Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Adaptive filtering in wavelet frames: application to echoe (multiple) suppression in geophysics S. Ventosa, S. Le Roy, I. Huard, A. Pica, H. Rabeson, P. Ricarte, L. Duval, M.-Q. Pham, C. Chaux, J.-C. Pesquet IFPEN laurent.duval [ad] ifpen.fr Journ´ ees images & signaux
2014/03/18
1/44
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
2/44
In just one slide: on echoes and morphing Wavelet frame coefficients: data and model 2 2000
Scale
4 1500 8
1000
16
500 0 2.8
3
3.2
3.4
3.6
3.8
4
4.2
Time (s) 2 2000
Scale
4 1500 8
1000
16
500 0 2.8
3
3.2
3.4
3.6
3.8
4
4.2
Time (s)
Figure 1: Morphing and adaptive subtraction required 2/44
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
3/44
Agenda 1. Issues in geophysical signal processing 2. Problem: multiple reflections (echoes) • adaptive filtering with approximate templates
3. Continuous, complex wavelet frames • how they (may) simplify adaptive filtering • and how they are discretized (back to the discrete world)
4. Adaptive filtering (morphing) • no constraint: unary filters • with constraints: proximal tools
5. Conclusions
3/44
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
4/44
Issues in geophysical signal processing
Figure 2: Seismic data acquisition and wave fields 4/44
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
5/44
Issues in geophysical signal processing a)
Receiver number 1500
1600
1700
1800
1900
1.5
2
2.5
Time (s)
3
3.5
4
4.5
5
5.5
Figure 3: Seismic data: aspect & dimensions (time, offset)
5/44
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
6/44
Issues in geophysical signal processing Shot number 1.8
2200
2000
1800
1600
1400
1200
2 2.2
Time (s)
2.4 2.6 2.8 3 3.2 3.4
Figure 4: Seismic data: aspect & dimensions (time, offset) 6/44
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
7/44
Issues in geophysical signal processing Reflection seismology: • seismic waves propagate through the subsurface medium • seismic traces: seismic wave fields recorded at the surface • primary reflections: geological interfaces • many types of distortions/disturbances • processing goal: extract relevant information for seismic data • led to important signal processing tools: • ℓ1 -promoted deconvolution (Claerbout, 1973) • wavelets (Morlet, 1975) • exabytes (106 gigabytes) of incoming data • need for fast, scalable (and robust) algorithms
7/44
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
8/44
Multiple reflections and templates
Figure 5: Seismic data acquisition: focus on multiple reflections
8/44
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
8/44
Multiple reflections and templates a)
Time (s)
b)
Receiver number 1500
1600
1700
1800
Receiver number
1900
1500
1.5
1.5
2
2
2.5
2.5
3
3
3.5
3.5
4
4
4.5
4.5
5
5
5.5
5.5
1600
1700
1800
1900
Figure 5: Reflection data: shot gather and template
8/44
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
9/44
Multiple reflections and templates Multiple reflections: • seismic waves bouncing between layers • one of the most severe types of interferences • obscure deep reflection layers • high cross-correlation between primaries (p) and multiples (m) • additional incoherent noise (n) • dptq “ pptq`mptq`nptq • with approximate templates: r1 ptq, r2 ptq,. . . rJ ptq
• Issue: how to adapt and subtract approximate templates?
9/44
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
10/44
Multiple reflections and templates
Amplitude
−5
Data Model
0
5 2.8
(a)
3
3.2
3.4
3.6
3.8
4
4.2
Time (s)
Figure 6: Multiple reflections: data trace d and template r1
10/44
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
11/44
Multiple reflections and templates Multiple filtering: • multiple prediction (correlation, wave equation) has limitations • templates are not accurate ř • mptq « j hj ˙ rj ? • standard: identify, apply a matching filer, subtract 2
hopt “ arg min }d ´ h ˙ r} hPRl
• primaries and multiples are not (fully) uncorrelated • same (seismic) source • similarities/dissimilarities in time/frequency • variations in amplitude, waveform, delay • issues in matching filter length: • short filters and windows: local details • long filters and windows: large scale effects 11/44
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
12/44
Multiple reflections and templates Amplitude
−5
Data Model
0
5 2.8
3
3.2
3.4
3.6
3.8
4
4.2
Time (s)
(a)
Amplitude
−2 Filtered Data (+) Filtered Model (−)
−1 0 1 2.8
(b)
3
3.2
3.4
3.6
3.8
4
4.2
Time (s)
Figure 7: Multiple reflections: data trace, template and adaptation 12/44
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
13/44
Multiple reflections and templates Shot number 2200
2000
1800
1600
Shot number 1400
1200
1.8
2
2
2.2
2.2
2.4
2.4
Time (s)
Time (s)
1.8
2.6 2.8
2200
2000
1800
2200
2000
1800
3 3.2
3.4
2200
2000
1800
1600
1200
Shot number 1400
1200
1.8
2
2
2.2
2.2
2.4
2.4
Time (s)
Time (s)
1400
3.4
Shot number
2.8
1200
2.8
3.2
2.6
1400
2.6
3
1.8
1600
1600
2.6 2.8
3
3
3.2
3.2
3.4
3.4
Figure 8: Multiple reflections: data trace and templates, 2D version 13/44
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
14/44
Multiple reflections and templates • A long history of multiple filtering methods • general idea: combine adaptive filtering and transforms • data transforms: Fourier, Radon • enhance the differences between primaries, multiples and noise • reinforce the adaptive filtering capacity • intrication with adaptive filtering? • might be complicated (think about inverse transform)
• First simple approach: • exploit the non-stationary in the data • naturally allow both large scale & local detail matching
ñ Redundant wavelet frames
• intermediate complexity in the transform • simplicity in the (unary/FIR) adaptive filtering
14/44
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
15/44
Hilbert transform and pairs Reminders [Gabor-1946][Ville-1948] {upωq “ ´ı signpωqfppωq Htf 1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−4
−3
−2
−1
0
1
2
Figure 9: Hilbert pair 1 15/44
3
What else?
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
15/44
Hilbert transform and pairs Reminders [Gabor-1946][Ville-1948] {upωq “ ´ı signpωqfppωq Htf 1
0.5
0
−0.5 −4
−3
−2
−1
0
1
2
Figure 9: Hilbert pair 2 15/44
3
What else?
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
15/44
Hilbert transform and pairs Reminders [Gabor-1946][Ville-1948] {upωq “ ´ı signpωqfppωq Htf 2
1.5
1
0.5
0
−0.5
−1
−1.5
−2 −4
−3
−2
−1
0
1
2
Figure 9: Hilbert pair 3 15/44
3
4
What else?
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
15/44
Hilbert transform and pairs Reminders [Gabor-1946][Ville-1948] {upωq “ ´ı signpωqfppωq Htf 3
2
1
0
−1
−2
−4
−3
−2
−1
0
1
2
Figure 9: Hilbert pair 4 15/44
3
What else?
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
16/44
Continuous & complex wavelets 0.5
0.5
0
0
−0.5 −3
−0.5 −2
−1
0 1 Real part
2
3
−3
−2
−1 0 1 Imaginary part
2
3
0.5 0 −0.5 0.5 0 −0.5 Imaginary part
−3
−2
−1
0
1
2
3
Real part
Figure 10: Complex wavelets at two different scales — 1
16/44
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
17/44
Continuous & complex wavelets 0.5
0.5
0
0
−0.5
−0.5 −5
0 Real part
5
−5
0 Imaginary part
5
0.5 0 −0.5 0.5 0 −0.5 Imaginary part
−8
−6
−4
−2
0
2
4
6
8
Real part
Figure 11: Complex wavelets at two different scales — 2
17/44
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
18/44
Continuous wavelets • Transformation group:
affine = translation (τ ) + dilation (a) • Basis functions:
1 ψτ,a ptq “ ? ψ a • • • •
t´τ a
˙
a ą 1: dilation aă ? 1: contraction 1{ a: energy normalization multiresolution (vs monoresolution in STFT/Gabor) FT
ψτ,a ptq ÝÑ 18/44
ˆ
? aΨpaf qe´ı2πf τ
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
19/44
Continuous wavelets • Definition
Cs pτ, aq “
ż
˚ sptqψτ,a ptqdt
• Vector interpretation
Cs pτ, aq “ xsptq, ψτ,a ptqy projection onto time-scale atoms (vs STFT time-frequency) • Redundant transform: τ Ñ τ ˆ a “samples” • Parseval-like formula
Cs pτ, aq “ xSpf q, Ψτ,a pf qy ñ sounder time-scale domain operations! (cf. Fourier) 19/44
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
20/44
Continuous wavelets Introductory example
Data
Real part
Modulus
Imaginary part
Figure 12: Noisy chirp mixture in time-scale & sampling 20/44
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
21/44
Continuous wavelets Noise spread & feature simplification (signal vs wiggle) 2 1 0 −1 −2 50
100
150
200
250
300
350
400
4 2 0 −2 −4 5 0 −5
300
350
400
450
500
550
600
650
700
300
350
400
450
500
550
600
650
700
2 0 −2 2 0 −2
Figure 13: Noisy chirp mixture in time-scale: zoomed scaled wiggles 21/44
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
22/44
Continuous wavelets
Amplitude
−5
Data Model
0
5 2.8
3
3.2
3.4
3.6
3.8
4
4.2
Time (s)
(a) 2
2 2000
2000 4
1500 8
1000
16
500
Scale
Scale
4
1500 8
1000
16
500
0 2.8
3
3.2
3.4
3.6
Time (s)
3.8
4
4.2
0 2.8
3
3.2
3.4
3.6
3.8
4
Time (s)
Figure 14: Which morphing is easier: time or time-scale? 22/44
4.2
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
23/44
Continuous wavelets • Inversion with another wavelet φ
sptq “
ij
Cs pu, aqφu,a ptq
duda a2
ñ time-scale domain processing! (back to the trace signal) • Scalogram |Cs pt, aq|2 • Energy conversation
E“ • Parseval-like formula
xs1 , s2 y “ 23/44
ij
ij
|Cs pt, aq|2
dtda a2
Cs1 pt, aqCs˚2 pt, aq
dtda a2
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
24/44
Continuous wavelets • Wavelet existence: admissibility criterion
ż 0 p˚ ż `8 p ˚ Φ pνqΨpνq Φ pνqΨpνq dν “ dν ă 8 0 ă Ah “ ν ν ´8 0 generally normalized to 1 • Easy to satisfy (common freq. support midway 0 & 8) • With ψ “ φ, induces band-pass property: • necessary condition: |Φp0q| “ 0, or zero-average shape • amplitude spectrum neglectable w.r.t. |ν| at infinity • Example: Morlet-Gabor (not truly admissible)
ψptq “ ? 24/44
1 2πσ 2
t2
e´ 2σ2 e´ı2πf0 t
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
25/44
Discretization and redundancy Being practical again: dealing with discrete signals • Can one sample in time-scale (CWT) domain:
Cs pτ, aq “
ż
˚ sptqψτ,a ptqdt,
1 ψτ,a ptq “ ? ψ a
ˆ
t´τ a
˙
with cj,k “ Cs pkb0 aj0 , aj0 q, pj, kq P Z and still be able to recover sptq? • Result 1 (Daubechies, 1984): there exists a wavelet frame if
a0 b0 ă C, (depending on ψ). A frame is generally redundant
• Result 2 (Meyer, 1985): there exist an orthonormal basis for a
specific ψ (non trivial, Meyer wavelet) and a0 “ 2 b0 “ 1
Now: how to choose the practical level of redundancy? 25/44
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
26/44
Discretization and redundancy 8
7
6
5
4
3
2
1
0
20
40
60
80
100
120
Figure 15: Wavelet frame sampling: J “ 21, b0 “ 1, a0 “ 1.1 26/44
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
26/44
Discretization and redundancy 8
7
6
5
4
3
2
1
0
20
40
60
80
100
120
Figure 15: Wavelet frame sampling: J “ 5, b0 “ 2, a0 “ 26/44
? 2
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
26/44
Discretization and redundancy 8
7
6
5
4
3
2
1
0
20
40
60
80
100
120
Figure 15: Wavelet frame sampling: J “ 3, b0 “ 1, a0 “ 2 26/44
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
27/44
Discretization and redundancy 0.15 primary multiple noise sum
true multiple adapted multiple 0.1
0.05
0
−0.05
−0.1 0
0.5
1
1.5
2
2.5
3
3.5
4
0
0.5
1
1.5
Time (s)
2
2.5
3
3.5
4
Time (s)
Median S/Nadapt (dB)
19 20
18
18
17
16
16 15
14
14 12 13 10 20
12 15 10
S/N (dB)
5 4
6
8
10
12
14
16
11 10
Redundancy
Figure 16: Redundancy selection with variable noise experiments 27/44
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
28/44
Discretization and redundancy • Complex Morlet wavelet:
ψptq “ π ´1{4 e´iω0 t e´t
2 {2
, ω0 : central frequency
• Discretized time r, octave j, voice v: v ψr,j rns
“?
1 2j`v{V
ˆ
nT ´ r2j b0 ψ 2j`v{V
˙
, b0 : sampling at scale zero
• Time-scale analysis:
@ D ÿ v v rns drnsψr,j d “ dvr,j “ drns, ψr,j rns “ n
28/44
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
29/44
Discretization and redundancy 2
2 2000
2000 4
1500 8
1000
16
Scale
Scale
4
1500 8
1000
16
500
500
0 2.8
3
3.2
3.4
3.6
3.8
4
0
4.2
2.8
3
3.2
3.4
Time (s)
3.6
3.8
4
4.2
Time (s)
2
2 2000
2000 4
1500 8
1000
16
500
Scale
Scale
4
1500 8
1000
16
500
0 2.8
3
3.2
3.4
3.6
Time (s)
3.8
4
4.2
0 2.8
3
3.2
3.4
3.6
3.8
4
4.2
Time (s)
Figure 17: Morlet wavelet scalograms, data and templates
Take advantage from the closest similarity/dissimilarity: • remember wiggles: on sliding windows, at each scale, a single complex coefficient compensates amplitude and phase 29/44
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
30/44
Unary filters • Windowed unary adaptation: complex unary filter h (aopt )
compensates delay/amplitude mismatches: › ›2 › › ÿ › › aopt “ arg min ›d ´ aj rk › › › taj upjPJq j
• Vector Wiener equations for complex signals:
xd, rm y “
ÿ j
aj xrj , rm y
• Time-scale synthesis:
ˆ “ drns 30/44
ÿÿ r j,v
v dˆvr,j ψrr,j rns
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
31/44
Results
2
2 2000
2000 4
1500 8
1000
16
Scale
Scale
4
1500 8
1000
16
500
500
0 2.8
3
3.2
3.4
3.6
3.8
4
0
4.2
2.8
3
3.2
3.4
Time (s)
3.6
3.8
4
4.2
Time (s)
2
2 2000
2000 4
1500 8
1000
16
500
Scale
Scale
4
1500 8
1000
16
500
0 2.8
3
3.2
3.4
3.6
Time (s)
3.8
4
4.2
0 2.8
3
3.2
3.4
3.6
3.8
4
Time (s)
Figure 18: Wavelet scalograms, data and templates, after unary adaptation
31/44
4.2
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
32/44
Results (reminders)
2
2 2000
2000 4
1500 8
1000
16
Scale
Scale
4
1500 8
1000
16
500
500
0 2.8
3
3.2
3.4
3.6
3.8
4
0
4.2
2.8
3
3.2
3.4
Time (s)
3.6
3.8
4
4.2
Time (s)
2
2 2000
2000 4
1500 8
1000
16
500
Scale
Scale
4
1500 8
1000
16
500
0 2.8
3
3.2
3.4
3.6
Time (s)
3.8
4
4.2
0 2.8
3
3.2
3.4
3.6
Time (s)
Figure 19: Wavelet scalograms, data and templates
32/44
3.8
4
4.2
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
33/44
Results Shot number 1.8
2200
2000
1800
1600
1400
2 2.2
Time (s)
2.4 2.6 2.8 3 3.2 3.4
Figure 20: Original data 33/44
1200
What else?
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
34/44
Results Shot number 1.8
2200
2000
1800
1600
1400
1200
2 2.2
Time (s)
2.4 2.6 2.8 3 3.2 3.4
Figure 21: Filtered data, “best” template 34/44
What else?
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
35/44
Results Shot number 1.8
2200
2000
1800
1600
1400
1200
2 2.2
Time (s)
2.4 2.6 2.8 3 3.2 3.4
Figure 22: Filtered data, three templates 35/44
What else?
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
36/44
Going a little further Impose geophysical data related assumptions: e.g. sparsity 1 4/3 3/2 2 3 4
Figure 23: Generalized Gaussian modeling of seismic data wavelet frame decomposition with different power laws. 36/44
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
37/44
Variational approach minimize xPH
J ÿ
j“1
fj pLj xq
with lower-semicontinuous proper convex functions fj and bounded linear operators Lj .
37/44
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
37/44
Variational approach minimize xPH
J ÿ
j“1
fj pLj xq
with lower-semicontinuous proper convex functions fj and bounded linear operators Lj .
• fj can be related to noise (e.g. a quadratic term when the
noise is Gaussian),
37/44
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
37/44
Variational approach minimize xPH
J ÿ
j“1
fj pLj xq
with lower-semicontinuous proper convex functions fj and bounded linear operators Lj .
• fj can be related to noise (e.g. a quadratic term when the
noise is Gaussian), • fj can be related to some a priori on the target solution (e.g.
an a priori on the wavelet coefficient distribution),
37/44
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
37/44
Variational approach minimize xPH
J ÿ
j“1
fj pLj xq
with lower-semicontinuous proper convex functions fj and bounded linear operators Lj .
• fj can be related to noise (e.g. a quadratic term when the
noise is Gaussian), • fj can be related to some a priori on the target solution (e.g.
an a priori on the wavelet coefficient distribution), • fj can be related to a constraint (e.g. a support constraint),
37/44
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
37/44
Variational approach minimize xPH
J ÿ
j“1
fj pLj xq
with lower-semicontinuous proper convex functions fj and bounded linear operators Lj .
• fj can be related to noise (e.g. a quadratic term when the
noise is Gaussian), • fj can be related to some a priori on the target solution (e.g.
an a priori on the wavelet coefficient distribution), • fj can be related to a constraint (e.g. a support constraint), • Lj can model a blur operator,
37/44
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
37/44
Variational approach minimize xPH
J ÿ
j“1
fj pLj xq
with lower-semicontinuous proper convex functions fj and bounded linear operators Lj .
• fj can be related to noise (e.g. a quadratic term when the
noise is Gaussian), • fj can be related to some a priori on the target solution (e.g.
an a priori on the wavelet coefficient distribution), • fj can be related to a constraint (e.g. a support constraint), • Lj can model a blur operator, • Lj can model a gradient operator (e.g. total variation), 37/44
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
37/44
Variational approach minimize xPH
J ÿ
j“1
fj pLj xq
with lower-semicontinuous proper convex functions fj and bounded linear operators Lj .
• fj can be related to noise (e.g. a quadratic term when the
noise is Gaussian), • fj can be related to some a priori on the target solution (e.g. • • • • 37/44
an a priori on the wavelet coefficient distribution), fj can be related to a constraint (e.g. a support constraint), Lj can model a blur operator, Lj can model a gradient operator (e.g. total variation), Lj can model a frame operator.
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
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Problem re-formulation pkq lodomoon
observed signal
pkq pkq pkq “ lopo¯mo ¯omo on ` lon omoon on ` lom primary
multiple
noise
Assumption: templates linked to m ¯ pkq throughout time-varying (FIR) filters: m ¯ pkq “
J´1 ÿÿ j“0 p
¯ ppq pkqrpk´pq h j j
where ¯ pkq : unknown impulse response of the filter corresponding to • h j template j and time k, then: d on loomo
observed signal 38/44
¯ on ` loomo “ loomo h p¯ on `R loomo n on primary
filter
noise
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
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Assumptions • F is a frame, p¯ is a realization of a random vector P :
fP ppq9 expp´ϕpF pqq, ¯ is a realization of a random vector H: • h fH phq9 expp´ρphqq, • n is a realization of a random vector N , of probability density:
fN pnq9 expp´ψpnqq, • slow variations along time and concentration of the filters pn`1q
|hj 39/44
pnq
ppq ´ hj ppq| ď εj,p ;
J´1 ÿ j“0
ρrj phj q ď τ
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
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Results: synthetics minimize
yPRN ,hPRN P
` ˘ z ´ Rh ´ y ψ loooooooomoooooooon `
fidelity: noise-realted
`
ϕpF yq loomoon
loρphq omoon
a priori on signal
a priori on filters
• ϕk “ κk | ¨ | (ℓ1 -norm) where κk ą 0 • ρrj phj q: }hj }ℓ1 , }hj }2ℓ2 or }hj }ℓ1,2 ` ˘ • ψ z ´ Rh ´ y : quadratic (Gaussian noise) 540
350
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450
500
550
600
650
700
350
400
450
500
550
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560
580
650
Figure 24: Simulated results with heavy noise.
600
700
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
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Results: potential on real data
Figure 25: (a) Unary filters (b) Proximal FIR filters.
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Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
42/44
Conclusions Take-away messages: • Practical side • Competitive with more standard 2D processing • Very fast (unary part): industrial integration
• Technical side • Lots of choices, insights from 1D or 1.5D • Non-stationary, wavelet-based, adaptive multiple filtering • Take good care of cascaded processing • Present work • Going 2D: crucial choices on redundancy, directionality
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Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
43/44
Conclusions Now what’s next: curvelets, shearlets, dual-tree complex wavelets?
Figure 26: From T. Lee (TPAMI-1996): 2D Gabor filters (odd and even) or Weyl-Heisenberg coherent states
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Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
What else?
Conclusions
44/44
References Ventosa, S., S. Le Roy, I. Huard, A. Pica, H. Rabeson, P. Ricarte, and L. Duval, 2012, Adaptive multiple subtraction with wavelet-based complex unary Wiener filters: Geophysics, 77, V183–V192; http://arxiv.org/abs/1108.4674 Pham, M. Q., C. Chaux, L. Duval, L. and J.-C. Pesquet, 2014, A Primal-Dual Proximal Algorithm for Sparse Template-Based Adaptive Filtering: Application to Seismic Multiple Removal: IEEE Trans. Signal Process., accepted; http://tinyurl.com/proximal-multiple Jacques, L., L. Duval, C. Chaux, and G. Peyr´e, 2011, A panorama on multiscale geometric representations, intertwining spatial, directional and frequency selectivity: Signal Process., 91, 2699–2730; http://arxiv.org/abs/1101.5320 44/44
