Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Adaptive filtering in wavelet frames for echo (multiple reflections) 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 S´ eminaire Signal Image, IMS, IMB, LaBRI, U. Bordeaux
2014/09/18
1/48
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
2/48
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 2/48
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
3/48
Agenda 1. Issues in geophysical signal processing 2. Problem: multiple reflections (echoes) • adaptive filtering with approximate templates
3. (complex) wavelet frames • how they (may) simplify adaptive filtering • and how they are discretized (back to the discrete world)
4. Adaptive filtering (morphing) • without constraint: unary filters (on analytic signals) • with constraints: proximal tools
5. Conclusions
3/48
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
4/48
Issues in geophysical signal processing
Figure 2: Seismic data acquisition and wave fields 4/48
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
5/48
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)
5/48
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
6/48
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, receiver) 6/48
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Conclusions
7/48
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
7/48
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
8/48
Multiple reflections and templates
Figure 5: Seismic data acquisition: focus on multiple reflections
8/48
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
8/48
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
8/48
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Conclusions
9/48
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?
9/48
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
10/48
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
10/48
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Conclusions
11/48
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 11/48
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
12/48
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 12/48
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
13/48
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 13/48
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Conclusions
14/48
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 complex wavelet frames
• intermediate complexity in the transform • simplicity in the (unary/FIR) adaptive filtering
14/48
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
15/48
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 15/48
3
Going proximal
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
15/48
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 15/48
3
Going proximal
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
15/48
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 15/48
3
4
Going proximal
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
15/48
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 15/48
3
Going proximal
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
16/48
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
16/48
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
17/48
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
17/48
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
18/48
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 18/48
ˆ
? aΨpaf qe´ı2πf τ
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
19/48
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) 19/48
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
20/48
Continuous wavelets Introductory example
Data
Real part
Modulus
Imaginary part
Figure 12: Noisy chirp mixture in time-scale & sampling 20/48
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Conclusions
21/48
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 21/48
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Conclusions
22/48
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? 22/48
4.2
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
23/48
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 “ 23/48
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
Going proximal
Conclusions
24/48
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 “ ? 24/48
1 2πσ 2
t2
e´ 2σ2 e´ı2πf0 t
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Conclusions
25/48
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? 25/48
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
26/48
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 26/48
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
26/48
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 “ 26/48
? 2
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
26/48
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 26/48
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Conclusions
27/48
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 27/48
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Conclusions
28/48
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
28/48
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Conclusions
29/48
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 29/48
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
30/48
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 30/48
ÿÿ r j,v
v dˆvr,j ψrr,j rns
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Conclusions
31/48
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
31/48
4.2
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Conclusions
32/48
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
32/48
3.8
4
4.2
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
33/48
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 33/48
1200
Going proximal
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
34/48
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 34/48
Going proximal
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
35/48
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 35/48
Going proximal
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Conclusions
36/48
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. 36/48
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
37/48
Variational approach minimize xPH
J ÿ
j“1
fj pLj xq
with lower-semicontinuous proper convex functions fj and bounded linear operators Lj .
37/48
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
37/48
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),
37/48
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Conclusions
37/48
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),
37/48
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Conclusions
37/48
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),
37/48
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Conclusions
37/48
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,
37/48
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Conclusions
37/48
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), 37/48
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Conclusions
37/48
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. • • • • 37/48
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
Going proximal
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 38/48
¯ on ` loomo “ loomo h p¯ on `R loomo n on primary
filter
noise
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Conclusions
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Results: synthetics (noise: σ “ 0.08) y
Original signal y
yˆ
r2
Estimated signal yp
s
Model r2
r1
Model r1
Original multiple s sˆ
Estimated multiple sp Observed signal z
z 100
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200
300
400
500
600
700
800
900
1000
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
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 40/48
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
Going proximal
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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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400
450
500
550
600
650
700
350
400
450
500
550
600
560
580
650
Figure 24: Simulated results with heavy noise.
600
700
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Conclusions
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Results: synthetics
σ \ ρr 0.01 0.02 0.04 0.08
ℓ1 20.90 20.89 19.00 17.55
SNRy ℓ2 21.23 21.16 19.90 16.81
ℓ1,2 23.57 23.51 20.67 17.34
ℓ1 24.36 22.53 20.15 16.96
SNRs ℓ2 24.68 23.02 20.14 16.56
ℓ1,2 26.74 23.76 19.84 15.96
Signal-to-noise ratios (SNR, averaged over 100 noise realizations)
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Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
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Results: synthetics, w/ a significance index 2
tf /b = 0.5
1 0 14
2
16
18
20
22
24
26
18
20
22
24
26
18
20
22
24
26
18
20
22
24
26
tf /b = 1
1 0 14
2
16
tf /b = 2
1 0 14
2
16
tf /b = 4
1 0 14
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16
Going proximal
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
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Results: synthetics, w/ a significance index
σ 0.01 0.02 0.04 0.08
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ρr
L 3 4 3 4 3 4 3 4
ℓ1 f {b 7.3 4.2 3.0 2.6 3.4 3.5 3.5 3.8
Haar ℓ2 f {b 4.3 2.8 2.7 2.3 3.5 3.7 3.5 3.8
ℓ1,2 f {b 7.7 5.1 3.4 2.5 3.2 3.3 3.5 3.7
tf {b for primaries Daubechies ℓ1 ℓ2 ℓ1,2 f {b f {b f {b 2.6 3.0 4.9 4.2 1.9 3.3 1.5 2.1 1.6 3.0 2.9 2.7 3.0 3.9 2.7 3.2 3.8 2.8 3.5 3.9 3.3 3.4 3.6 3.2
ℓ1 f {b 5.6 5.3 2.8 3.0 3.2 3.3 3.8 3.8
Symmlet ℓ2 ℓ1,2 f {b f {b 2.6 7.2 1.3 5.1 2.1 3.1 1.7 4.3 3.8 3.2 3.7 3.3 4.2 3.7 4.1 4.2
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
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Results: potential on real data
Figure 25: Portion of a receiver gather: recorded data. 45/48
Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
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Results: potential on real data
Figure 25: Low noise: (a) Unary filters (b) Proximal FIR filters.
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Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
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Results: potential on real data
Figure 25: High noise: (a) Unary filters (b) Proximal FIR filters.
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Conclusions
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Conclusions
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Conclusions Take-away messages: • Practical side • Competitive with more standard 2D processing • Very fast (unary part): industrial integration
• Technical side • Non-stationary, wavelet-based, adaptive multiple filtering • Take good care in cascaded processing • Present work • Other applications: (image) pattern matching, (voice) echo cancellation, (speech) exemplar search, ultrasonic/acoustic emissions • Going 2D: crucial choices on *-lets: redundancy, directionality • Better “sparsity” penalizations: ℓ1 or ℓℓ1 ? 2 46/48
Context
Multiple filtering
Wavelets
Discretization, unary filters
Results
Going proximal
Conclusions
47/48
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
Going proximal
Conclusions
48/48
References
S. Ventosa, 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 Transactions on Signal Processing, August 2014; http://arxiv.org/abs/1405.1081 L. Jacques, L. Duval, C. Chaux, and G. Peyr´ e, 2011, A panorama on multiscale geometric representations, intertwining spatial, directional and frequency selectivity: Signal Processing, 91, 2699–2730; http://arxiv.org/abs/1101.5320 A. Repetti, M. Q. Pham, L. Duval, E. Chouzenoux and J.-C. Pesquet, 2014, Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed ℓ1 {ℓ2 Regularization: IEEE Signal Processing Letter, accepted http://arxiv.org/abs/1407.5465
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