Adaptive filtering in wavelet frames: application to …...Context Multiple ﬁltering Wavelets Discretization, unary ﬁlters Results Going proximal Conclusions 2/46 In just one slide:

Context Multiple filtering Wavelets Discretization, unary filters Results Going proximal Conclusions

Adaptive filtering in wavelet frames: applicationto echo (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

IFPENlaurent.duval [ad] ifpen.fr

SIERRA 2014, Saint-Etienne

2014/03/25

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In just one slide: on echoes and morphingWavelet frame coefficients: data and model

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Figure 1: Morphing and adaptive subtraction required

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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)• without constraint: unary filters (on analytic signals)• with constraints: proximal tools

5. Conclusions

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Issues in geophysical signal processing

Figure 2: Seismic data acquisition and wave fields

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Figure 3: Seismic data: aspect & dimensions (time, offset)

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Issues in geophysical signal processingShot number

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Figure 4: Seismic data: aspect & dimensions (time, offset)

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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

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Multiple reflections and templates

Figure 5: Seismic data acquisition: focus on multiple reflections

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Figure 5: Reflection data: shot gather and template

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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: r1ptq, r2ptq,. . . rJptq

• Issue: how to adapt and subtract approximate templates?

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Figure 6: Multiple reflections: data trace d and template r1

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Multiple reflections and templatesMultiple filtering:

• multiple prediction (correlation, wave equation) has limitations

• templates are not accurate• mptq « ř

j hj ˙ rj?• standard: identify, apply a matching filer, subtract

hopt “ argminhPRl

}d´ h ˙ r}2

• 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

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Figure 7: Multiple reflections: data trace, template and adaptation

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Multiple reflections and templatesShot number

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Figure 8: Multiple reflections: data trace and templates, 2D version

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• 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

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Hilbert transform and pairsReminders [Gabor-1946][Ville-1948]

{Htfupωq “ ´ı signpωq pfpωq

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Figure 9: Hilbert pair 1

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−4 −3 −2 −1 0 1 2 3

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Continuous & complex wavelets

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Figure 10: Complex wavelets at two different scales — 1

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Continuous & complex wavelets

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Figure 11: Complex wavelets at two different scales — 2

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Continuous wavelets

• Transformation group:

affine = translation (τ) + dilation (a)

• Basis functions:

ψτ,aptq “ 1?aψ

ˆt´ τ

a

˙

• a ą 1: dilation• a ă 1: contraction• 1{?

a: energy normalization• multiresolution (vs monoresolution in STFT/Gabor)

ψτ,aptq FTÝÑ?aΨpafqe´ı2πfτ

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Continuous wavelets

• Definition

Cspτ, aq “żsptqψ˚

τ,aptqdt

• Vector interpretation

Cspτ, aq “ xsptq, ψτ,aptqy

projection onto time-scale atoms (vs STFT time-frequency)

• Redundant transform: τ Ñ τ ˆ a “samples”

• Parseval-like formula

Cspτ, aq “ xSpfq,Ψτ,apfqy

ñ sounder time-scale domain operations! (cf. Fourier)

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Continuous waveletsIntroductory example

Data Real part

Imaginary partModulus

Figure 12: Noisy chirp mixture in time-scale & sampling

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Continuous waveletsNoise spread & feature simplification (signal vs wiggle)

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Figure 13: Noisy chirp mixture in time-scale: zoomed scaled wiggles

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Continuous wavelets

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Figure 14: Which morphing is easier: time or time-scale?

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Continuous wavelets

• Inversion with another wavelet φ

sptq “ĳ

Cspu, aqφu,aptqdudaa2

ñ time-scale domain processing! (back to the trace signal)

• Scalogram|Cspt, aq|2

• Energy conversation

E “ĳ

|Cspt, aq|2dtdaa2

• Parseval-like formula

xs1, s2y “ĳ

Cs1pt, aqC˚s2

pt, aqdtdaa2

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Continuous wavelets

• Wavelet existence: admissibility criterion

0 ă Ah “ż `8

0

pΦ˚pνqΨpνqν

dν “ż 0

´8

pΦ˚pνqΨpνqν

dν ă 8

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 “ 1?2πσ2

e´ t2

2σ2 e´ı2πf0t

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Discretization and redundancy

Being practical again: dealing with discrete signals

• Can one sample in time-scale (CWT) domain:

Cspτ, aq “żsptqψ˚

τ,aptqdt, ψτ,aptq “ 1?aψ

ˆt´ τ

a

˙

with cj,k “ Cspkb0aj0, aj0q, pj, kq P Z and still be able to

recover sptq?• Result 1 (Daubechies, 1984): there exists a wavelet frame ifa0b0 ă C, (depending on ψ). A frame is generally redundant

• Result 2 (Meyer, 1985): there exist an orthonormal basis for aspecific ψ (non trivial, Meyer wavelet) and a0 “ 2 b0 “ 1

Now: how to choose the practical level of redundancy?

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Figure 15: Wavelet frame sampling: J “ 21, b0 “ 1, a0 “ 1.1

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Figure 15: Wavelet frame sampling: J “ 5, b0 “ 2, a0 “?2

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Figure 15: Wavelet frame sampling: J “ 3, b0 “ 1, a0 “ 2

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0 0.5 1 1.5 2 2.5 3 3.5 4Time (s)

primarymultiplenoisesum

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true multipleadapted multiple

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Med

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adap

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Figure 16: Redundancy selection with variable noise experiments

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• Complex Morlet wavelet:

ψptq “ π´1{4e´iω0te´t2{2, ω0: central frequency

• Discretized time r, octave j, voice v:

ψvr,jrns “ 1?

2j`v{Vψ

ˆnT ´ r2jb0

2j`v{V

˙, b0: sampling at scale zero

• Time-scale analysis:

d “ dvr,j “@drns, ψv

r,jrnsD

“ÿ

n

drnsψvr,jrns

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Figure 17: Morlet wavelet scalograms, data and templates

Take advantage from the closest similarity/dissimilarity:

• remember wiggles: on sliding windows, at each scale, a singlecomplex coefficient compensates amplitude and phase

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Unary filters

• Windowed unary adaptation: complex unary filter h (aopt)compensates delay/amplitude mismatches:

aopt “ argmintajupjPJq

›››››d ´ÿ

j

ajrk

›››››

2

• Vector Wiener equations for complex signals:

xd, rmy “ÿ

j

aj xrj , rmy

• Time-scale synthesis:

drns “ÿ

r

ÿ

j,v

dvr,jrψvr,jrns

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Results

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Figure 18: Wavelet scalograms, data and templates, after unary adaptation

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Results (reminders)

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Figure 19: Wavelet scalograms, data and templates

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ResultsShot number

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Figure 20: Original data

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ResultsShot number

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Figure 21: Filtered data, “best” template

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ResultsShot number

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Figure 22: Filtered data, three templates

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Going a little furtherImpose geophysical data related assumptions: e.g. sparsity

1

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Figure 23: Generalized Gaussian modeling of seismic data wavelet frame

decomposition with different power laws.

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Variational approach

minimizexPH

Jÿ

j“1

fjpLjxq

with lower-semicontinuous proper convex functions fj and bounded linear

operators Lj .

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minimizexPH

Jÿ

j“1

fjpLjxq


operators Lj .

• fj can be related to noise (e.g. a quadratic term when thenoise is Gaussian),

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minimizexPH

Jÿ

j“1

fjpLjxq


operators Lj .


• fj can be related to some a priori on the target solution (e.g.an a priori on the wavelet coefficient distribution),

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minimizexPH

Jÿ

j“1

fjpLjxq


operators Lj .



• fj can be related to a constraint (e.g. a support constraint),

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minimizexPH

Jÿ

j“1

fjpLjxq


operators Lj .




• Lj can model a blur operator,

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minimizexPH

Jÿ

j“1

fjpLjxq


operators Lj .





• Lj can model a gradient operator (e.g. total variation),

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minimizexPH

Jÿ

j“1

fjpLjxq


operators Lj .





• Lj can model a gradient operator (e.g. total variation),

• Lj can model a frame operator.37/46


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Problem re-formulation

dpkqloomoonobserved signal

“ ppkqloomoonprimary

` mpkqloomoonmultiple

` npkqloomoonnoise

Assumption: templates linked to mpkq throughout time-varying(FIR) filters:

mpkq “J´1ÿ

j“0

ÿ

p

hppqj pkqrpk´pq

j

where• h

pkqj : unknown impulse response of the filter corresponding to

template j and time k, then:

dloomoonobserved signal

“ ploomoonprimary

`R hloomoonfilter

` nloomoonnoise

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Results: synthetics (noise: σ “ 0.08)

100 200 300 400 500 600 700 800 900 1000

s

z

s

r2

r1

y

y

Original signal y

Estimated signal pyModel r1

Model r2

Original multiple s

Estimated multiple psObserved signal z

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Assumptions

• F is a frame, p is a realization of a random vector P :

fP ppq9 expp´ϕpFpqq,

• h is a realization of a random vector H:

fHphq9 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

|hpn`1qj ppq ´ h

pnqj ppq| ď εj,p ;

J´1ÿ

j“0

rρjphjq ď τ

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Results: synthetics

minimizeyPRN ,hPRNP

ψ`z ´ Rh ´ y

˘loooooooomoooooooonfidelity: noise-realted

` ϕpFyqloomoona priori on signal

` ρphqloomoona priori on filters

• ϕk “ κk| ¨ | (ℓ1-norm) where κk ą 0

• rρjphjq: }hj}ℓ1 , }hj}2ℓ2 or }hj}ℓ1,2• ψ

`z ´ Rh ´ y

˘: quadratic (Gaussian noise)

350 400 450 500 550 600 650 700 350 400 450 500 550 600 650 700

540 560 580 600

Figure 24: Simulated results with heavy noise.41/46


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Results: synthetics

SNRy SNRs

σ \ rρ ℓ1 ℓ2 ℓ1,2 ℓ1 ℓ2 ℓ1,20.01 20.90 21.23 23.57 24.36 24.68 26.74

0.02 20.89 21.16 23.51 22.53 23.02 23.76

0.04 19.00 19.90 20.67 20.15 20.14 19.84

0.08 17.55 16.81 17.34 16.96 16.56 15.96

Signal-to-noise ratios (SNR, averaged over 100 noise realisations)

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Results: potential on real data

Figure 25: Portion of a receiver gather: recorded data.

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Results: potential on real data

Figure 25: (a) Unary filters (b) Proximal FIR filters.

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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 in cascaded processing

• Present work• Other applications: pattern matching, (voice) echo

cancellation, ultrasonic/acoustic emissions with home-madetemplates

• Going 2D: crucial choices on redundancy, directionality

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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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References

Ventosa, S., S. Le Roy, I. Huard, A. Pica, H. Rabeson, P. Ricarte,and L. Duval, 2012, Adaptive multiple subtraction withwavelet-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, APrimal-Dual Proximal Algorithm for Sparse Template-BasedAdaptive Filtering: Application to Seismic Multiple Removal: IEEETrans. Signal Process., accepted;http://tinyurl.com/proximal-multiple

Jacques, L., L. Duval, C. Chaux, and G. Peyre, 2011, A panoramaon multiscale geometric representations, intertwining spatial,directional and frequency selectivity: Signal Process., 91,2699–2730; http://arxiv.org/abs/1101.5320

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http://arxiv.org/abs/1108.4674

http://tinyurl.com/proximal-multiple

http://arxiv.org/abs/1101.5320

Documents

Adaptive filtering in wavelet frames: application to …...Context Multiple ﬁltering Wavelets Discretization, unary ﬁlters Results Going proximal Conclusions 2/46 In just one slide: