This document discusses using adaptive filtering in continuous wavelet frames to suppress multiple reflections in geophysical signals. It introduces the problem of multiple reflections obscuring deeper geological layers in seismic data. Adaptive filtering with approximate templates is challenging due to variations in primaries and multiples. Continuous, complex wavelet frames can simplify adaptive filtering by spreading noise and highlighting signal features in the time-scale domain. The document explores using wavelet frames to adaptively filter and subtract multiples from seismic data through morphing in the time-scale domain rather than time domain.

1. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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
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2. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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In just one slide: on echoes and morphing
Wavelet frame coefficients: data and model
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Figure 1: Morphing and adaptive subtraction required
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3. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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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)
• no constraint: unary filters
• with constraints: proximal tools
5. Conclusions
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4. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Issues in geophysical signal processing
Figure 2: Seismic data acquisition and wave fields
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5. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Issues in geophysical signal processing
Receiver number
Time(s)
1500 1600 1700 1800 1900
1.5
2
2.5
3
3.5
4
4.5
5
5.5
a)
Figure 3: Seismic data: aspect & dimensions (time, offset)
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Issues in geophysical signal processing
Shot number
Time(s) 120014001600180020002200
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
Figure 4: Seismic data: aspect & dimensions (time, offset)
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7. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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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
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Multiple reflections and templates
Figure 5: Seismic data acquisition: focus on multiple reflections
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Multiple reflections and templates
Receiver number
Time(s)
1500 1600 1700 1800 1900
1.5
2
2.5
3
3.5
4
4.5
5
5.5
a) Receiver number
1500 1600 1700 1800 1900
1.5
2
2.5
3
3.5
4
4.5
5
5.5
b)
Figure 5: Reflection data: shot gather and template
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10. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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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: r1ptq, r2ptq,. . . rJ ptq
• Issue: how to adapt and subtract approximate templates?
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Multiple reflections and templates
2.8 3 3.2 3.4 3.6 3.8 4 4.2
−5
0
5
Amplitude
Time (s)
Data
Model
(a)
Figure 6: Multiple reflections: data trace d and template r1
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12. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Multiple reflections and templates
Multiple 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 “ arg min
hPRl
}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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13. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Multiple reflections and templates
2.8 3 3.2 3.4 3.6 3.8 4 4.2
−5
0
5
Amplitude
Time (s)
Data
Model
(a)
2.8 3 3.2 3.4 3.6 3.8 4 4.2
−2
−1
0
1
Amplitude
Time (s)
Filtered Data (+)
Filtered Model (−)
(b)
Figure 7: Multiple reflections: data trace, template and adaptation
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14. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Multiple reflections and templates
Shot numberTime(s)
120014001600180020002200
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2
2.2
2.4
2.6
2.8
3
3.2
3.4
Shot number
Time(s)
120014001600180020002200
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2
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Shot number
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Shot number
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120014001600180020002200
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2
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3
3.2
3.4
Figure 8: Multiple reflections: data trace and templates, 2D version
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15. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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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
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16. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Hilbert transform and pairs
Reminders [Gabor-1946][Ville-1948]
{Htfupωq “ ´ı signpωq pfpωq
−4 −3 −2 −1 0 1 2 3
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Figure 9: Hilbert pair 1
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17. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Hilbert transform and pairs
Reminders [Gabor-1946][Ville-1948]
{Htfupωq “ ´ı signpωq pfpωq
−4 −3 −2 −1 0 1 2 3
−0.5
0
0.5
1
Figure 9: Hilbert pair 2
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18. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Hilbert transform and pairs
Reminders [Gabor-1946][Ville-1948]
{Htfupωq “ ´ı signpωq pfpωq
−4 −3 −2 −1 0 1 2 3 4
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Figure 9: Hilbert pair 3
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19. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Hilbert transform and pairs
Reminders [Gabor-1946][Ville-1948]
{Htfupωq “ ´ı signpωq pfpωq
−4 −3 −2 −1 0 1 2 3
−2
−1
0
1
2
3
Figure 9: Hilbert pair 4
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20. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Continuous & complex wavelets
−3 −2 −1 0 1 2 3
−0.5
0
0.5
Real part
−3 −2 −1 0 1 2 3
−0.5
0
0.5
Imaginary part
−3 −2 −1 0 1 2 3−0.5
0
0.5
−0.5
0
0.5
Real part
Imaginary part
Figure 10: Complex wavelets at two different scales — 1
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Continuous & complex wavelets
−5 0 5
−0.5
0
0.5
Real part
−5 0 5
−0.5
0
0.5
Imaginary part
−8 −6 −4 −2 0 2 4 6 8−0.5
0
0.5
−0.5
0
0.5
Real part
Imaginary part
Figure 11: Complex wavelets at two different scales — 2
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22. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Continuous wavelets
• Transformation group:
affine = translation (τ) + dilation (a)
• Basis functions:
ψτ,aptq “
1
?
a
ψ
ˆ
t ´ τ
a
˙
• a ą 1: dilation
• a ă 1: contraction
• 1{
?
a: energy normalization
• multiresolution (vs monoresolution in STFT/Gabor)
ψτ,aptq
FT
ÝÑ
?
aΨpafqe´ı2πfτ
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23. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Continuous wavelets
• Definition
Cspτ, aq “
ż
sptqψ˚
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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24. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Continuous wavelets
Introductory example
Data Real part
Imaginary partModulus
Figure 12: Noisy chirp mixture in time-scale & sampling
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25. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Continuous wavelets
Noise spread & feature simplification (signal vs wiggle)
50 100 150 200 250 300 350 400
−2
−1
0
1
2
300 350 400 450 500 550 600 650 700−5
0
5
−4
−2
0
2
4
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
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26. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Continuous wavelets
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−5
0
5
Amplitude
Time (s)
Data
Model
(a)
Time (s)
Scale
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4
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16
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500
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Figure 14: Which morphing is easier: time or time-scale?
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27. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Continuous wavelets
• Inversion with another wavelet φ
sptq “
ij
Cspu, aqφu,aptq
duda
a2
ñ time-scale domain processing! (back to the trace signal)
• Scalogram
|Cspt, aq|2
• Energy conversation
E “
ij
|Cspt, aq|2 dtda
a2
• Parseval-like formula
xs1, s2y “
ij
Cs1 pt, aqC˚
s2
pt, aq
dtda
a2
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28. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Continuous wavelets
• Wavelet existence: admissibility criterion
0 ă Ah “
ż `8
0
pΦ˚pνqΨpνq
ν
dν “
ż 0
´8
pΦ˚pνqΨpνq
ν
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 “
ż
sptqψ˚
τ,aptqdt, ψτ,aptq “
1
?
a
ψ
ˆ
t ´ τ
a
˙
with cj,k “ Cspkb0aj
0, aj
0q, pj, kq P Z and still be able to
recover sptq?
• Result 1 (Daubechies, 1984): there exists a wavelet frame if
a0b0 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?
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Discretization and redundancy
0 20 40 60 80 100 120
1
2
3
4
5
6
7
8
Figure 15: Wavelet frame sampling: J “ 21, b0 “ 1, a0 “ 1.1
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Discretization and redundancy
0 20 40 60 80 100 120
1
2
3
4
5
6
7
8
Figure 15: Wavelet frame sampling: J “ 5, b0 “ 2, a0 “
?
2
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Discretization and redundancy
0 20 40 60 80 100 120
1
2
3
4
5
6
7
8
Figure 15: Wavelet frame sampling: J “ 3, b0 “ 1, a0 “ 2
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Discretization and redundancy
0 0.5 1 1.5 2 2.5 3 3.5 4
Time (s)
primary
multiple
noise
sum
0 0.5 1 1.5 2 2.5 3 3.5 4
−0.1
−0.05
0
0.05
0.1
0.15
Time (s)
true multiple
adapted multiple
4
6
8
10
12
14
16
5
10
15
20
10
12
14
16
18
20
RedundancyS/N (dB)
MedianS/Nadapt
(dB)
10
11
12
13
14
15
16
17
18
19
Figure 16: Redundancy selection with variable noise experiments
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34. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Discretization and redundancy
• Complex Morlet wavelet:
ψptq “ π´1{4
e´iω0t
e´t2{2
, ω0: central frequency
• Discretized time r, octave j, voice v:
ψv
r,jrns “
1
?
2j`v{V
ψ
ˆ
nT ´ r2jb0
2j`v{V
˙
, b0: sampling at scale zero
• Time-scale analysis:
d “ dv
r,j “
@
drns, ψv
r,jrns
D
“
ÿ
n
drnsψv
r,jrns
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35. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Discretization and redundancy
Time (s)
Scale
2.8 3 3.2 3.4 3.6 3.8 4 4.2
2
4
8
16
0
500
1000
1500
2000
Time (s)
Scale
2.8 3 3.2 3.4 3.6 3.8 4 4.2
2
4
8
16
0
500
1000
1500
2000
Time (s)
Scale
2.8 3 3.2 3.4 3.6 3.8 4 4.2
2
4
8
16
0
500
1000
1500
2000
Time (s)
Scale
2.8 3 3.2 3.4 3.6 3.8 4 4.2
2
4
8
16
0
500
1000
1500
2000
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
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36. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Unary filters
• Windowed unary adaptation: complex unary filter h (aopt)
compensates delay/amplitude mismatches:
aopt “ arg min
tajupjPJq
›
›
›
›
›
d ´
ÿ
j
ajrk
›
›
›
›
›
2
• Vector Wiener equations for complex signals:
xd, rmy “
ÿ
j
aj xrj, rmy
• Time-scale synthesis:
ˆdrns “
ÿ
r
ÿ
j,v
ˆdv
r,j
rψv
r,jrns
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Results
Time (s)
Scale
2.8 3 3.2 3.4 3.6 3.8 4 4.2
2
4
8
16
0
500
1000
1500
2000
Time (s)
Scale
2.8 3 3.2 3.4 3.6 3.8 4 4.2
2
4
8
16
0
500
1000
1500
2000
Time (s)
Scale
2.8 3 3.2 3.4 3.6 3.8 4 4.2
2
4
8
16
0
500
1000
1500
2000
Time (s)
Scale
2.8 3 3.2 3.4 3.6 3.8 4 4.2
2
4
8
16
0
500
1000
1500
2000
Figure 18: Wavelet scalograms, data and templates, after unary adaptation
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Results (reminders)
Time (s)
Scale
2.8 3 3.2 3.4 3.6 3.8 4 4.2
2
4
8
16
0
500
1000
1500
2000
Time (s)
Scale
2.8 3 3.2 3.4 3.6 3.8 4 4.2
2
4
8
16
0
500
1000
1500
2000
Time (s)
Scale
2.8 3 3.2 3.4 3.6 3.8 4 4.2
2
4
8
16
0
500
1000
1500
2000
Time (s)
Scale
2.8 3 3.2 3.4 3.6 3.8 4 4.2
2
4
8
16
0
500
1000
1500
2000
Figure 19: Wavelet scalograms, data and templates
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Results
Shot number
Time(s) 120014001600180020002200
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
Figure 20: Original data
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40. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Results
Shot number
Time(s) 120014001600180020002200
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
Figure 21: Filtered data, “best” template
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41. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Results
Shot number
Time(s) 120014001600180020002200
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
Figure 22: Filtered data, three templates
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42. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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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.
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43. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Variational approach
minimize
xPH
Jÿ
j“1
fjpLjxq
with lower-semicontinuous proper convex functions fj and bounded linear
operators Lj.
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44. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Variational approach
minimize
xPH
Jÿ
j“1
fjpLjxq
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),
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45. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Variational approach
minimize
xPH
Jÿ
j“1
fjpLjxq
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),
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46. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Variational approach
minimize
xPH
Jÿ
j“1
fjpLjxq
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),
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47. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Variational approach
minimize
xPH
Jÿ
j“1
fjpLjxq
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,
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48. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Variational approach
minimize
xPH
Jÿ
j“1
fjpLjxq
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),
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49. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
37/44
Variational approach
minimize
xPH
Jÿ
j“1
fjpLjxq
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),
• Lj can model a frame operator.
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50. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Problem re-formulation
dpkq
loomoon
observed signal
“ ¯ppkq
loomoon
primary
` ¯mpkq
loomoon
multiple
` npkq
loomoon
noise
Assumption: templates linked to ¯mpkq throughout time-varying
(FIR) filters:
¯mpkq
“
J´1ÿ
j“0
ÿ
p
¯h
ppq
j pkqr
pk´pq
j
where
• ¯h
pkq
j : unknown impulse response of the filter corresponding to
template j and time k, then:
dloomoon
observed signal
“ ¯ploomoon
primary
`R ¯hloomoon
filter
` nloomoon
noise
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51. 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´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
|h
pn`1q
j ppq ´ h
pnq
j ppq| ď εj,p ;
J´1ÿ
j“0
rρjphjq ď τ
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52. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Results: synthetics
minimize
yPRN ,hPRNP
ψ
`
z ´ Rh ´ y
˘
loooooooomoooooooon
fidelity: noise-realted
` ϕpFyqloomoon
a priori on signal
` ρphqloomoon
a priori on filters
• ϕk “ κk| ¨ | (ℓ1-norm) where κk ą 0
• rρjphjq: }hj}ℓ1 , }hj}2
ℓ2
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.40/44

53. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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Results: potential on real data
Figure 25: (a) Unary filters (b) Proximal FIR filters.
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54. Context Multiple filtering Wavelets Discretization, unary filters Results What else? 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
• 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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55. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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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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56. Context Multiple filtering Wavelets Discretization, unary filters Results What else? Conclusions
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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 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
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