Embed Size (px)
Citation preview
Empirical Wavelet Transform
Jérôme Gilles
Department of Mathematics, [email protected]
Adaptive Data Analysis and Sparsity WorkshopJanuary 31th, 2013
Empirical Wavelet Transform
Outline
Introduction - EMD1D Empirical Wavelets
DefinitionExperiments
2D Extensions
Tensor product caseRidgelet caseExperiments
Empirical Wavelet Transform
Time-frequency signal analysis
Empirical Wavelet Transform
Time-Frequency representations are useful to analyze signals.
Short-time Fourier transform:FW
f (m,n) =∫
f (s)g(s − nt0)e−ımω0sds.Wavelet transform:WT f (m,n) = a−m/2
0
∫f (t)ψ(a−m
0 t − nb0)dt .Wigner-Ville transform (quadratic→ nonlinear + interferenceterms).Hilbert-Huang transform (EMD + Hilbert transform)
Time-frequency signal analysis
Empirical Wavelet Transform
Time-Frequency representations are useful to analyze signals.
Short-time Fourier transform:FW
f (m,n) =∫
f (s)g(s − nt0)e−ımω0sds.
Wavelet transform:WT f (m,n) = a−m/2
0
∫f (t)ψ(a−m
0 t − nb0)dt .Wigner-Ville transform (quadratic→ nonlinear + interferenceterms).Hilbert-Huang transform (EMD + Hilbert transform)
Time-frequency signal analysis
Empirical Wavelet Transform
Time-Frequency representations are useful to analyze signals.
Short-time Fourier transform:FW
f (m,n) =∫
f (s)g(s − nt0)e−ımω0sds.Wavelet transform:WT f (m,n) = a−m/2
0
∫f (t)ψ(a−m
0 t − nb0)dt .
Wigner-Ville transform (quadratic→ nonlinear + interferenceterms).Hilbert-Huang transform (EMD + Hilbert transform)
Time-frequency signal analysis
Empirical Wavelet Transform
Time-Frequency representations are useful to analyze signals.
Short-time Fourier transform:FW
f (m,n) =∫
f (s)g(s − nt0)e−ımω0sds.Wavelet transform:WT f (m,n) = a−m/2
0
∫f (t)ψ(a−m
0 t − nb0)dt .Wigner-Ville transform (quadratic→ nonlinear + interferenceterms).
Hilbert-Huang transform (EMD + Hilbert transform)
Time-frequency signal analysis
Empirical Wavelet Transform
Time-Frequency representations are useful to analyze signals.
Short-time Fourier transform:FW
f (m,n) =∫
f (s)g(s − nt0)e−ımω0sds.Wavelet transform:WT f (m,n) = a−m/2
0
∫f (t)ψ(a−m
0 t − nb0)dt .Wigner-Ville transform (quadratic→ nonlinear + interferenceterms).Hilbert-Huang transform (EMD + Hilbert transform)
Empirical Mode Decomposition (EMD): Principle
1The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series
analysis, Proc. Royal Society London A., vol.454, pp.903–995,1998
Empirical Wavelet Transform
Goal: decompose a signal f (t) into a finite sum of Intrinsic ModeFunctions (IMF) fk (t):
f (t) =N∑
k=0
fk (t)
where an IMF is an AM-FM signal:
fk (t) = Fk (t) cos (ϕk (t)) where Fk (t), ϕ′k (t) > 0 ∀t .
Main assumption: Fk and ϕ′k vary much slower than ϕk .
Huang et al.1 propose a pure algorithmic method to extract thedifferent IMF.
Empirical Mode Decomposition (EMD): Algorithm
Empirical Wavelet Transform
Initialization: f 0 = fwhile all IMF are no extracted do
r k0 = f k
while r kn is not an IMF (Sifting process) do
Upper envelope u(t) (maxima + spline) of r kn (t)
Lower envelope l(t) (minima + spline) of r kn (t)
Mean envelope m(t) = (u(t) + l(t))/2IMF candidate r k
n+1(t) = r kn (t)−m(t)
end whilef k+1 = f k − r k
n+1end while
0.2 0.4 0.6 0.8 1.0
-2
2
4
6
Empirical Mode Decomposition (EMD): Algorithm
Empirical Wavelet Transform
Initialization: f 0 = fwhile all IMF are no extracted do
r k0 = f k
while r kn is not an IMF (Sifting process) do
Upper envelope u(t) (maxima + spline) of r kn (t)
Lower envelope l(t) (minima + spline) of r kn (t)
Mean envelope m(t) = (u(t) + l(t))/2IMF candidate r k
n+1(t) = r kn (t)−m(t)
end whilef k+1 = f k − r k
n+1end while
0.2 0.4 0.6 0.8 1.0
-2
2
4
6
Empirical Mode Decomposition (EMD): Algorithm
Empirical Wavelet Transform
Initialization: f 0 = fwhile all IMF are no extracted do
r k0 = f k
while r kn is not an IMF (Sifting process) do
Upper envelope u(t) (maxima + spline) of r kn (t)
Lower envelope l(t) (minima + spline) of r kn (t)
Mean envelope m(t) = (u(t) + l(t))/2IMF candidate r k
n+1(t) = r kn (t)−m(t)
end whilef k+1 = f k − r k
n+1end while
0.2 0.4 0.6 0.8 1.0
-2
2
4
6
Empirical Mode Decomposition (EMD): Algorithm
Empirical Wavelet Transform
Initialization: f 0 = fwhile all IMF are no extracted do
r k0 = f k
while r kn is not an IMF (Sifting process) do
Upper envelope u(t) (maxima + spline) of r kn (t)
Lower envelope l(t) (minima + spline) of r kn (t)
Mean envelope m(t) = (u(t) + l(t))/2IMF candidate r k
n+1(t) = r kn (t)−m(t)
end whilef k+1 = f k − r k
n+1end while
0.2 0.4 0.6 0.8 1.0
-2
2
4
6
Example of EMD: input signals
Empirical Wavelet Transform
f Sig
1(t
)=
6t+
cos(
8πt)
+0.
5co
s(40π
t) 0.2 0.4 0.6 0.8 1.0
2
4
6
0.2 0.4 0.6 0.8 1.0
1
2
3
4
5
6
0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.5
1.0
0.2 0.4 0.6 0.8 1.0
-0.4
-0.2
0.2
0.4
f Sig
2(t
)=
6t2
+co
s(10π
t+
10π
t2)
+χ
(t>
0.5)
cos(
80π
t−
15π
)+χ
(t≤
0.5)
cos(
60π
t)
0.2 0.4 0.6 0.8 1.0
-2
2
4
6
0.2 0.4 0.6 0.8 1.0
1
2
3
4
5
6
0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.5
1.0
0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.5
1.0
f Sig
3(t
)=
11.
2+co
s(2π
t)+
11.
5+si
n(2π
t)co
s(32π
t+
cos(
64π
t))
0.2 0.4 0.6 0.8 1.0-1
1
2
3
4
5
0.2 0.4 0.6 0.8 1.0
1
2
3
4
5
0.2 0.4 0.6 0.8 1.0
-2
-1
1
2
Example of EMD: fSig1
Empirical Wavelet Transform
0.2 0.4 0.6 0.8 1.0
2
4
6
0.2 0.4 0.6 0.8 1.0
1
2
3
4
5
6
0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.5
1.0
0.2 0.4 0.6 0.8 1.0
-0.4
-0.2
0.2
0.4
0.2 0.4 0.6 0.8
234567
0.2 0.4 0.6 0.8-1.0-0.5
0.51.01.52.02.5
0.2 0.4 0.6 0.8
-1.0
-0.5
0.5
1.0
0.2 0.4 0.6 0.8
-1.0-0.5
0.51.0
0.2 0.4 0.6 0.8
-0.6-0.4-0.2
0.20.40.6
Example of EMD: fSig2
Empirical Wavelet Transform
0.2 0.4 0.6 0.8 1.0
-2
2
4
6
0.2 0.4 0.6 0.8 1.0
1
2
3
4
5
6
0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.5
1.0
0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.5
1.0
0.2 0.4 0.6 0.8-1
123456
0.2 0.4 0.6 0.8-0.5
0.5
1.0
0.2 0.4 0.6 0.8
-1.0
-0.5
0.5
1.0
0.2 0.4 0.6 0.8
-1.0-0.5
0.51.01.5
0.2 0.4 0.6 0.8
-2
-1
1
2
0.2 0.4 0.6 0.8
-1.5-1.0-0.5
0.51.01.5
0.2 0.4 0.6 0.8
-1.5-1.0-0.5
0.51.01.52.0
0.2 0.4 0.6 0.8
-1.0
-0.5
0.5
1.0
Example of EMD: fSig3
Empirical Wavelet Transform
0.2 0.4 0.6 0.8 1.0-1
1
2
3
4
5
0.2 0.4 0.6 0.8 1.0
1
2
3
4
5
0.2 0.4 0.6 0.8 1.0
-2
-1
1
2
0.2 0.4 0.6 0.8
0.900.951.001.051.10
0.2 0.4 0.6 0.8
-1.0
-0.5
0.5
1.0
0.2 0.4 0.6 0.8
-1.0-0.5
0.51.01.5
0.2 0.4 0.6 0.8-0.5
0.5
1.0
0.2 0.4 0.6 0.8-0.5
0.5
1.0
1.5
0.2 0.4 0.6 0.8
-1.0
-0.5
0.5
1.0
0.2 0.4 0.6 0.8
-2
-1
1
2
Example of EMD: fSig4 - ECG
Empirical Wavelet Transform
1000 2000 3000 4000
-0.2
0.2
0.4
0.6
Hilbert-Huang Transform
Empirical Wavelet Transform
Hilbert transform
Hf (t) =1π
p.v .∫ +∞
−∞
f (τ)t − τ dτ
Property: if fk (t) = Fk (t) cos (ϕk (t)) then
f ∗k (t) = fk (t) + ıHfk (t) = Fk (t)eıϕk (t)
⇒ we can extract Fk (t) and the instantaneous frequency dϕkdt (t).
Hilbert-Huang Transform
Empirical Wavelet Transform
Hilbert transform
Hf (t) =1π
p.v .∫ +∞
−∞
f (τ)t − τ dτ
Property: if fk (t) = Fk (t) cos (ϕk (t)) then
f ∗k (t) = fk (t) + ıHfk (t) = Fk (t)eıϕk (t)
⇒ we can extract Fk (t) and the instantaneous frequency dϕkdt (t).
HHT
For each IMF k , we extract Fk and dϕkdt (t) and accumulate the
information in the time-frequency plane.
HHT of fsig2
Empirical Wavelet Transform
freq
uenc
y (r
ad/s
)
time0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0246
HHT of fsig4 - ECG
Empirical Wavelet Transform
freq
uenc
y (r
ad/s
)
time0 500 1000 1500 2000 2500 3000 3500 4000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 500 1000 1500 2000 2500 3000 3500 4000−0.2
00.20.40.6
EMD: Issues and Properties
Useful to analyze real signals.Implementation dependent.Problem: it’s a nonlinear algorithm which has nomathematical theory⇒ difficult to predict and understandits output and behavior in the general case.Experimental property: seems to behave as an adaptivefilter bank (Flandrin et al.2)
2Empirical mode decomposition as a filter bank, IEEE Signal Processing Letters, vol.11, No.2, pp.112–114,
2004
Empirical Wavelet Transform
Key ideas about wavelets
Empirical Wavelet Transform
Wavelets⇔ filtering
WT f (m,n) = a−m/20
∫f (t)ψ(a−m
0 t − nb0)dt
= a−m/20
∫f (t)ψ
(t − nam
0 b0
am0
)dt
= (f ? ψm)(nam0 b0)
where ψm(s) = ψ
(−s
a−m0
).
Key ideas about wavelets
Empirical Wavelet Transform
Wavelets⇔ filtering
WT f (m,n) = a−m/20
∫f (t)ψ(a−m
0 t − nb0)dt
= a−m/20
∫f (t)ψ
(t − nam
0 b0
am0
)dt
= (f ? ψm)(nam0 b0)
where ψm(s) = ψ
(−s
a−m0
).
⇒Wavelets can be built both in the temporal or Fourier domains.
Empirical wavelet transform (EWT): Concept
Empirical Wavelet Transform
Idea:Combining the strength of wavelet’s formalism with the adaptability of EMD.
Empirical wavelet transform (EWT): Concept
Empirical Wavelet Transform
Idea:Combining the strength of wavelet’s formalism with the adaptability of EMD.
Wavelets are equivalent to filter banks→ (dyadic) decomposition of the Fourier line
ωππ/2π/4π/8. . .
Empirical wavelet transform (EWT): Concept
Empirical Wavelet Transform
Idea:Combining the strength of wavelet’s formalism with the adaptability of EMD.
Wavelets are equivalent to filter banks→ (dyadic) decomposition of the Fourier line
ωππ/2π/4π/8. . .
Does not necessarily correspond to “modes” positions.
Empirical wavelet transform (EWT): Concept
Empirical Wavelet Transform
Idea:Combining the strength of wavelet’s formalism with the adaptability of EMD.
Wavelets are equivalent to filter banks→ (dyadic) decomposition of the Fourier line
ωππ/2π/4π/8. . .
Does not necessarily correspond to “modes” positions.
EWT→ adaptive decomposition of the Fourier line
πω1 ω2 ω3 ωn ωn+1oo oo
EWT: finding the modes
Empirical Wavelet Transform
Fourier spectrum segmentation:Find the local maxima.Take support boundaries as the middle between successivemaxima.
fsig1 0.05 0.10 0.15
1000
2000
3000
4000
5000
6000
EWT: finding the modes
Empirical Wavelet Transform
Fourier spectrum segmentation:Find the local maxima.Take support boundaries as the middle between successivemaxima.
fsig1 0.05 0.10 0.15
1000
2000
3000
4000
5000
6000
EWT: finding the modes
Empirical Wavelet Transform
Fourier spectrum segmentation:Find the local maxima.Take support boundaries as the middle between successivemaxima.
fsig2 0.2 0.4 0.6 0.8
1000
2000
3000
4000
EWT: filter bank construction (1/3)
Empirical Wavelet Transform
Notationsωn: support boundariesτn: half the length of the “transition phase”
πω1 ω2 ω3 ωn ωn+1oo
2τ1 2τ2 2τ3 2τn 2τn+1 τN
1
oo
EWT: filter bank construction (1/3)
Empirical Wavelet Transform
Notationsωn: support boundariesτn: half the length of the “transition phase”
πω1 ω2 ω3 ωn ωn+1oo
2τ1 2τ2 2τ3 2τn 2τn+1 τN
1
oo
In practice we choose τn = γωn
EWT: filter bank construction (2/3)
Empirical Wavelet Transform
Scaling function spectrum
φn(ω) =
1 if |ω| ≤ (1− γ)ωn
cos[π2 β(
12γωn
(|ω| − (1− γ)ωn))]
if (1− γ)ωn ≤ |ω| ≤ (1 + γ)ωn
0 otherwise
Wavelet spectrum
ψn(ω) =
1 if (1 + γ)ωn ≤ |ω| ≤ (1− γ)ωn+1
e−ıω2 cos[π2 β(
12γωn+1
(|ω| − (1− γ)ωn+1))]
if (1− γ)ωn+1 ≤ |ω| ≤ (1 + γ)ωn+1
e−ıω2 sin[π2 β(
12γωn
(|ω| − (1− γ)ωn))]
if (1− γ)ωn ≤ |ω| ≤ (1 + γ)ωn
0 otherwise
EWT: filter bank construction (3/3)
Empirical Wavelet Transform
Scaling function spectrum for ωn = 1 and γ = 0.5
-Π -Π
2-1 0 1
Π
2Π
0.2
0.4
0.6
0.8
1.0
Wavelet spectrum for ωn = 1, ωn+1 = 2.5 and γ = 0.2
-Π -Π
2-1 0 1
Π
2Π
0.2
0.4
0.6
0.8
1.0
EWT: property and example (1/2)
Empirical Wavelet Transform
Proposition
If γ < minn
(ωn+1−ωnωn+1+ωn
), then the set {φ1(t), {ψn(t)}Nn=1} is an
orthonormal basis of L2(R).
EWT: property and example (1/2)
Empirical Wavelet Transform
Proposition
If γ < minn
(ωn+1−ωnωn+1+ωn
), then the set {φ1(t), {ψn(t)}Nn=1} is an
orthonormal basis of L2(R).
Filter Bank for ωn ∈ {0,1.5,2,2.8, π} with γ = 0.05 < 0.057
-3 -2 -1 1 2 3
0.2
0.4
0.6
0.8
1.0
EWT: property and example (2/2)Detail coefficients:
WEf (n, t) = 〈f , ψn〉 =
∫f (τ)ψn(τ − t)dτ
=(
f (ω)ψn(ω))∨
,
Approximation coefficients (conventionWEf (0, t):
WEf (0, t) = 〈f , φ1〉 =
∫f (τ)φ1(τ − t)dτ
=(
f (ω)φ1(ω))∨
,
The reconstruction:
f (t) =WEf (0, t) ? φ1(t) +
N∑n=1
WEf (n, t) ? ψn(t)
=
(WE
f (0, ω)φ1(ω) +N∑
n=1
WEf (n, ω)ψn(ω)
)∨
.
Empirical Wavelet Transform
EWT: algorithm
Input: f , N (number of scales)1 Fourier transform of f → f .2 Compute the local maxima of f on [0, π] and find the set{ωn}.
3 Choose γ < minn
(ωn+1−ωnωn+1+ωn
).
4 Build the filter bank.5 Filter the signal to get each component.
Empirical Wavelet Transform
Experiment: fSig1
Empirical Wavelet Transform
0.2 0.4 0.6 0.8 1.0
2
4
6
0.2 0.4 0.6 0.8 1.0
1
2
3
4
5
6
0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.5
1.0
0.2 0.4 0.6 0.8 1.0
-0.4
-0.2
0.2
0.4
0.2 0.4 0.6 0.8 1.0
123456
0.2 0.4 0.6 0.8 1.0
-1.0-0.5
0.51.0
0.2 0.4 0.6 0.8 1.0
-0.4-0.2
0.20.40.6
Experiment of EMD: fSig2
Empirical Wavelet Transform
0.2 0.4 0.6 0.8 1.0
-2
2
4
6
0.2 0.4 0.6 0.8 1.0
1
2
3
4
5
6
0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.5
1.0
0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.5
1.0
0.2 0.4 0.6 0.8 1.0
123456
0.2 0.4 0.6 0.8 1.0
-1.0-0.5
0.51.0
0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.5
1.0
0.2 0.4 0.6 0.8 1.0
-1.5-1.0-0.5
0.51.0
Experiment of EMD: fSig3
Empirical Wavelet Transform
0.2 0.4 0.6 0.8 1.0-1
1
2
3
4
5
0.2 0.4 0.6 0.8 1.0
1
2
3
4
5
0.2 0.4 0.6 0.8 1.0
-2
-1
1
2
0.2 0.4 0.6 0.8 1.0
0.5
1.0
1.5
0.2 0.4 0.6 0.8 1.0-1
1
2
3
0.2 0.4 0.6 0.8 1.0
-2
-1
1
2
Experiment of EMD: ECG
Empirical Wavelet Transform
1000 2000 3000 4000
-0.2
0.2
0.4
0.6
0.02 0.04 0.06 0.08 0.10 0.12 0.14
50
100
150
200
250
0.030.05
-0.15
-0.05
0.05
0.15
-0.05
0.05
-0.05
0.05
-0.06
-0.02
0.02
0.06
-0.1
0.1
0.4
Time-Frequency representation of fsig2
Empirical Wavelet Transform
freq
uenc
y (r
ad/s
)
time0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0246
Time-Frequency representation of fsig4
Empirical Wavelet Transform
freq
uenc
y (r
ad/s
)
time0 500 1000 1500 2000 2500 3000 3500 4000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 500 1000 1500 2000 2500 3000 3500 4000−0.2
00.20.40.6
2D - Extension
joint work with Giang Tran and Stan Osher
Empirical Wavelet Transform
2D Extension - “Tensor product” approach
Empirical Wavelet Transform
Like the “classic” wavelet transform→ process rows then columns
but. . .
2D Extension - “Tensor product” approach
Empirical Wavelet Transform
Like the “classic” wavelet transform→ process rows then columns
but. . .
The number of detected scales can be different for each row
The position of the Fourier boundaries can vary a lot from onerow to the next (⇔ “information discontinuity”)
2D Extension - “Tensor product” approach
Empirical Wavelet Transform
Like the “classic” wavelet transform→ process rows then columns
but. . .
The number of detected scales can be different for each rowThe position of the Fourier boundaries can vary a lot from onerow to the next (⇔ “information discontinuity”)
2D Extension - “Tensor product” approach
Empirical Wavelet Transform
Like the “classic” wavelet transform→ process rows then columns
but. . .
The number of detected scales can be different for each rowThe position of the Fourier boundaries can vary a lot from onerow to the next (⇔ “information discontinuity”)
⇒ Idea: “Mean Filter Banks”
2D Extension - Tensor product algorithm
Empirical Wavelet Transform
2D Extension - Tensor product algorithm
Empirical Wavelet Transform
1DFFT−−−−→
2D Extension - Tensor product algorithm
Empirical Wavelet Transform
1DFFT−−−−→ Average−−−−−→50 100 150 200 250 300 350 400 450 500
1
2
3
4
5
6x 10
4
2D Extension - Tensor product algorithm
Empirical Wavelet Transform
1DFFT−−−−→ Average−−−−−→50 100 150 200 250 300 350 400 450 500
1
2
3
4
5
6x 10
4
→ MFBR
2D Extension - Tensor product algorithm
Empirical Wavelet Transform
1DFFT−−−−→ Average−−−−−→50 100 150 200 250 300 350 400 450 500
1
2
3
4
5
6x 10
4
→ MFBR
↓ 1DFFT
↓ Average
50 100 150 200 250 300 350 400 450 500
1
2
3
4
5
6x 10
4
↗
MFB
C
2D Extension - Tensor product algorithm
Empirical Wavelet Transform
1DFFT−−−−→ Average−−−−−→50 100 150 200 250 300 350 400 450 500
1
2
3
4
5
6x 10
4
→ MFBR
↓ 1DFFT
↓ Average
50 100 150 200 250 300 350 400 450 500
1
2
3
4
5
6x 10
4
↗
MFB
C
MFBR
MFBC MFBC MFBC. . .
. . .
... ... ...
2D Extension - Example
Empirical Wavelet Transform
2D Extension - Ridgelet approach
Classic Ridgelets
ω t
fθ
FP W1,t
WRf
F∗1,ω
θ
(j, t)
θ
ωt
θ
W∗1,t F1,ω
θ
(j, t)
θ
F∗P f
WRf
Empirical Wavelet Transform
2D Extension - Ridgelet approach
Empirical Ridgelets
ω
fθ
FP WE
WERf
ω
θ
t
θ
WE∗
θ
ω
F∗P f
WERf
averageθ
mfb
ω
Empirical Wavelet Transform
2D Extension - Ridgelet: a first example
Empirical Wavelet Transform
2D Extension - Ridgelet: a first example
Empirical Wavelet Transform
0-1-2-3-45-6-7-8-9
10-11-12-13
2D Extension - Ridgelet: a first example
Empirical Wavelet Transform
2D Extension - Ridgelet: a first example
Empirical Wavelet Transform
2D Extension - Ridgelet: a noisy example
Empirical Wavelet Transform
2D Extension - Ridgelet: a noisy example
Empirical Wavelet Transform
0-1-2-3-45-6-7-8-9
10-11-12-13
2D Extension - Ridgelet: a noisy example
Empirical Wavelet Transform
Conclusion - Future work
Empirical Wavelet Transform
ConclusionGet the ability of EMD under the Wavelet theory.
Fast algorithm.
Adaptive wavelets is not a completely new idea: Malvar-Wilson wavelets, Brushlets.
Future work
Investigate different strategies to “segment” the spectrum.
Possibility of using a “best-basis” pursuit approach to find the optimal number ofsubbands.
Generalization to any kind of Fourier based wavelets (e.g. Splines).
2D (nD) extension: finish ridgelet idea, curvelet, “true” spectrum segmentation.
Explore the applications (denoising, deconvolution, . . . ).
Solve PDEs (cf. Stan’s talk).
Impact on existing functional spaces (Sobolev, Besov, Triebel-Lizorkin, . . . ); adaptiveharmonic analysis.
. . .
Conclusion - Future work
Empirical Wavelet Transform
ConclusionGet the ability of EMD under the Wavelet theory.
Fast algorithm.
Adaptive wavelets is not a completely new idea: Malvar-Wilson wavelets, Brushlets.
Future workInvestigate different strategies to “segment” the spectrum.
Possibility of using a “best-basis” pursuit approach to find the optimal number ofsubbands.
Generalization to any kind of Fourier based wavelets (e.g. Splines).
2D (nD) extension: finish ridgelet idea, curvelet, “true” spectrum segmentation.
Explore the applications (denoising, deconvolution, . . . ).
Solve PDEs (cf. Stan’s talk).
Impact on existing functional spaces (Sobolev, Besov, Triebel-Lizorkin, . . . ); adaptiveharmonic analysis.
. . .
Conclusion - Future work
Empirical Wavelet Transform
ConclusionGet the ability of EMD under the Wavelet theory.
Fast algorithm.
Adaptive wavelets is not a completely new idea: Malvar-Wilson wavelets, Brushlets.
Future workInvestigate different strategies to “segment” the spectrum.
Possibility of using a “best-basis” pursuit approach to find the optimal number ofsubbands.
Generalization to any kind of Fourier based wavelets (e.g. Splines).
2D (nD) extension: finish ridgelet idea, curvelet, “true” spectrum segmentation.
Explore the applications (denoising, deconvolution, . . . ).
Solve PDEs (cf. Stan’s talk).
Impact on existing functional spaces (Sobolev, Besov, Triebel-Lizorkin, . . . ); adaptiveharmonic analysis.
. . .
Conclusion - Future work
Empirical Wavelet Transform
ConclusionGet the ability of EMD under the Wavelet theory.
Fast algorithm.
Adaptive wavelets is not a completely new idea: Malvar-Wilson wavelets, Brushlets.
Future workInvestigate different strategies to “segment” the spectrum.
Possibility of using a “best-basis” pursuit approach to find the optimal number ofsubbands.
Generalization to any kind of Fourier based wavelets (e.g. Splines).
2D (nD) extension: finish ridgelet idea, curvelet, “true” spectrum segmentation.
Explore the applications (denoising, deconvolution, . . . ).
Solve PDEs (cf. Stan’s talk).
Impact on existing functional spaces (Sobolev, Besov, Triebel-Lizorkin, . . . ); adaptiveharmonic analysis.
. . .
Conclusion - Future work
Empirical Wavelet Transform
ConclusionGet the ability of EMD under the Wavelet theory.
Fast algorithm.
Adaptive wavelets is not a completely new idea: Malvar-Wilson wavelets, Brushlets.
Future workInvestigate different strategies to “segment” the spectrum.
Possibility of using a “best-basis” pursuit approach to find the optimal number ofsubbands.
Generalization to any kind of Fourier based wavelets (e.g. Splines).
2D (nD) extension: finish ridgelet idea, curvelet, “true” spectrum segmentation.
Explore the applications (denoising, deconvolution, . . . ).
Solve PDEs (cf. Stan’s talk).
Impact on existing functional spaces (Sobolev, Besov, Triebel-Lizorkin, . . . ); adaptiveharmonic analysis.
. . .
Conclusion - Future work
Empirical Wavelet Transform
ConclusionGet the ability of EMD under the Wavelet theory.
Fast algorithm.
Adaptive wavelets is not a completely new idea: Malvar-Wilson wavelets, Brushlets.
Future workInvestigate different strategies to “segment” the spectrum.
Possibility of using a “best-basis” pursuit approach to find the optimal number ofsubbands.
Generalization to any kind of Fourier based wavelets (e.g. Splines).
2D (nD) extension: finish ridgelet idea, curvelet, “true” spectrum segmentation.
Explore the applications (denoising, deconvolution, . . . ).
Solve PDEs (cf. Stan’s talk).
Impact on existing functional spaces (Sobolev, Besov, Triebel-Lizorkin, . . . ); adaptiveharmonic analysis.
. . .
Conclusion - Future work
Empirical Wavelet Transform
ConclusionGet the ability of EMD under the Wavelet theory.
Fast algorithm.
Adaptive wavelets is not a completely new idea: Malvar-Wilson wavelets, Brushlets.
Future workInvestigate different strategies to “segment” the spectrum.
Possibility of using a “best-basis” pursuit approach to find the optimal number ofsubbands.
Generalization to any kind of Fourier based wavelets (e.g. Splines).
2D (nD) extension: finish ridgelet idea, curvelet, “true” spectrum segmentation.
Explore the applications (denoising, deconvolution, . . . ).
Solve PDEs (cf. Stan’s talk).
Impact on existing functional spaces (Sobolev, Besov, Triebel-Lizorkin, . . . ); adaptiveharmonic analysis.
. . .
Conclusion - Future work
Empirical Wavelet Transform
ConclusionGet the ability of EMD under the Wavelet theory.
Fast algorithm.
Adaptive wavelets is not a completely new idea: Malvar-Wilson wavelets, Brushlets.
Future workInvestigate different strategies to “segment” the spectrum.
Possibility of using a “best-basis” pursuit approach to find the optimal number ofsubbands.
Generalization to any kind of Fourier based wavelets (e.g. Splines).
2D (nD) extension: finish ridgelet idea, curvelet, “true” spectrum segmentation.
Explore the applications (denoising, deconvolution, . . . ).
Solve PDEs (cf. Stan’s talk).
Impact on existing functional spaces (Sobolev, Besov, Triebel-Lizorkin, . . . ); adaptiveharmonic analysis.
. . .
Conclusion - Future work
Empirical Wavelet Transform
ConclusionGet the ability of EMD under the Wavelet theory.
Fast algorithm.
Adaptive wavelets is not a completely new idea: Malvar-Wilson wavelets, Brushlets.
Future workInvestigate different strategies to “segment” the spectrum.
Possibility of using a “best-basis” pursuit approach to find the optimal number ofsubbands.
Generalization to any kind of Fourier based wavelets (e.g. Splines).
2D (nD) extension: finish ridgelet idea, curvelet, “true” spectrum segmentation.
Explore the applications (denoising, deconvolution, . . . ).
Solve PDEs (cf. Stan’s talk).
Impact on existing functional spaces (Sobolev, Besov, Triebel-Lizorkin, . . . ); adaptiveharmonic analysis.
. . .
THANK YOU!
Empirical Wavelet Transform
PS: Jack, I’m from UCLA and on the job market ;-)
