Initiation au traitement du signal et applications Notes de cours
Sparse Signal Processing Parcimonie en Traitement du...
Transcript of Sparse Signal Processing Parcimonie en Traitement du...
Sparse Signal Processing Parcimonie en Traitement du Signal
Rémi Gribonval INRIA Rennes - Bretagne Atlantique, France
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
Two inverse problems in audio processing
• Source localization ✓ S. Nam
• Audio inpainting✓ A. Adler, N. Bertin, V. Emiya,
M. Elad, C.Guichaoua, M. Jafari, M. Plumbley
2
echange.inria.fr
small-project.eu
lundi 12 novembre 12
November 8th 2012 - R. GRIBONVAL - Let’s Imagine the Future
Source localizationwith S. Nam
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
Localization with few microphones
• Possible goals
✓ localize emitting sources✓ reconstruct emitted signals✓ extrapolate acoustic field
• Linear inverse problem
• Need a model
4
y = Mx
time-seriesrecorded
at sensors
(discretized) spatio-temporal acoustic field
2 Rm 2 RN
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
Localization with few microphones
• Possible goals
✓ localize emitting sources✓ reconstruct emitted signals✓ extrapolate acoustic field
• Linear inverse problem
• Need a model
4
y = Mx
time-seriesrecorded
at sensors
(discretized) spatio-temporal acoustic field
2 Rm 2 RN
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
Physics-driven design of model
• Pressure field
• Wave equation on a domain
• Boundary + initial conditions, e.g.
5
(�p� 1c2
�2
�t2 p)(�r, t) = s(�r, t), �r ⇥ D
�p
�n(⇥r, t) = 0, ⇥r � �D
p(�r, t)
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
Physics-driven design of model
• Pressure field
• Wave equation on a domain
• Boundary + initial conditions, e.g.
5
(�p� 1c2
�2
�t2 p)(�r, t) = s(�r, t), �r ⇥ D
�p
�n(⇥r, t) = 0, ⇥r � �D
p(�r, t)
} �x = z
x
Discretization
sources& boundaries
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
Group sparse source model
• Few non-moving sources = spatially sparse
6
space
time t
�r
z�r,t
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
Group sparse regularization
• Inverse problem
• Regularization with mixed norm
✦ Convex optimization: efficient & provably convergent algorithms
✦ Promotes group sparsity, cf Kowalski & Torresani 2009, Eldar & Mishali 2009, Baraniuk & al 2010, Jenatton & al 2011
7
y = Mx
x = arg minx
12ky �Mxk2
2 + �k�xk1,2
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
• Setting✓ 2D+t vibrating plate 77x77 ✓ 2 sources, random location✓ 6 microphones, random location✓ known complex boundaries✓ ground truth generated with naive
discretization
• Results
Sparse Field Reconstruction
8
Ground truth
Sparse reconstruction
S. Nam and R. Gribonval. Physics-driven structured cosparse modeling for source localization, ICASSP 2012
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
• Setting✓ 2D+t vibrating plate 77x77 ✓ 2 sources, random location✓ 6 microphones, random location✓ known complex boundaries✓ ground truth generated with naive
discretization
• Results
Sparse Field Reconstruction
8
Ground truth
Sparse reconstruction
S. Nam and R. Gribonval. Physics-driven structured cosparse modeling for source localization, ICASSP 2012
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
Localizing the source next door
• Domain, Source and Microphones
9
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
Localizing the source next door
• Domain, Source and Microphones
• Sparse source localization
9
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
Localizing the source next door
• Domain, Source and Microphones
• Sparse source localization
9
Reasons of success•sparsity of sources•known room shape•known boundaries
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
Localizing the source next door
• Domain, Source and Microphones
• Sparse source localization
9
Reasons of success•sparsity of sources•known room shape•known boundaries
What if shape is unknown ?
lundi 12 novembre 12
November 8th 2012 - R. GRIBONVAL - Let’s Imagine the Future
Audio inpaintingwith A. Adler, V. Emiya, M. Elad, M. Jafari, M. Plumbley
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
Declipping as a linear inverse problem
11
• Original (unknown) samples
• Clipped (observed) samples
• Subset of reliable samples
• Linear inverse problem
M
x
y
yreliable
yreliable =
x
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
Sparse audio models
• Time domain • Time-frequency domain
(Black = zero)
12
Analysis
Synthesis
x ⇡ Dz
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
• Model ✓ sparsity in time-frequency dictionary
• Algorithm: ✓ find sparse coefficients such that
✦ (Orthonormal) Matching Pursuit (Mallat & Zhang 93)✓ + ensure compatibility with clipping constraint
✦ Convex optimization✓ estimate
A. Adler, V. Emiya, M. Jafari, M. Elad, R. Gribonval and M. D. Plumbley, Audio Inpainting, IEEE Trans Audio Speech and Language Proc., 2012
Audio Declipping
13
0 0.01 0.02 0.03 0.04 0.05
−0.5
0
0.5
time (s)
Ampl
itude
x = Dz
y = MDzz
x = Dz
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
• Model ✓ sparsity in time-frequency dictionary
• Algorithm: ✓ find sparse coefficients such that
✦ (Orthonormal) Matching Pursuit (Mallat & Zhang 93)✓ + ensure compatibility with clipping constraint
✦ Convex optimization✓ estimate
A. Adler, V. Emiya, M. Jafari, M. Elad, R. Gribonval and M. D. Plumbley, Audio Inpainting, IEEE Trans Audio Speech and Language Proc., 2012
Audio Declipping
13
0 0.01 0.02 0.03 0.04 0.05
−0.5
0
0.5
time (s)
Ampl
itude
x = Dz
y = MDzz
x = Dz
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
• Model ✓ sparsity in time-frequency dictionary
• Algorithm: ✓ find sparse coefficients such that
✦ (Orthonormal) Matching Pursuit (Mallat & Zhang 93)✓ + ensure compatibility with clipping constraint
✦ Convex optimization✓ estimate
A. Adler, V. Emiya, M. Jafari, M. Elad, R. Gribonval and M. D. Plumbley, Audio Inpainting, IEEE Trans Audio Speech and Language Proc., 2012
Audio Declipping
13
0 0.01 0.02 0.03 0.04 0.05
−0.5
0
0.5
time (s)
Ampl
itude
x = Dz
y = MDzz
x = Dz
Clipped
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
• Model ✓ sparsity in time-frequency dictionary
• Algorithm: ✓ find sparse coefficients such that
✦ (Orthonormal) Matching Pursuit (Mallat & Zhang 93)✓ + ensure compatibility with clipping constraint
✦ Convex optimization✓ estimate
A. Adler, V. Emiya, M. Jafari, M. Elad, R. Gribonval and M. D. Plumbley, Audio Inpainting, IEEE Trans Audio Speech and Language Proc., 2012
Audio Declipping
13
0 0.01 0.02 0.03 0.04 0.05
−0.5
0
0.5
time (s)
Ampl
itude
Declipped
x = Dz
y = MDzz
x = Dz
Clipped
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
• Model ✓ sparsity in time-frequency dictionary
• Algorithm: ✓ find sparse coefficients such that
✦ (Orthonormal) Matching Pursuit (Mallat & Zhang 93)✓ + ensure compatibility with clipping constraint
✦ Convex optimization✓ estimate
A. Adler, V. Emiya, M. Jafari, M. Elad, R. Gribonval and M. D. Plumbley, Audio Inpainting, IEEE Trans Audio Speech and Language Proc., 2012
Audio Declipping
13
0 0.01 0.02 0.03 0.04 0.05
−0.5
0
0.5
time (s)
Ampl
itude
Declipped
x = Dz
y = MDzz
x = Dz
Clipped Original
lundi 12 novembre 12
November 8th 2012 - R. GRIBONVAL - Let’s Imagine the Future
Summary & next challenges
lundi 12 novembre 12
R. GRIBONVAL - Let’s Imagine the Future November 8th 2012
Inverse problems ... and sparse models
15
Observation Domain
lundi 12 novembre 12
R. GRIBONVAL - Let’s Imagine the Future November 8th 2012
Inverse problems ... and sparse models
16
Observation Domain
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
Choosing a model
17
• Expert knowledge (Fourier / wavelets)✓ Harmonic analysis / physics✓ Evolution of species
• Training from corpus ✓ Dictionary learning✓ Individual experience
• «Online» training / adaptivity ?✓ Blind Calibration & Deconvolution✓ Adaptation to new environment
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
Data Jungle
• New data beyond signals and images
18
✓HyperspectralSatellite imaging
✓Spherical geometryCosmology, HRTF (3D audio)
✓GraphsSocial networksBrain connectivity
✓Vector valued Diffusion tensor
Key problem
Versatile low-dimensional models
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
What’s next, please ?
• Unified efficient data processing ✦ Signal processing✦ Machine Learning
• Ground-breaking advances✦ Compressive acquisition and compressive learning✦ Sparse models beyond dictionaries
• Upcoming applications✦ Inpainting / super-resolution (image/video/audio)✦ Distributed video coding✦ Astronomical imaging (interferometry)✦ Low-dose biomedical imaging (CT & IRM) ✦ Audio recording @ high spatial resolution✦ Low-power compressive-sensors ✦ Dynamic high-resolution brain imaging ✦ ...
19
lundi 12 novembre 12
November 8th 2012R. GRIBONVAL - Let’s Imagine the Future
• Frédéric Bimbot• Nancy Bertin, Emmanuel Vincent• Current Docs & Postdocs:✓ Alexis Benichoux, Anthony Bourrier, Srdjan Kitic,
Lei Yu, Cagdas Bilen, ...
• Stéphanie Lemaile• Jules Espiau
22
SPECIAL TH NKS
lundi 12 novembre 12