Sparse Signal Processing Parcimonie en Traitement du Signalvideos.rennes.inria.fr/JP-Banatre/Marie-Doumic/Gribonval.pdf · Sparse Signal Processing Parcimonie en Traitement du Signal

Sparse Signal Processing Parcimonie en Traitement du Signal

Rémi Gribonval INRIA Rennes - Bretagne Atlantique, France

[email protected]

lundi 12 novembre 12

mailto:[email protected]

mailto:[email protected]

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


November 8th 2012 - R. GRIBONVAL - Let’s Imagine the Future

Source localizationwith S. Nam



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



Localization with few microphones

• Possible goals

✓ localize emitting sources✓ reconstruct emitted signals✓ extrapolate acoustic field


• Need a model

4

y = Mx

time-seriesrecorded

at sensors

(discretized) spatio-temporal acoustic field

2 Rm 2 RN



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)



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



Group sparse source model

• Few non-moving sources = spatially sparse

6

space

time t

�r

z�r,t



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



• 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



• 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



Localizing the source next door

• Domain, Source and Microphones

9





• Sparse source localization

9






9

Reasons of success•sparsity of sources•known room shape•known boundaries






9

Reasons of success•sparsity of sources•known room shape•known boundaries

What if shape is unknown ?



Audio inpaintingwith A. Adler, V. Emiya, M. Elad, M. Jafari, M. Plumbley



Declipping as a linear inverse problem

11

• Original (unknown) samples

• Clipped (observed) samples

• Subset of reliable samples


M

x

y

yreliable

yreliable =

x



Sparse audio models

• Time domain • Time-frequency domain

(Black = zero)

12

Analysis

Synthesis

x ⇡ Dz



• 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


http://hal.inria.fr/inria-00577079/en








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










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










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










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





Summary & next challenges


R. GRIBONVAL - Let’s Imagine the Future November 8th 2012

Inverse problems ... and sparse models

15

Observation Domain


R. GRIBONVAL - Let’s Imagine the Future November 8th 2012

Inverse problems ... and sparse models

16

Observation Domain



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



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



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



TH NKSJEAN-PIERRE



• 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


November 8th 2012R. GRIBONVAL - Let’s Imagine the Future 23


Documents

Sparse Signal Processing Parcimonie en Traitement du Signalvideos.rennes.inria.fr/JP-Banatre/Marie-Doumic/Gribonval.pdf · Sparse Signal Processing Parcimonie en Traitement du Signal