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Wavelet packetsDiego Milone
Análisis y Procesamiento Avanzado de Señales
Doctorado en Ingeniería, FICH-UNLMaestría en Ingeniería Biomédica, FI-UNER
Maestría en Computación Aplicada a la Ciencia y la Ingeniería, FICH-UNL
30 de octubre de 2015
Introduction WP definitions Splitting algorithm More basis...
Outline
Introduction
Wavelet packetsDefinition of basic wavelet packetsGeneral wavelet packetsWavelet packet treesSupport and localization
The splitting algorithmSplitting the treeFinite signals and computational complexity
More basis...Some (fixed) wavelet packets basesLocal cosine bases
Introduction WP definitions Splitting algorithm More basis...
Outline
Introduction
Wavelet packetsDefinition of basic wavelet packetsGeneral wavelet packetsWavelet packet treesSupport and localization
The splitting algorithmSplitting the treeFinite signals and computational complexity
More basis...Some (fixed) wavelet packets basesLocal cosine bases
Introduction WP definitions Splitting algorithm More basis...
Linear expansions in signal processing
We write a signal f belonging to a space S as
f =∑i
αiψi
where the set {ψi}i∈Z is complete in S.
[3]
Introduction WP definitions Splitting algorithm More basis...
Linear expansions in signal processing
We write a signal f belonging to a space S as
f =∑i
αiψi
where the set {ψi}i∈Z is complete in S.
For the orthogonal bases case, there exists a complete set{ψi}i∈Z such that
〈ψi, ψj〉 = δi,j
and then
f =∑i
〈ψi, f〉ψi
[3]⇑
Introduction WP definitions Splitting algorithm More basis...
Linear expansions in signal processingWe write a signal f belonging to a space S as
f =∑i
αiψi
where the set {ψi}i∈Z is complete in S.
For the orthogonal bases case, there exists a complete set{ψi}i∈Z such that
〈ψi, ψj〉 = δi,j
and then
f =∑i
〈ψi, f〉ψi
Introduction WP definitions Splitting algorithm More basis...
Linear expansions in signal processingIn the bi-orthogonal case, there exists a dual set {ψi}i∈Z suchthat ⟨
ψi, ψj
⟩= δi,j
and we have
f =∑i
⟨ψ, f
⟩ψi =
∑i
〈ψ, f〉 ψi
Introduction WP definitions Splitting algorithm More basis...
Multirate filter banks and structured basisAssume a sequence x[n] in `2(Z), the space of finite energysequences.
Structured basis are obtained when a finite set gi[n] is used togenerate the infinite basis, for example
ψi+kN [n] = gi[n− kN ].
Introduction WP definitions Splitting algorithm More basis...
Multirate filter banks and structured basisAssume a sequence x[n] in `2(Z), the space of finite energysequences.
Structured basis are obtained when a finite set gi[n] is used togenerate the infinite basis, for example
ψi+kN [n] = gi[n− kN ].
Expansion in such basis can be computed using multirate filterbanks, with filters hi[n] = gi[−n] (a filter bank to obtain theexpansion coefficients) followed by subsampling.
Introduction WP definitions Splitting algorithm More basis...
Multirate filter banks and structured basisThe synthesis filter bank consist of the upsampling followed byinterpolation with impulse response gi[n]... For N = 2 we have:
Introduction WP definitions Splitting algorithm More basis...
Multirate filter banks and structured basisThe synthesis filter bank consist of the upsampling followed byinterpolation with impulse response gi[n]... For N = 2 we have:
Because gi[n] and their translates by integer multiples of Nform an orthonormal basis, we have
S0 ⊕ S1 ⊕ · · · ⊕ SN−1 = `2(Z)
Introduction WP definitions Splitting algorithm More basis...
Wavelet basesInstead of (Fourier) modulation, a very different family oforthonormal bases is obtained by choosing a single prototypebut using shifting and scaling...
ψm,n(t) = 2−m/2ψ
(t− 2n
2m
)
Introduction WP definitions Splitting algorithm More basis...
Wavelet basesInstead of (Fourier) modulation, a very different family oforthonormal bases is obtained by choosing a single prototypebut using shifting and scaling
ψm,n(t) = 2−m/2ψ
(t− 2n
2m
)
Introduction WP definitions Splitting algorithm More basis...
Wavelet basesInstead of (Fourier) modulation, a very different family oforthonormal bases is obtained by choosing a single prototypebut using shifting and scaling
ψm,n(t) = 2−m/2ψ
(t− 2n
2m
)
Introduction WP definitions Splitting algorithm More basis...
Completing the wavelet tree...[5]
Introduction WP definitions Splitting algorithm More basis...
Completing the wavelet tree...
Introduction WP definitions Splitting algorithm More basis...
Alternative tiling of the time-frequency plane
Introduction WP definitions Splitting algorithm More basis...
Alternative tiling of the time-frequency plane
Introduction WP definitions Splitting algorithm More basis...
Alternative tiling of the time-frequency plane
Introduction WP definitions Splitting algorithm More basis...
Alternative tiling of the time-frequency plane
Introduction WP definitions Splitting algorithm More basis...
Alternative tiling of the time-frequency plane
Introduction WP definitions Splitting algorithm More basis...
Alternative tiling of the time-frequency plane
Introduction WP definitions Splitting algorithm More basis...
Alternative tiling of the time-frequency plane
...?
Introduction WP definitions Splitting algorithm More basis...
Wavelet packet decomposition
Introduction WP definitions Splitting algorithm More basis...
Wavelet packet decomposition
DWT WP
Introduction WP definitions Splitting algorithm More basis...
Wavelet packet decomposition
DWT WP
Introduction WP definitions Splitting algorithm More basis...
Wavelet packet decomposition
DWT WP
Introduction WP definitions Splitting algorithm More basis...
Wavelet packet decomposition
DWT WP
Introduction WP definitions Splitting algorithm More basis...
Wavelet packet decomposition
DWT WP
Introduction WP definitions Splitting algorithm More basis...
Outline
Introduction
Wavelet packetsDefinition of basic wavelet packetsGeneral wavelet packetsWavelet packet treesSupport and localization
The splitting algorithmSplitting the treeFinite signals and computational complexity
More basis...Some (fixed) wavelet packets basesLocal cosine bases
Introduction WP definitions Splitting algorithm More basis...
Definition of basic wavelet packets
Consider two finite trigonometric sums:
√2m0(ξ) =
2N−1∑0
h[k]e−ikξ
√2m1(ξ) =
2N−1∑0
g[k]e−ikξ
[1]
Introduction WP definitions Splitting algorithm More basis...
Definition of basic wavelet packets
Consider two finite trigonometric sums:
√2m0(ξ) =
2N−1∑0
h[k]e−ikξ
√2m1(ξ) =
2N−1∑0
g[k]e−ikξ
[1]⇑
Introduction WP definitions Splitting algorithm More basis...
Definition of basic wavelet packetsConsider two finite trigonometric sums:
√2m0(ξ) =
2N−1∑0
h[k]e−ikξ
√2m1(ξ) =
2N−1∑0
g[k]e−ikξ
Introduction WP definitions Splitting algorithm More basis...
Definition of basic wavelet packetsConsider two finite trigonometric sums:
√2m0(ξ) =
2N−1∑0
h[k]e−ikξ
√2m1(ξ) =
2N−1∑0
g[k]e−ikξ
...satisfying the following conditions:
g[k] = (−1)k+1h[2N −1−k]
or m1(ξ) = ej(2N−1)ξm0(ξ+π)
m0(0) = 1; m0(ξ) 6= 0 on [−π/3, π/3]
|m0(ξ)|2 + |m0(ξ + π)|2 = 1.
Introduction WP definitions Splitting algorithm More basis...
Definition of basic wavelet packetsConsider two finite trigonometric sums:
√2m0(ξ) =
2N−1∑0
h[k]e−ikξ
√2m1(ξ) =
2N−1∑0
g[k]e−ikξ
...satisfying the following conditions:
g[k] = (−1)k+1h[2N −1−k] or m1(ξ) = ej(2N−1)ξm0(ξ+π)
m0(0) = 1; m0(ξ) 6= 0 on [−π/3, π/3]
|m0(ξ)|2 + |m0(ξ + π)|2 = 1.
Introduction WP definitions Splitting algorithm More basis...
Definition of basic wavelet packetsConsider two finite trigonometric sums:
√2m0(ξ) =
2N−1∑0
h[k]e−ikξ
√2m1(ξ) =
2N−1∑0
g[k]e−ikξ
...satisfying the following conditions:
g[k] = (−1)k+1h[2N −1−k] or m1(ξ) = ej(2N−1)ξm0(ξ+π)
m0(0) = 1; m0(ξ) 6= 0 on [−π/3, π/3]
|m0(ξ)|2 + |m0(ξ + π)|2 = 1.
Introduction WP definitions Splitting algorithm More basis...
Definition of basic wavelet packetsConsider two finite trigonometric sums:
√2m0(ξ) =
2N−1∑0
h[k]e−ikξ
√2m1(ξ) =
2N−1∑0
g[k]e−ikξ
...satisfying the following conditions:
g[k] = (−1)k+1h[2N −1−k] or m1(ξ) = ej(2N−1)ξm0(ξ+π)
m0(0) = 1; m0(ξ) 6= 0 on [−π/3, π/3]
|m0(ξ)|2 + |m0(ξ + π)|2 = 1.
Introduction WP definitions Splitting algorithm More basis...
Definition of basic wavelet packets
The wavelet packets wn(x) are defined by induction onn = 0, 1, 2, 3, ... using the two identities
w2n(x) =√
22N−1∑
0
h[k]wn(2x− k)
w2n+1(x) =√
2
2N−1∑0
g[k]wn(2x− k)
and by the condition w0(x) ∈ L1(R),∫∞−∞w0(x)dx = 1.
Introduction WP definitions Splitting algorithm More basis...
Definition of basic wavelet packetsThe wavelet packets wn(x) are defined by induction onn = 0, 1, 2, 3, ... using the two identities
w2n(x) =√
2
2N−1∑0
h[k]wn(2x− k)
w2n+1(x) =√
2
2N−1∑0
g[k]wn(2x− k)
and by the condition w0(x) ∈ L1(R),∫∞−∞w0(x)dx = 1.
Introduction WP definitions Splitting algorithm More basis...
With n = 0 we have
w0(x) =√
2
2N−1∑0
h[k]w0(2x− k)
thus, ϕ(x) = w0(x) appears as a fixed point of the operator Tdefined by
T [f(x)] =√
2
2N−1∑0
h[k]f(2x− k)
example...
Introduction WP definitions Splitting algorithm More basis...
With n = 0 we have
w0(x) =√
2
2N−1∑0
h[k]w0(2x− k)
thus, ϕ(x) = w0(x) appears as a fixed point of the operator Tdefined by
T [f(x)] =√
2
2N−1∑0
h[k]f(2x− k)
example...
Introduction WP definitions Splitting algorithm More basis...
Example: interatively building ϕ(x)
A possible choice for h[k] is given by
√2h[0] = (1 +
√3)/4,
√2h[1] = (3 +
√3)/4
√2h[2] = (3−
√3)/4,
√2h[3] = (1−
√3)/4
Then we have...
Introduction WP definitions Splitting algorithm More basis...
Example: interatively building ϕ(x)
Introduction WP definitions Splitting algorithm More basis...
Example: interatively building ϕ(x)
...and the sequence fj converges uniformly to the fixed point ϕ(x).
Introduction WP definitions Splitting algorithm More basis...
⇒ ϕ(x) is the “father” wavelet in the construction oforthonormal wavelet bases.
Introduction WP definitions Splitting algorithm More basis...
⇒ ϕ(x) is the “father” wavelet in the construction oforthonormal wavelet bases.
Then, using w2n+1(x) =√
2∑2N−1
0 g[k]wn(2x− k) with n = 0we obtain ψ = w1.
⇒ ψ(x) is the “mother” wavelet in the construction oforthonormal wavelet bases.
Introduction WP definitions Splitting algorithm More basis...
⇒ ϕ(x) is the “father” wavelet in the construction oforthonormal wavelet bases.
Then, using w2n+1(x) =√
2∑2N−1
0 g[k]wn(2x− k) with n = 0we obtain ψ = w1.
⇒ ψ(x) is the “mother” wavelet in the construction oforthonormal wavelet bases.
Next, with n = 1 we obtain w2(x) and w3(x), and by repeatingthe process we generate all of the wavelet packets.
Introduction WP definitions Splitting algorithm More basis...
An important result from this construction is that the doblesequence (on n and k)
wn(x− k); n ∈ N0; k ∈ Z
is an orthonormal basis for L2(R).
Introduction WP definitions Splitting algorithm More basis...
An important result from this construction is that the doblesequence (on n and k)
wn(x− k); n ∈ N0; k ∈ Z
is an orthonormal basis for L2(R).
That is, the subsequence derived from here by taking2j ≤ n < 2j+1 is an orthonormal basis for the orthogonalcomplement Wj of Vj in Vj+1.
Introduction WP definitions Splitting algorithm More basis...
An important result from this construction is that the doblesequence (on n and k)
wn(x− k); n ∈ N0; k ∈ Z
is an orthonormal basis for L2(R).
That is, the subsequence derived from here by taking2j ≤ n < 2j+1 is an orthonormal basis for the orthogonalcomplement Wj of Vj in Vj+1.
Recall: in the multiresolution wavelet analysis 2j/2ψ(2jx− k)generates Wj (and 2j/2ϕ(2jx− k) generates Vj).
Introduction WP definitions Splitting algorithm More basis...
An important result from this construction is that the doblesequence (on n and k)
wn(x− k); n ∈ N0; k ∈ Z
is an orthonormal basis for L2(R).
That is, the subsequence derived from here by taking2j ≤ n < 2j+1 is an orthonormal basis for the orthogonalcomplement Wj of Vj in Vj+1.
Recall: in the multiresolution wavelet analysis 2j/2ψ(2jx− k)generates Wj (and 2j/2ϕ(2jx− k) generates Vj).
Thus, this construction of wavelet packets appears as a changeof orthonormal basis inside each Wj .
Introduction WP definitions Splitting algorithm More basis...
General wavelet packets
The general wavelet packets are the functions
2j/2wn(2jx− k), n ∈ N j, k ∈ Z
Introduction WP definitions Splitting algorithm More basis...
General wavelet packets
The general wavelet packets are the functions
2j/2wn(2jx− k), n ∈ N j, k ∈ Z
if j = 0 ...
Introduction WP definitions Splitting algorithm More basis...
General wavelet packets
The general wavelet packets are the functions
2j/2wn(2jx− k), n ∈ N j, k ∈ Z
if j = 0 we have the previous basic wavelet packets definition
Introduction WP definitions Splitting algorithm More basis...
General wavelet packets
The general wavelet packets are the functions
2j/2wn(2jx− k), n ∈ N j, k ∈ Z
if j = 0 we have the previous basic wavelet packets definition
if n = 1 ...
Introduction WP definitions Splitting algorithm More basis...
General wavelet packets
The general wavelet packets are the functions
2j/2wn(2jx− k), n ∈ N j, k ∈ Z
if j = 0 we have the previous basic wavelet packets definition
if n = 1 we have an standard orthogonal wavelet basis
Introduction WP definitions Splitting algorithm More basis...
But.... which of these sets of wavelet packets makes uporthogonal bases for L2(R)?
Introduction WP definitions Splitting algorithm More basis...
But.... which of these sets of wavelet packets makes uporthogonal bases for L2(R)?
Let E be a set of pairs (j, n) such that the correspondingfrequency intervals 2jn ≤ ξ < 2j(n+ 1) constitute a partition of[0,∞), up to a contable set. Then the subsequence
2j/2wn(2jx− k), n ∈ N j, k ∈ Z
is an orthogonal basis for L2(R).
Introduction WP definitions Splitting algorithm More basis...
But.... which of these sets of wavelet packets makes uporthogonal bases for L2(R)?
Let E be a set of pairs (j, n) such that the correspondingfrequency intervals 2jn ≤ ξ < 2j(n+ 1) constitute a partition of[0,∞), up to a contable set. Then the subsequence
2j/2wn(2jx− k), n ∈ N j, k ∈ Z
is an orthogonal basis for L2(R).
(...filter different frequency bands and then cut these bands intemporal windows to study their energy variations...)
Introduction WP definitions Splitting algorithm More basis...
Trees and spaces
Any node of the binary tree is labeled by (j, p), where:j − L is the depth of the node in the tree, andp is the number of the node (at the same depth)
[4]
Introduction WP definitions Splitting algorithm More basis...
Admissible trees
We call admissible tree any binary tree where each node haseither 0 or 2 children.
Basic examples...
Introduction WP definitions Splitting algorithm More basis...
Alternative tiling of the time-frequency plane[R]
Introduction WP definitions Splitting algorithm More basis...
Number of WP basesThe number of different wavelet packet orthogonal bases of VLis equal to the number of different admissible binary trees...
Introduction WP definitions Splitting algorithm More basis...
Number of WP basesThe number of different wavelet packet orthogonal bases of VLis equal to the number of different admissible binary trees...
The number BJ of wavelet packet bases in a full wavelet packetbinary tree of depth J satisfies
22J−1 ≤ BJ ≤ 2
542J−1
Homework: Proof [4] 8.1.1
Introduction WP definitions Splitting algorithm More basis...
Trees and Heisenberg boxes
The wavelet packet tree and the corresponding Heisenbergboxes of a wavelet packet basis.
Introduction WP definitions Splitting algorithm More basis...
Trees and Heisenberg boxes
The wavelet packet tree and the corresponding Heisenbergboxes of a wavelet packet basis.
More examples...?• The Heisenberg boxes from the WP tree...• The WP tree from the Heisenberg boxes...
Introduction WP definitions Splitting algorithm More basis...
Time support
Suppose that for some N ∈ N, the scaling function w0(x) issupported in [0, 2N − 1] and that the scaling filter h[k] satisfiesh[k] = 0 for k < 0 and k ≥ 2N .
[2]
Introduction WP definitions Splitting algorithm More basis...
Time supportSuppose that for some N ∈ N, the scaling function w0(x) issupported in [0, 2N − 1] and that the scaling filter h[k] satisfiesh[k] = 0 for k < 0 and k ≥ 2N .
Introduction WP definitions Splitting algorithm More basis...
Time supportSuppose that for some N ∈ N, the scaling function w0(x) issupported in [0, 2N − 1] and that the scaling filter h[k] satisfiesh[k] = 0 for k < 0 and k ≥ 2N .
Then for each n ∈ N, wn(x) is supported in [0, 2N − 1].
Introduction WP definitions Splitting algorithm More basis...
Nominal frequency
|wn(ξ)| for n = 0, 1, . . . , 7 corresponding to the Daubechies12 filter.
The location of the dominant peak is referred to as the nominalfrequency of wn(x).
Introduction WP definitions Splitting algorithm More basis...
Nominal frequency
|wn(ξ)| for n = 0, 1, . . . , 7 corresponding to the Daubechies12 filter.
The location of the dominant peak is referred to as the nominalfrequency of wn(x).
Introduction WP definitions Splitting algorithm More basis...
Frequency order
Frequency ordered wavelet packets computed with theDaubechies5 filter, at the depth j − L = 3 of the wavelet packettree, with L = 0.
[4]
Introduction WP definitions Splitting algorithm More basis...
Frequency order
Frequency ordered wavelet packets computed with a Haar filter,at the depth j − L = 3 of the wavelet packet tree, with L = 0.
Introduction WP definitions Splitting algorithm More basis...
Outline
Introduction
Wavelet packetsDefinition of basic wavelet packetsGeneral wavelet packetsWavelet packet treesSupport and localization
The splitting algorithmSplitting the treeFinite signals and computational complexity
More basis...Some (fixed) wavelet packets basesLocal cosine bases
Introduction WP definitions Splitting algorithm More basis...
Decomposition
d2pj+1[k] = dpj ∗ h[−2k] d2p+1j+1 [k] = dpj ∗ g[−2k]
Homework: What is the frequency localization of signals for each node aftersubsampling?
[4]
Introduction WP definitions Splitting algorithm More basis...
Decomposition
d2pj+1[k] = dpj ∗ h[−2k] d2p+1j+1 [k] = dpj ∗ g[−2k]
Homework: What is the frequency localization of signals for each node aftersubsampling?
Introduction WP definitions Splitting algorithm More basis...
Decomposition
d2pj+1[k] = dpj ∗ h[−2k] d2p+1j+1 [k] = dpj ∗ g[−2k]
Homework: What is the frequency localization of signals for each node aftersubsampling?
Introduction WP definitions Splitting algorithm More basis...
Reconstruction
dpj [k] = d2pj+1 ∗ h[k] + d2p+1j+1 [k] ∗ g[k]
Homework: What is the frequency localization of signals for each node afterinterpolation?
Introduction WP definitions Splitting algorithm More basis...
Reconstruction
dpj [k] = d2pj+1 ∗ h[k] + d2p+1j+1 [k] ∗ g[k]
Homework: What is the frequency localization of signals for each node afterinterpolation?
Introduction WP definitions Splitting algorithm More basis...
Finite signals
d2pj+1[k] = dpj ∗ h[−2k]
d2p+1j+1 [k] = dpj ∗ g[−2k]
dpj [k] = d2pj+1 ∗ h[k] + d2p+1j+1 [k] ∗ g[k]
Standard or circular convolutions?
Introduction WP definitions Splitting algorithm More basis...
Finite signals
d2pj+1[k] = dpj ∗ h[−2k]
d2p+1j+1 [k] = dpj ∗ g[−2k]
dpj [k] = d2pj+1 ∗ h[k] + d2p+1j+1 [k] ∗ g[k]
Standard or circular convolutions?
Introduction WP definitions Splitting algorithm More basis...
Coefficients in the full tree
Introduction WP definitions Splitting algorithm More basis...
Coefficients in the full tree
For each node p in the scale j the signal dpj has 2−j samples.
Introduction WP definitions Splitting algorithm More basis...
Coefficients in the full tree
For each node p in the scale j the signal dpj has 2−j samples.
At any depth j − L of the tree, the wavelet packet signals{dpj}0≤p<2j−L include a total of N coefficients.
Introduction WP definitions Splitting algorithm More basis...
Coefficients in the full tree
For each node p in the scale j the signal dpj has 2−j samples.
At any depth j − L of the tree, the wavelet packet signals{dpj}0≤p<2j−L include a total of N coefficients.
Since the maximum depth is log2N , there are at most Nlog2Ncoefficients in a full wavelet packet tree.
Basic examples...
Introduction WP definitions Splitting algorithm More basis...
Computational complexityIn a full wavelet packet tree of depth log2N all coefficients arecomputed by iterating
d2pj+1[k] = dpj ∗ h[−2k]
d2p+1j+1 [k] = dpj ∗ g[−2k]
for L ≤ j < 0.
Homework: How many wavelet packet bases are included in this wavelet packetstree?
Introduction WP definitions Splitting algorithm More basis...
Computational complexityIn a full wavelet packet tree of depth log2N all coefficients arecomputed by iterating
d2pj+1[k] = dpj ∗ h[−2k]
d2p+1j+1 [k] = dpj ∗ g[−2k]
for L ≤ j < 0.If h and g have K non-zero coefficients, this requires about
KNlog2N
additions and multiplications.
Homework: How many wavelet packet bases are included in this wavelet packetstree?
Introduction WP definitions Splitting algorithm More basis...
Computational complexity
And for the reconstruction algorithm...
dpj [k] = d2pj+1 ∗ h[k] + d2p+1j+1 [k] ∗ g[k]
• the computational complexity in the worst case?• the computational complexity in the simplest case?• and for the reconstruction of a simple (dyadic) wavelet
tree?
Introduction WP definitions Splitting algorithm More basis...
Computational complexity
And for the reconstruction algorithm...
dpj [k] = d2pj+1 ∗ h[k] + d2p+1j+1 [k] ∗ g[k]
What is...• the computational complexity in the worst case?
• the computational complexity in the simplest case?• and for the reconstruction of a simple (dyadic) wavelet
tree?
Introduction WP definitions Splitting algorithm More basis...
Computational complexity
And for the reconstruction algorithm...
dpj [k] = d2pj+1 ∗ h[k] + d2p+1j+1 [k] ∗ g[k]
What is...• the computational complexity in the worst case?KNlog2N
• the computational complexity in the simplest case?• and for the reconstruction of a simple (dyadic) wavelet
tree?
Introduction WP definitions Splitting algorithm More basis...
Computational complexity
And for the reconstruction algorithm...
dpj [k] = d2pj+1 ∗ h[k] + d2p+1j+1 [k] ∗ g[k]
What is...• the computational complexity in the worst case?• the computational complexity in the simplest case?
• and for the reconstruction of a simple (dyadic) wavelettree?
Introduction WP definitions Splitting algorithm More basis...
Computational complexity
And for the reconstruction algorithm...
dpj [k] = d2pj+1 ∗ h[k] + d2p+1j+1 [k] ∗ g[k]
What is...• the computational complexity in the worst case?• the computational complexity in the simplest case?
2KN
• and for the reconstruction of a simple (dyadic) wavelettree?
Introduction WP definitions Splitting algorithm More basis...
Computational complexity
And for the reconstruction algorithm...
dpj [k] = d2pj+1 ∗ h[k] + d2p+1j+1 [k] ∗ g[k]
What is...• the computational complexity in the worst case?• the computational complexity in the simplest case?• and for the reconstruction of a simple (dyadic) wavelet
tree?
Introduction WP definitions Splitting algorithm More basis...
Computational complexity
And for the reconstruction algorithm...
dpj [k] = d2pj+1 ∗ h[k] + d2p+1j+1 [k] ∗ g[k]
What is...• the computational complexity in the worst case?• the computational complexity in the simplest case?• and for the reconstruction of a simple (dyadic) wavelet
tree?2KN
Introduction WP definitions Splitting algorithm More basis...
Outline
Introduction
Wavelet packetsDefinition of basic wavelet packetsGeneral wavelet packetsWavelet packet treesSupport and localization
The splitting algorithmSplitting the treeFinite signals and computational complexity
More basis...Some (fixed) wavelet packets basesLocal cosine bases
Introduction WP definitions Splitting algorithm More basis...
sinc wavelet packetsinc wavelet packets are computed with perfect discretelow-pass and high-pass filters
|h(ξ)| ={ √
2 if ξ ∈ [−π/2 + 2kπ, π/2 + 2kπ]0 otherwise
|g(ξ)| ={ √
2 if ξ ∈ [π/2 + 2kπ, 3π/2 + 2kπ]0 otherwise
therefore...
Introduction WP definitions Splitting algorithm More basis...
sinc wavelet packetsinc wavelet packets are computed with perfect discretelow-pass and high-pass filters
|h(ξ)| ={ √
2 if ξ ∈ [−π/2 + 2kπ, π/2 + 2kπ]0 otherwise
|g(ξ)| ={ √
2 if ξ ∈ [π/2 + 2kπ, 3π/2 + 2kπ]0 otherwise
therefore...
ψpj (t) = 2−j/2+1 sin(2−jπt/2)
2−jπtcos(2−jπ(k + 1/2)(t− τj,p))
Introduction WP definitions Splitting algorithm More basis...
Walsh wavelet packetWalsh wavelet packets are generated by the Haar conjugatemirror filter
h[n] =
{1√2
if n = 0, 1
0 otherwise
Introduction WP definitions Splitting algorithm More basis...
(pseudo) local cosine wavelet packet
{ψkJ(t− 2Jn)
}n∈Z,0≤k<2J−L
Introduction WP definitions Splitting algorithm More basis...
(pseudo) local cosine wavelet packet
{ψkJ(t− 2Jn)
}n∈Z,0≤k<2J−L
Introduction WP definitions Splitting algorithm More basis...
mel wavelet packet0 -
2 5 0
2 5 0 -
5 0 0
5 0 0 -
6 2 5
6 2 5 -
7 5 0
7 5 0 -
8 7 5
8 7 5 -
1 0 0 0
1 0
0 0 - 1
1 2 5
1 1 2 5
- 1 2 5
0 1 2
5 0 - 1
3 7 5
1 3 7 5
- 1 5 0
0 1 5
0 0 - 1
6 2 5
1 6 2 5
- 1 7 5
0 1 7
5 0 - 1
8 7 5
1 8 7 5
- 2 0 0
0 2 0
0 0 - 2
2 5 0
2 2 5 0
- 2 5 0
0
2 5 0 0
- 3 0 0
0
3 0 0 0
- 3 5 0
0
3 5 0 0
- 4 0 0
0
4 0 0 0
- 5 0 0
0
5 0 0 0
- 6 0 0
0
6 0 0 0
- 7 0 0
0
7 0 0 0
- 8 0 0
0
Introduction WP definitions Splitting algorithm More basis...
Block transforms
Let g = 1[0,1] and the interval [ap, ap+1] covered by
gp(t) = 1[ap,ap+1](t) = g
(t− ap`p
)with `p = ap+1 − ap.
If {ek}k∈Z is an orthonormal basis of L2[0, 1], then{gp,k(t) = gp(t)
1√`pek
(t− ap`p
)}(p,k)∈Z
is a block orthonormal basis of L2(R).
Introduction WP definitions Splitting algorithm More basis...
Block transforms
Let g = 1[0,1] and the interval [ap, ap+1] covered by
gp(t) = 1[ap,ap+1](t) = g
(t− ap`p
)with `p = ap+1 − ap.
If {ek}k∈Z is an orthonormal basis of L2[0, 1], then{gp,k(t) = gp(t)
1√`pek
(t− ap`p
)}(p,k)∈Z
is a block orthonormal basis of L2(R).
Introduction WP definitions Splitting algorithm More basis...
Discrete block bases
Given a discrete rectangular window
gp[n] = 1[ap,ap+1−1](n)
If {ek,`}0≤k<` is an orthonormal basis of C` for any ` > 0, thenthe family {
gp,k[n] = gp[n]ek,`p [n− ap]}0≤k<`p,p∈Z
is a block orthonormal basis of `2(Z).
Introduction WP definitions Splitting algorithm More basis...
Discrete block bases
Given a discrete rectangular window
gp[n] = 1[ap,ap+1−1](n)
If {ek,`}0≤k<` is an orthonormal basis of C` for any ` > 0, thenthe family {
gp,k[n] = gp[n]ek,`p [n− ap]}0≤k<`p,p∈Z
is a block orthonormal basis of `2(Z).
Introduction WP definitions Splitting algorithm More basis...
Discrete block cosine transform
Given the discrete windows gp[n] = 1[ap,ap−1](n) to divide Z, theblock basis
{gp,k[n] = gp[n]λk
√2
`pcos
[kπ
`p
(n+
1
2− ap
)]}0≤k<N,p∈Z
is an orthonormal basis of `2(Z).
(The standard for JPEG image compression)
Introduction WP definitions Splitting algorithm More basis...
Discrete block cosine transform
Given the discrete windows gp[n] = 1[ap,ap−1](n) to divide Z, theblock basis
{gp,k[n] = gp[n]λk
√2
`pcos
[kπ
`p
(n+
1
2− ap
)]}0≤k<N,p∈Z
is an orthonormal basis of `2(Z).
(The standard for JPEG image compression)
Introduction WP definitions Splitting algorithm More basis...
Lapped orthogonal transforms
Introduction WP definitions Splitting algorithm More basis...
Lapped orthogonal transforms
Introduction WP definitions Splitting algorithm More basis...
Lapped orthogonal bases
If {ek}k∈N is an orthonormal basis of L2[0, 1], and ek(t) is anextension over R, symmetric with repect to 0 andantisymmetric with respecto to 1,
then
{gp,k(t) = gp(t)
1√`pek
(t− ap`p
)}k∈N,p∈Z
is an orthonormal basis of L2(R).
Introduction WP definitions Splitting algorithm More basis...
Lapped orthogonal bases
If {ek}k∈N is an orthonormal basis of L2[0, 1], and ek(t) is anextension over R, symmetric with repect to 0 andantisymmetric with respecto to 1, then
{gp,k(t) = gp(t)
1√`pek
(t− ap`p
)}k∈N,p∈Z
is an orthonormal basis of L2(R).
Introduction WP definitions Splitting algorithm More basis...
Local cosine basis
The family of local cosine functions
{gp,k(t) = gp(t)
√2
`pcos
[π
(k +
1
2
)t− ap`p
]}k∈N,p∈Z
is an orthonormal basis of L2(R).
Introduction WP definitions Splitting algorithm More basis...
Local cosine vectors in time-frequency
Introduction WP definitions Splitting algorithm More basis...
Local cosine trees
Admissible tree of (dyadic) local cosine
Introduction WP definitions Splitting algorithm More basis...
It can be automatically found the wavelet packet basis that“best fits” or is “best adapted to” a given signal?
DWT WP
Introduction WP definitions Splitting algorithm More basis...
It can be automatically found the wavelet packet basis that“best fits” or is “best adapted to” a given signal?
• Single tree algorithm• Double tree algorithm (Frame-adaptive WP)• Wavelet packet and local cosine best basis• Basis pursuit• Matching pursuit• Best orthogonal basis• Method of frames• ...• ...• ...
Introduction WP definitions Splitting algorithm More basis...
Bibliography
[1] Meyer Y., Wavelets, algorithms and applications, SIAM,1993.
[2] Walnut D., An introduction to wavelet analysis,Birkhauser, 2002.
[3] Ramchandran K., Vetterli M., Herley C., “Wavelets,subband coding, and best bases,” Proceedings of the IEEE,vol.84, no.4, 1996.
[4] Mallat S., A wavelet tour of signal processing, AcademicPress, 1999.
[5] Addison P., The Illustrated Wavelet Transform Handbook,IOP, 2002.
[6] Hess-Nielsen N., Wickerhauser M.V., Wavelets andtime-frequency analysis, Proceedings of the IEEE, vol.84,no.4, 1996.
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