z with #Λ > m. Then there exists a
solution {cn}n∈Λ with (#Λ − m) free parameters to
∑
n∈Λ
cne−inξ =
e iξ/2ˆa(ξ/2 + π)
a(ξ/2), ξ → 0.
In particular, if #Λ = m, a unique solution {cn}n∈Λ;(S2) construct a filter a coset by coset as follows:
a(2k) = a(2k) −∑
n∈Λ
cn a(1 + 2n − 2k), k ∈ Z,
a(1 + 2k) = a(1 + 2k) +∑
n∈Λ
cn a(2n − 2k), k ∈ Z.
Then a is a dual filter of a with m sum rules.Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 62 / 72
Symmetry
LemmaLet (a, a) be a pair of biorthogonal filters such that a hassymmetry: Sa(ξ) = ǫe−icξ for some ǫ ∈ {−1, 1} andc ∈ Z. Define
aS(ξ) := (ˆa(ξ) + ǫe−icξˆa(−ξ))/2.
Then aS is also a dual filter of a and aS has the samesymmetry type as a: S aS = Sa.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 63 / 72
ExampleLet a = aI
4 = {− 132, 0,
932,
12 ,
932, 0,− 1
32}[−3,3]. Then a = δ
is a dual filter of a such that Sa(ξ)Sˆa(ξ) = 1.
ha,a(0) =1
2, ha,a(1) = 0, ha,a(2) =
9
64, ha,a(3) =
45
128, ha,a(4) =
For m = 1 and Λm = {0}, then t0 = 12 is a solution and
a = { 1
64, 0,−1
8,1
4,23
32,1
4,−1
8, 0,
1
64}[−4,4].
For m = 2 and Λm = {0, 1}, then {t0 = 916, t1 = − 1
16}
a = {− 1
512, 0,
9
256,− 1
32,− 63
512,
9
32,
87
128,
9
32,− 63
512,− 1
32,
9
256, 0,− 1
512}[−6,6].
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 64 / 72
Example
a = a6,4 = {− 3
256,− 1
32,
3
64,
9
32,
55
128,
9
32,
3
64,− 1
32,− 3
256}[−4,4].
Then a = { 980,− 3
10, 3
16, 1, 3
16,− 3
10, 9
80}[−3,3] is a dual filter
of a.
ha,a(0) = −1
10, ha,a(1) = −
3
5, ha,a(2) = −
15
16, ha,a(3) = −
4389
640, ha,a(4) = −
78117
1024.
For m = 2 and Λm = {0, 1}, then {t0 = 316, t1 = −23
80} is
a solution.
a = {69
20480,−
23
2560, −
321
20480,
111
1280, −
91
20480,−
681
2560,
5463
20480,
561
640,
5463
20480,
−681
2560,−
91
20480,
111
1280,−
321
20480, −
23
2560,
69
20480}[−7,7].
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 65 / 72
Table
CBC Algorithm sr ‖ · ‖l2(Z) Var sm Ofi Ofi(a, b) Fsi(a, b)
a = aI4 4 0.410156 0.428571 2.440765 0.0650482
a with m = 1 2 0.673340 0.382886 0.593223 0.249223 0.0339170 0.188788a with m = 2 4 0.654892 0.514178 1.179370 0.218591 0.0101275 0.141911
a = a6,4 6 0.349457 0.565889 4.098191 0.176972
a with m = 2 4 1.06795 0.906996 0.349587 3.29133 0.0463589 0.135131a with m = 3 6 1.03806 1.01883 0.649332 3.08117 0.0320641 0.112711
a = a5,4 5 0.376099 0.496998 3.259609 0.122794
a with m = 2 3 0.846221 0.543673 0.346291 1.07161 0.0566240 0.183283a with m = 3 5 0.801151 0.740546 1.042980 0.904897 0.0193244 0.121141
The smoothness exponents, variances, frequencyseparation indicators, and orthogonal filter indicators of aand a in Examples. Note that Ofi(a, b) = Ofi(a, b) andFsi(a, b) = Fsi(a, b).
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 66 / 72
Graphs
−3 −2 −1 0 1 2 3
0
0.2
0.4
0.6
0.8
1
(a) φa with a = aI4
−3 −2 −1 0 1 2 3 4
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
(b) ψa,b with m =1
−4 −3 −2 −1 0 1 2 3 4
−1
−0.5
0
0.5
1
1.5
2
2.5
3
(c) φa with m = 1
−3 −2 −1 0 1 2 3 4
−4
−3
−2
−1
0
1
2
3
(d) ψa,b, m = 1
−3 −2 −1 0 1 2 3
0
0.2
0.4
0.6
0.8
1
(e) φa with a = aI4
−4 −3 −2 −1 0 1 2 3 4 5
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
(f) ψa,b with m =2
−6 −4 −2 0 2 4 6
−0.5
0
0.5
1
1.5
(g) φa with m = 2
−4 −3 −2 −1 0 1 2 3 4 5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
(h) ψa,b, m = 2
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 67 / 72
Polyphase and Laurent Polynomials
For a sequence u = {u(k)}k∈Z,
u(z) :=∑
k∈Z
u(k)zk, z ∈ C\{0}. (1)
u⋆(k) := u(−k)T, k ∈ Z.
u(k) = ǫu(2c − k) for all k ∈ Z ⇐⇒Su(z) := u(z)
u(1/z) = ǫz2c .
u has m linear-phase moments with phase c ⇐⇒u(z) = z c + O(|z − 1|m), z → 1.
biorthogonality condition becomes
a⋆(z)a(z) + a⋆(−z)a(−z) = 1, z ∈ C\{0}.Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 68 / 72
Biorthogonal Filters
PropositionFor a ∈ l0(Z), a has at least one finitely supported dualfilter, if and only if, a(z) and a(−z) have no commonzeros in C\{0}.
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 69 / 72
Chain Structure
TheoremLet a ∈ l0(Z) be a filter such that a has at least onefinitely supported dual filter a. Assume that the filtersupport of a is [m, n] for some integers m and n. Iflen(a) = n −m > 0, then there exists a unique dual filtera of a such that fsupp(a) ⊆ [m, n − 1].
(i) if len(a) is a positive even integer, then there existsa unique dual filter a of a such thatfsupp(a) ⊆ [m + 1, n − 1];
(ii) if len(a) is odd, then there exists a unique dual filtera of a such that fsupp(a) ⊆ [m, n − 2];
(iii) if len(a) = 1, a = 1
2a(m)δ(· − m), a = 1
2a(n)δ(· − n).
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 70 / 72
Dual Framelet Filter Banks
({a; b1, . . . , bs}, a; b1, . . . , bs})Θ is an OEP-baseddual framelet filter bank if[
b1(z) · · · bs(z)
b1(−z) · · · bs(−z)
] [b1(z) · · · bs(z)
b1(−z) · · · bs(−z)
]⋆
= MΘ,a,a(z),
where MΘ,a,a(z) :=[Θ(z) − Θ(z2)a(z)a⋆(z) −Θ(z2)a(z)a⋆(−z)−Θ(z2)a(−z)a⋆(z) Θ(−z) − Θ(z2)a(−z)a⋆(−z)
].
Equivalently,[b
[0]1 (z) · · · b
[0]s (z)
b[1]1 (z) · · · b
[1]s (z)
][b
[0]1 (z) · · · b
[0]s (z)
b[1]1 (z) · · · b
[1]s (z)
]⋆
= NΘ,a,a(z),
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 71 / 72
Tight Framelet Filter Banks
{a; b1, . . . , bs}θ is an OEP-based tight framelet filterbank fif
[b
[0]1 (z) · · · b
[0]s (z)
b[1]1 (z) · · · b
[1]s (z)
][b
[0]1 (z) · · · b
[0]s (z)
b[1]1 (z) · · · b
[1]s (z)
]⋆
= NΘ,a(z),
where NΘ,a(z) :=[
12Θ[0](z) − Θ(z)a[0](z)(a[0](z))⋆ 1
2zΘ[1](z) − Θ(z)a[0](z)(a[1](z))⋆
12Θ[1](z) − Θ(z)a[1](z)(a[0](z))⋆ 1
2Θ[0](z) − Θ(z)a[1](z)(a[1](z))⋆
].
Bin Han (University of Alberta) Algorithms of Wavelets and Framelets Summer School, Edmonton 72 / 72